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\begin{center}
\vspace{2.5cm}
{\bf \LARGE   QMath-8. Mathematical Results in Quantum Mechanics }

\vspace{2.5cm}

{\bf \LARGE Abstracts}

\vspace{5cm}
{\LARGE Editor: Ricardo Weder

\vspace{5cm}
Taxco, M\'exico. December 10-14, 2001}

\end{center}
\newpage

\vspace{20cm}
\begin{center}
\noindent {\bf \LARGE Featured Topics:}
 
\vspace{3cm}


\n Bound state problems and scattering theory for Schr\"{o}dinger operators.
 Inverse spectral and scattering theory  of Schr\"{o}dinger operators. Nonlinear Schr\"{o}dinger equations. 
Quantum chaos, quantum dots and wave guides. Parameter dependent Hamiltonians. 
Spectral and localization properties of  Sch\"{o}dinger operators
\end{center}

\newpage

\begin{center}

{\bf \LARGE Contents}
\end{center}


\n R. Adami: Blow-Up solutions for schroedinger equation with concentrated   nonlinearity \dotfill 7 

\n M.A. Antonets: Negative discrete spectrum of Dirac operator for  contact
interaction  with  plane
\dotfill 7


\n  B.Apagyi: Potentials from phase shifts at fixed energies by a non-linear
inversion method \dotfill 8

\noindent  N. Atakishyev: Mellin transforms for some families of q-polynomials \d 8


\n  B. Bodmann: Transformation formula relating resolvents of Berezin-Toeplitz  operators \linebreak
by an  invariance  property of Brownian motion \d 8

\n  U. Carlsson: Schroedinger operators with several potential wells in the
semi-classical limit \d 8

\n  M. Castagnino: The Classical limit\d 9

\n  M. Combescure: Semiclassical results in the linear response theory \d 10

\n M. Demuth: Capacities and equilibrium potentials in scattering theory \d 11

\n  J. Dittrich:  Quantum waveguides with combined boundary conditions \d 11


\n V. Enss: Quantum scattering in a rotating potential\d 11

\n P. B. Espinoza: Moyal-like evolution equations for SU(2) quasi-distribution functions \d 12

\n P. Exner: A norm-resolvent approximation of strongly singular interactions \d 12

\n  S. Fournais: Regularity of molecular eigenfunctions and densities  \d 12


\n S. Fulling: The quantum theory of ceilings and floors \d  13


\n E. Giere: Asymptotic completeness for functions of the Laplacian perturbed  by potentials or 
obstacles   \d  13


\n C. Hainzl: General decomposition of radial functions on ${\bf R}^n$ and  refined conditions 
for positive definiteness \d 13

\n B. Hall: Coherent states on spheres \d 13


\n I. Herbst: Quantum scattering with potentials independent of $|x|$ \d 14

\n M. Hitrik: Eigenfrecuencies and expansions for dissipative wave equations \d 14


\n  M. Horv\'ath: A necessary and sufficient condition
in inverse eigenvalue problems \d  14

\n V. K. Ignatovich: Apocrypha of the standard scattering theory \d 15 

\n V. Ivrri: Function in the box and logarithmic uncertainty principle \d 15

\n E. Kaikina: Asymptotic behavior for Korteweg-de Vries-Burgers equation on half-line
 without smallness of initial data \d 15

\n A. Komech: On attractor of a nonlinear U(1)-invariant 1D Klein-Gordon equation \d 16

\n M. Kovalyov: Spectral decomposition of potential for the one-dimensional Schroedinger
\linebreak operator \d  16

\n D. Krejcirik: Bound states in curved quantum layers \d   17

\n  I. Krasovski: Spectral estimates for discrete one-dimensional periodic Schr\"odinger  operators \d  17

\n V. V. Kucherenko: Spectral asymptotics for the $N$ particles Schr\"odinger equation with 
periodic binary potential \d  17

\n  L. Kuciuk: On quantum smaradache paradoxes \d   19

\n M. Kudryavtsev: The Cauchy problem for the Toda lattice with a class of non-stabilized initial data\d  20

\n  E. Ley-Koo: Variational theorem for the solution of the many-electron relativistic 
quantum \linebreak theory \d  21

\n  E. H. Lieb: Stability of a model of relativistic quantum electrodynamics \d 21

\n J. R. Mercado-Escalante: Inverse problem for the similarity exponents \d   22

\n B. Mityagin: Counting Lemma for  $1D$ periodic Dirac Operator \d   23
\linebreak

\newpage
\n P. I. Naumkin:Large time asymptotics for quadratic nonlinear Schr\"odinger equations in one space \linebreak dimension \d  24


\n F. Nicoleau: A scattering inverse problem with the Aharonov-Bohm effect \d   24

\n   R. H. Parmar: Super symmetric quantum mechanics with vector super potential \d   25
 
\n D. Pearson: Recent developments in value distribution theory for 
Schr\"odinger operators\d   27


\n G. S. Pogosyan:  Coulomb-oscillator duality in spaces of constant curvature \d  27

\n R. Quezada: A class of non-conservative quantum dynamical semigroups \d   28

\n V. Rabinovich: Pseudodifferential operators with analytic symbols and estimates for  
eigenfunctions of Schr\"{o}dinger operators \d   28



\n R. del R\'{\i}o : Inverse spectral results for Dirac systems \d  29

\n  Robert: Long time propagation  for quantum observables and coherent states in the 
semiclassical r\'egime \d    29


\n Gert Roepstorff: On a Class of anomaly free gauge theories \d      29


\n M. B. Ruskai: Comment on efficiency of adiabatic quantum algorithms \d    30



\n A. Rybkin: On the generalized spectral shift function for one-dimensional 
Schrodinger operators with slowly decaying potentials and trace formulas\d    33

\n A. Sachetti:  Tunneling instability for a double well Schroedinger operator with a non-linear 
perturbation \d     33


\n R. Schrader: Statistical ensembles and density of states \d  33

\n B. Schumayer: Painlev{\'e} test of coupled nonlinear Schr{\"{o}}dinger
       equations  and Bose-Einstein condensates \d   34


\n R. Seiringer: Gross-Pitaevskii Theory of the Rotating Bose Gas \d     36


\n E Sere: A ground state for the Dirac-Fock model\d    36

\n H. Siedentop: The Energy of relativistic one-electron atoms\d   36

\n F. Sobieckzky: When do digital images fragment into infinitely many segments \d   37

\n S. Sontz: �New results in reverse inequalities\d   37

\n  P. Stovicek: Weakly regular Floquet Hamiltonians with pure point spectrum\d    38

\n M. Tater: Scattering by a slab\d        38


\n A. Tip: Maxwell with a touch of Schr\"{o}dinger \d    39

\n J. Toloza: Exponentially Small Error Estimates of Quasiclassical Eigenvalues\d    41

\n A. Turbiner: Canonical Discretization (discretization as canonical transformation)\d    41

\n G. Uhlmann: On the local Dirichlet-to-Neumann map \d    42

\n C. van der Mee: The Inverse generalized Regge problem \d    42

\n  I. Veselic: Wegner estimate for alloy type models with non negative single site 
 potential of small
 support \d    43


\n C. Villegas Blas: The Bargmann transform, canonical transformations and the
 Kepler\linebreak  problem\d  44

\n V. Vougalter: Pauli operator and Aharonov-Casher theorem for measure valued 
magnetic\linebreak fields \d  44

\n K. B. Wolf: Finite Quantum/Optical Systems    \d  45

\n K. Yajima: The local smoothing property and Strichartz inequality 
for Schr\"odinger equations  
with potentials superquadratic at infinity \d    45

\n K. Yoshitomi: Band spectrum of the laplacian on a slab with the Dirichlet 
boundary condition on a grid \d   47

\n G. Zhislin: Peculiarities of the structure of the spectrum of many particle hamiltonians of
 charged systems in a homogeneous magnetic field \d    47

\newpage
  

 


{\bf\noindent 1 \hspace{6mm} R. Adami: Blow-Up solutions for schroedinger equation with concentrated 
nonlinearity}

\b
\bigskip 
\noindent In this talk we discuss some results on a class of one and three dimensional nonlinear 
Schr\"{o}dinger equations. The nonlinear term consists of a finite sum of Dirac deltas whose coupling 
constants are functions of the solution. We show how a blow-up phenomenon can occurr and describe some 
different families of blow-up solutions. Finally, we give an estimate on the blow-up rate and investigate
 the links between this problem and the corresponding one for the standard nonlinear Schroedinger equation.\\

\bigskip

\noindent{\bf 2  \hspace{6mm} M.A. Antonets: Negative discrete spectrum of Dirac operator
for contact interaction with plane}


\bigskip 
\n The negative discrete spectrum of the Dirac operator is considered in the case of an electron interacting
 by contact way with a plane in the presence of tilted magnetic field. For resolvent of this operator 
corresponding Krein formula is established. Then it is used to prove existence of the negative discrete 
eigen-values and constructing of their asymptotics for large constant of the contact interaction. 
The work is based on some new results of the theory of pseudodifferential operators [1].
 
\medskip 
\noindent {\bf References.} 

 \noindent 1.  Antonets M.A. Initial-value problem for pseudodifferential operators. 
Journal of Mathematical Science,Vol. 98, No. 6,March 2000, p 629-653. 


\b 


\n {\bf 3 \h B.Apagyi: Potentials from phase shifts at fixed energies by a non-linear
inversion method}

\b

\n With Z. Harman and W. Scheid. We apply the Cox-Thompson (CT) quantum inversion method at fixed
energy to derive potentials from given sets of phase shifts. The
resulted non-linear equations are solved by the root-finder
package of the Mathematica, the inversion potentials are then
constructed with the aid of the Maple. The test examples involving
box and Woods-Saxon potentials show the superiority of the CT
method over the well known Newton-Sabatier inversion procedure
which assumes linear equations for the solution.

\b





\noindent {\bf 4 \h N. Atakishyev: Mellin transforms for some families of q-polynomials}

\b


 \n We discuss q-analogues of Euler's reflection formula and gamma integral. 
Ramanujan's q-extension of the Euler integral representation for the gamma function plays a central role in 
this work and it enables us to derive the Mellin integral transforms for some families of basic 
hypergeometric orthogonal polynomials from the Askey scheme.

\b
\n {\bf 5 \h B. Bodmann: Transformation formula relating resolvents of Berezin-Toeplitz operators 
by an invariance property of Brownian motion }
\bigskip


\n Using a stochastic representation provided by Wiener-regularized path integrals for the
 semigroups generated by certain Berezin-Toeplitz operators, a transformation formula for their resolvents 
is derived. The key property used in the transformation of the stochastic representation is that, up to a 
time change, Brownian motion is invariant under conformal mappings.

\b
\n {\bf 6 \h  U. Carlsson: Schroedinger operators with several potential wells in the
semi-classical limit }
\bigskip

\n Consider first a Schroedinger operator with exactly one point-well.
Then, if the Hessian is positive definite at the point, it is
well-known that the lowest eigenvalue is simple, and that the
corresponding eigenfunction is positive up to a non-zero constant
factor.
In the case of a double-well potential the eigenvalue splits.
When analyzing the spectrum of an operator with several wells
we use one-well operators as reference operators. In the case
of infinitely many wells conditions on the potential are needed
which in turn imply geometric restrictions on the distribution
of the wells.
Of special interest to us is the case when the wells have a
hierarchical configuration. Throughout the talk I restrict my
study to the structure of the bottom of the spectrum. One question
of interest to me is if the double-well splitting phenomenon also
appear repeatedly in the hierarchical case.

\b


\n{\bf 7 \b M. Castagnino: The Classical limit}

\b

\n The quantum to classical--statistical limit is studied. It is proved that the limit $\bar h \to 0$ 
coincides 
with this limit for the operators algebra but it is not so for the states convex set. In the later case, to 
obtain the classical limit, decoherence must be invoked.
It is well known that, once upon a time, the quantum to classical limit was considered just the limit 
$\bar h \to 0$ (as the relativistic to classical limit is just $c\to \infty)$. Bohr and Heisenberg 
stressed this analogy but Einstein was unmoved by such arguments which considered an oversimplification, 
since the statistical structure of quantum mechanics is completely different to the individual particle one 
of classical mechanics ( actually, Einstein was right, the experience showed that matters were much more 
difficult, since in the former limit many physical phenomena must be considered (decoherence, 
localization, correlations, etc.) making the subject quite complicated. 
See e. g.  [1]. Latter a  ``classical statistical" structure for quantum mechanics was found by 
Wigner [2] (i. e. quantum mechanics formulated on phase space) and it was shown that it is convenient 
to decompose this limit into the quantum  to classical statistical  limit and the classical statistical to 
classical  limit (see [3]). 
In this talk, using the methods of  papers [4], [5] and [6],
we will  consider the quantum to classical statistical limit for a system with continuous evolution spectrum.
The idea is to show, in the simplest but complete way, 
the essence of the limit, trying to clear the matter as much as possible.

\medskip 
\noindent {\bf References.} 
 
\begin{enumerate}

\item  L. E. Ballentine, {\it Quantum mechanics}, Prentice Hall, Englewoods Cliffs, 1990.

\item  M. Hillery, R.F. O'Connell, M. O. Scully, E.P. Wigner, Phys. Rep., {\bf 106}, 123,  1984.

\item  M. Castagnino, R. Laura, Phys. Rev. A, {\bf 62}, 022107, 2000.

\item  R. Laura, M. Castagnino, Phys. Rev. A, {\bf 57}, 4140, 1998.

\item  R. Laura, M. Castagnino. Phys. Rev. E, {\bf 57}, 3948, 1998.

\item  M. Castagnino, M. Gadella, R. Laura, R. Id Betan, Phys. Lett. A, {\bf 282}, 245, 2001.

\end{enumerate}

\b
\n {\bf 8 \h M. Combescure: Semiclassical results in the linear response theory}

\b

\n With Didier Robert. We consider a quantum system of non-interacting fermions at temperature T, in the 
framework of linear response theory. We show that semiclassical theory is an appropriate framework for
 describing some of their thermodynamic properties, in particular through exact expansions in Planck constant
 h of their thermodynamic susceptibilities. We show how the orbits of the classical motion in phase space 
manifest themselves in these expansions, in the regime where T is of order of h.

\b

\n{\bf 9 \h M. Demuth: Capacities and equilibrium potentials in scattering theory}

\b
\n Regular Dirichlet forms are associated to selfadjoint operators and to Hunt processes. 
The one-equilibrium potential realizes the infinium in the definition of the capacity. 
The equilibrium potential has a stochastic representation using the first hitting time of 
an obstacle. 
In obstacle scattering Dynkin's formula describes resolvent differences in terms of the 
harmonic extension operator. This can be used to study Hilbert-Schmidt or trace 
class properties of such resolvent differences. In scattering theory the scattering 
matrix gives the main link to physical applications. In case of perturbations by two 
different obstacles the difference of the scattering matrices is estimated by the 
capacity of the symmetric difference of the obstacles. The same follows for the 
scattering phases. 
This kind of quantitative estimates can be extended to large coupling problems where 
the decrease of the scattering phases depends on the Laplace transform of the 
occupation time. 


\b


\n{\bf  10  \h J. Dittrich:  Quantum waveguides with combined boundary conditions} 

\b

\n The existence of bound states is studied for two examples of straight quantum waveguides with 
Dirichlet and Neumann boundary conditions imposed on different parts of the boundary. Cases with 
and without bound states are found.

\b
\n{\bf 11 \h V. Enss: Quantum scattering in a rotating potential}

\b
\n We study scattering theory for a quantum particle under the influence of a
classical rotating body. The interaction is described by an explicitly
time-dependent potential obtained by a uniform rotation of a fixed potential.
Under mild regularity and suitable -- possibly unisotropic -- decay
assumptions we investigate existence and completeness of wave operators and,
in particular, the energy transfer between the particle and the rotating body.
We show that the kinetic energy is uniformly bounded on scattering states and
we determine the loss or gain of energy for special classes of states.

\n Joint work with Vadim Kostrykin and Robert Schrader.

\b
\n{\bf 12 \h P. B. Espinoza: Moyal-like evolution equations for SU(2) quasi-distribution functions.}

\b
\n We derive a differential form of the Star Product for a s-parametrised family of quasi-distributions of SU(2). 
We apply these results to find evolution equations for dynamical systems described by polynomial 
Hamiltonians of SU(2) algebra generators.

\b
\n{\bf 13 \h P. Exner: A norm-resolvent approximation of strongly singular interactions}

\b
\noindent We discuss a family Schr\"odinger operators with
potentials scaled on a specific nonlinear way which approximates
the $\delta'$-interaction Hamiltonian in the norm-resolvent sense.
This approximation, based on a formal scheme proposed by Cheon and
Shigehara, has nontrivial convergence properties which are in
several respects opposite to those of the Klauder phenomenon.

\b
\n {\bf 14 \h S. Fournais: Regularity of molecular eigenfunctions and densities }

\b

\n This is joint work with Maria Hoffmann-Ostenhof, Thomas Hoffmann-Ostenhof and Thomas \O stergaard 
S\o rensen.
We study the regularity of molecular eigenfunctions near singularities of the potential. Using a 
Jastrow-type Ansatz, we can describe precisely the singularities of the eigenfunction up to class
 $C^{1,1}$. As an application we prove that the electron densities of molecular eigenfunctions are 
smooth away from the nuclei.
 
\b
\n{\bf 15 \h S. Fulling: The quantum theory of ceilings and floors}

\b
\n The semiclassical analysis of the propagator of a particle subject to both a linear potential and a 
reflecting boundary is surprisingly nontrivial, and it provides a prototype for the study of systems with
 both potentials and boundaries, as the Airy problem does for ordinary turning-point problems. 

\b
\n{\bf 16 E. Giere: Asymptotic completeness for functions of the Laplacian perturbed by potentials or 
obstacles  }

\b
\n Using scattering theoretical methods we show that the absence of the singular continuous spectrum of 
perturbations of functions of the Laplacian. In an application we study the fractional
 Laplacian $(-\Delta)^{\alpha/2}$ and its perturbation by short range potentials or obstacles. For 
the latter we give an example with an unbounded set as an obstacle.

\b

\n{\bf 17 \h C. Hainzl: General decomposition of radial functions on ${\bf R}^n$ and  refined conditions for 
positive definiteness}

\b
 \n We present a generalization of the Fefferman-de la Llave decomposition of the Coulomb potential to quite 
arbitrary radial functions $V$ on ${\bf R}^n$ going to zero at infinity. As a byproduct, we obtain
 conditions for positive definiteness of $V$, thereby improving results of Askey. In a subsequent 
paper we use this to describe assymptotic exactness of Hartree-Fock approximation of Yukawa systems in
 the high density limit.



\b
\n{\bf 18 \h B. Hall: Coherent states on spheres}

\b
\n I will describe joint work with Jeffrey Mitchell on
coherent states for a quantum particle whose classical
configuration space is a sphere. These coherent states are not of
Perelomov type.

\b
\n{\bf 19 \h I. Herbst: Quantum scattering with potentials independent of $|x|$}
 
\b
\n Scattering theory is developed for an n dimensional particle under the influence of a potential which 
is homogeneous of degree zero for large x and such that V restricted to a large sphere is a Morse 
function. We describe the somewhat unusual asymptotic behavior of the wave function as it develops in time 
and define appropriate asyptotic dynamics in certain energy regimes. We prove asymptotic completeness of 
the corresponding wave operators. We describe similar but geometrically more complex behavior of quantum 
systems in magnetic fields which are homogeneous of degree -1. This work is joint with Erik Skibsted and
partially with Horia Cornean.

\b

\n{\bf 20 \h M. Hitrik: Eigenfrecuencies and expansions for dissipative wave equations} 

\b
\n We study eigenfrequencies and propagator expansions for damped wave equations on compact manifolds. In the strongly damped case, the propagator is shown to admit an expansion in terms of the finitely many eigenmodes near the real axis, with an error exponentially decaying in time. In the presence of an elliptic closed geodesic not meeting the support of the damping coefficient, we show that there exists a sequence of eigenfrequencies converging rapidly to the real axis. In the case of Zoll manifolds, the set of all eigenfrequencies is shown to exhibit a cluster structure determined by the Morse index of the closed geodesics and the damping coefficient averaged along the geodesic flow. We then show that the propagator can be expanded in terms of the clusters of eigenfrequencies in the entire spectral band. 		


\b

\n{\bf 21 \h  M. Horv\'ath: A necessary and sufficient condition
in inverse eigenvalue problems}

\b
\n Consider the Schr\"odinger operator  $Ly=-y''+q(x)y$  on  $x\in [0,\pi]$.
We investigate the problem of reconstructing the potential $q(x)$ by a
sequence of eigenvalues $\lambda_n$, $Ly_n=\lambda_ny_n$, where the
eigenfuctions satisfy a fixed right boundary condition at $\pi$ and
possibly different arbitrary left boundary conditions. Among others we
have the following
\hfill\break\noindent
$\underline {Theorem}$ {\it{Suppose that $y_n(\pi)=0$ and that the potential}}
{\it{is known on the subsegment $(0,a)$ with some $0\le a<\pi$. Let}}
$\mu\ne\pm\sqrt{\lambda_n}$. 
{\it{The eigenvalues $\lambda_n$, $n\ge 1$ determine the potential $q(x)$}}
{\it{if and only if the system}}
$\{\exp(\pm 2i\mu x),\exp(\pm 2i\sqrt{\lambda_n}x): n\ge 1\}$
{\it{is complete in $L_2(a-\pi,\pi-a)$.}}\hfill\break
By the aid of this statement most of the formerly known sufficient
conditions can be converted into standard completeness results.


\b
\n{\bf 22 \h  V. K. Ignatovich: Apocrypha of the standard scattering theory} 
\bigskip


\noindent  The standard scattering theory contradicts the principles of canonical
quantum mechanics. The scattering theory, which rigorously follows
principles of quantum mechanics gives not cross sections, but
dimensionless scattering probabilities. With dimensionless scattering
probabilities it is impossible to describe transmisson of a sample. To
describe the transmission we must introduce the area of wavefront of
scattered particle, which leads us to a nonspreading wave packet. 
The best candidate for such a packet is the Broglie singular wave packet.
Some physics for such a packet is presented.
Also some results for scattering of neutrons in a monoatomic gas are
presented that differ from predictions of standard theory.

\b
\n{\bf 23 \h V. Ivrri: Function in the box and logarithmic uncertainty principle}

\b
\n Microlocalization (and Microlocal Analysis in general) require logarithmic uncertainty 
principle: $\rho \times \gamma \ge Ch |\log h|$ where $\rho$, $\gamma$ are scales with respect to 
momenta and coordinates respectively, $h$ is Plank's constant.

\b
\n{\bf 24 \h E. Kaikina: Asymptotic behavior for Korteweg-de Vries-Burgers equation on half-line
 without smallness of initial data}

\b
\n We study the initial-boundary value problem for the Korteweg-de Vries-Burgers (KdVB) equation 
without smallness condition on the data. The KdVB equation has many physical applications. There
 are many works on the large time asymptotics of solutions in the case of small initial data for 
the Cauchy problem. By virtue of the difficulty of the study of not small solutions of nonlinear
 problem globally in time there are few results on the asymptotic behavior of solutions with large
 initial data. Due to the zero boundary value we will obtain that solutions have more rapid time decay 
in comparison with the corresponding Cauchy problem. Therefore using the symmetry property of the
nonlinearity we estimate the $L^�$ - norm of solutions without any assumption on the size of the initial 
data. Also we apply the energy method and estimates of Green function to prove smallness of  
the uniform norm of the solution after some time and prove global existence and asymptotic formulas for 
solutions.

\b
\n{\bf 25 \h A. Komech: On attractor of a nonlinear U(1)-invariant 1D Klein-Gordon equation } 

\b
\noindent An attractor is studied for all finite energy solutions to a model nonlinear U(1)-invariant 
1D Klein-Gordon equation. The attractor is a union of the solitary waves 
$\psi(x)\exp(i \omega t)$.

\b
\n{\bf 26 \h M. Kovalyov: Spectral decomposition of potential for the one-dimensional Schroedinger operator}

\b

\n If the spectrum of the 1-dim Schrodinger Operator (SO) consists of a finite number of points, the coresponding potential must be a nonliner superpositin of a finite number of solitons. Is a similar assertion valid for continuous spectrum, i.e. if the spectrum is continuous can the potential be represented as nonlinear superposition of some special soliton-like solutions? Can these solutions be used to construct a potential with required a priory given properties? These questions will be discussed and graphic illustrations will be shown.

\b
\n{\bf 27 \h D. Krejcirik: Bound states in curved quantum layers}

\b
\n We consider a quantum particle constrained to a tubular neighbourhood of constant width built over a curved non-compact surface embedded in~$\textrm{R}^3$. We suppose that the latter is asymptotically planar and that such a layer has the hard-wall boundary. Under these assumptions we find sufficient conditions which guarantee the existence of geometrically induced bound states. This is a joint work with Pierre Duclos and Pavel Exner.

\b

\n{\bf 28 \h  I. Krasovski: Spectral estimates for discrete 
one-dimensional periodic Schr\"odinger  operators}

\b
\n We obtain the estimates for the boundaries of the spectrum and the
integral estimates for the measure of the gaps of periodic operators
on $l^2(Z)$:
$(H\psi)(n)=\psi(n+1)+\psi(n-1)+V_n\psi(n)$,
where the real potential $V_n$ has a period $q$, that is 
$V_n=V_{n+q}$, for any $n\in Z$. 
We construct and study the quasimomentum $k(z)$ corresponding to this
operator. Our results are obtained from the properties of $k(z)$
considered as a conformal mapping of complex domains.


\b


\n{\bf 29 \h V. V. Kucherenko: Spectral asymptotics for the $N$ particles Schr\"odinger equation with 
periodic binary potential}

\b

\n With Salvador Arellano Balderas. We consider the Boson's eigenfunctions for the $N$ particles 
Schr\"odinger spinless equation on the torus $M=({\bf R}^3/{\bf Z}^3)^N$:


\begin{equation}
\sum_{j=1}^N-\epsilon^2\Delta_{x_j}\psi+\sum_{1\leq i< j\leq N}v(|x_i-x_0|)\psi=E\psi
\end{equation}

Let us $L_2^s(M)$ is the symmetrical subspace of $L_2(M)$; therefore $\psi\in L_2^s(M)$. We suppose that

\begin{equation}
v(|x-y|)=\sum_{|k|=0}\nu_k\mbox{exp}\{i 2\pi(k,(x-y))\}
\end{equation}

$$\mbox{Im}\nu_k=0,\nu_k=\nu_{-k},\nu_k\geq 0,\sum_{|k|=0}\nu_k|k|^4<\infty.$$

The supposition $\nu_k\geq 0$ corresponds to the repulsive case.
\vspace{5mm}

It is well known that any symmetrical function $\psi(x_1,\ldots,x_N)\in L_2^s(M)$ can be decomposed into the
 Fourier serie with respect to the symmetrized harmonics.
\vspace{5mm}

$$S\mbox{exp}\{i2\pi(k_1,x_1)+\cdots+i2\pi(k_N,x_N)\},$$

\noindent where $S$ is the summetrization operator. Between that Fourier harmonics we choosed [1] 
the functions

$$u_k=\frac{1}{\sqrt{N}}\sum_{j=1}^N\mbox{exp}\{i2\pi(k,x_j)\}. k\in{\bf Z}^3,x\in{\bf R}^3.$$

It was proved [1] that any symmetrical Fourier harmonic is a polinomial function from some 
finite number of the funcions $u_{k_1},\ldots,\bar{u}_{k_m}$. That means that we can look for the
 asymptotic solution to the problem (1) as a composite function of infinite number
 funtions $u_k$, $\bar{u}_k;k\in{\bf P}$; where ${\bf P}$ is the subset of ${\bf Z}^3\setminus
 0$ such that ${\bf P}\cup(-{\bf P})={\bf Z}^3\setminus 0$ and ${\bf P}\cap(-{\bf P})=\phi$. 
In reality we'll use the functions of the form



\begin{equation}
\mbox{exp}\left(\frac{\sqrt{N}}{\epsilon}\sum_{k\in p}\beta_ku_k\bar{u}_k\right)Q(\ldots u_k,\bar{u}_k,\ldots)
\end{equation}

\noindent where $\beta_k<$, $\Sigma|\beta_k|<\infty$ and $Q$ are some polinomial in finite numbers of functions $u_k,\bar{u}_k$. In the following we set

$$\beta_k=\frac{\epsilon}{2\sqrt{N}}\left(1-\sqrt{1+\frac{2\nu_k N}{k^24\pi^2\epsilon^2}}\right).$$

If is proved that at the functions $f(\ldots u_k,\bar{u}_k,\ldots)$ the problem (1) is reduced in the main 
term to the spectral problem of the sum of non interacting harmonic oscillators.
\vspace{5mm}

With the quasimodes (3) we constructed the asymptotics of some spectral series to the problem (1) 
and proved for the minimal eigenvalue of the problem (1) double sided estimate

$$\frac{\nu_0 N^2}{2}-\frac{N}{2}\sum_{||k|=1}^\infty\nu_k+2\epsilon\sqrt{N}c^2<\lambda_0\leq$$

$$\frac{\nu_0 N^2}{2}-\frac{N}{2}\sum_{|k|=1}\nu_k-8\epsilon\sqrt{N}\pi^2\Sigma \beta_k K^2$$

\noindent with some constant $c$. The details will be given at the talk.

\medskip 
\noindent {\bf References.} 


\begin{enumerate}
\item 1. Valeri V. Kucherenko ``Semiclassic for the $N$ partcles Schr\"odinger equation with binary potential'' International conference ``Differential equation and Related Topics''. Moscow may 22-27, 2001. Book of abstracts, p. 220-221.
\end{enumerate}


\b
\n{\bf 30 \h  L. Kuciuk: On quantum smaradache paradoxes}

\b

\n A survey on the below intriguing and funny quantum Smarandache paradoxes is presented, paradoxes that
 are interconnected with many aspects of quantum physics: 

1) Sorites Paradox (associated with Eubulides of Miletus (fourth century B.C.): 

Our visible world is composed of a totality of invisible particles. 

a) An invisible particle does not form a visible object, nor do two invisible particles, three invisible 
particles, etc. 
However, at some point, the collection of invisible particles becomes large enough to form a 
visible object, but there is apparently no definite point where this occurs. 

b) A similar paradox is developed in an opposite direction. It is always possible to remove a 
particle from an object in such a way that what is left is still a visible object. However, repeating 
and repeating this process, at some point, the visible object is decomposed so that the left part 
becomes invisible, but there is no definite point where this occurs. 

Generally, between <A> and <Non-A> there is no clear distinction, no exact frontier. Where 
does <A> really end and <Non-A> begin? One extends Zadeh's "fuzzy set" term to the 
"neutrosophic set" concept. 

2) Uncertainty Paradox: Large matter, which is under the 'determinist principle', is formed by a 
totality of elementary particles, which are under Heisenberg's 'indeterminacy principle'. 

3) Unstable Paradox: Stable matter is formed by unstable elementary particles (elementary 
particles decay when free). 

4) Short Time Living Paradox: Long time living matter is formed by very short time living 
elementary particles. 

\b
\n{\bf 31 \h M. Kudryavtsev: The Cauchy problem for the Toda lattice with a class of non-stabilized initial data}

\noindent We consider the Cauchy problem for the Toda lattice in the case when the corresponding $L$-operator is a Jacobi matrix with bounded elements, whose spectrum of multiplicity 2 is separated from its simple spectrum and contains an interval of absolutely continuous spectrum. A new type of spectral data, which are analogous for scattering data, is introduced for this matrix. An integral equation that allows us to reconstruct the matrix from this spectral data is obtained. We use this equation to solve the Cauchy problem for the Toda lattice with the initial data that are not stabilized. 

\b
\n{\bf 32 \h  E. Ley-Koo: Variational theorem for the solution of the many-electron relativistic quantum theory}

\b

\n With C.F. Bunge and R. J�uregui. The exact solutions of Dirac�s equation for the free-electron and for 
hydrogen-like atoms are reviewed with emphasis on the existence of the positive and negative energy 
states.  The problems associated with the solution of the Dirac-Coulomb equation for many-electron atoms, 
including the dissolution in the continium, first analyzed by Brown and Ravenhall in 1951, and the projection
 into positive energy-states in the no-pair Hamiltonian, advocated by Sucher, are also reviewed.  
The Goldman-Drake variational solution of the Dirac equation for the hydrogen-like atoms including
 both positive and negative energy solutions is also analyzed.  In this contribution a variational
 theorem is formulated and proven for the solutions of the Dirac-Coulomb equation for many electron-atoms
 including positive and negative energy states. 

\b

\n {\bf 33 \h E. H. Lieb: Stability of a model of relativistic quantum electrodynamics}

\b


\n The relativistic ``no pair'' model of quantum electrodynamics
uses the Dirac operator, $D(A)$ for the electron dynamics together with
the usual self-energy of the quantized ultraviolet cutoff
electromagnetic
field $A$ --- in the Coulomb gauge.  There are no positrons because the
electron wave functions are constrained to lie in the positive spectral
subspace of some Dirac operator, $D$, but the model is defined for any
number, $N$, of electrons, and hence describes a true many-body system.
In addition to the electrons there are a number, $K$, of fixed nuclei
with
charges $\leq Z$.  If the fields are not quantized but are classical,
it was shown earlier that such a model is always unstable (the ground
state energy $E=-\infty$) if one uses the customary $D(0)$to define
the electron space, but is stable ($E > -\mathrm{const.}(N+K)$) if one
uses $D(A)$ itself  (provided the fine structure constant $\alpha$ and
$Z$ are not too large).  This result is extended to quantized fields
here, and stability is proved for $\alpha =1/137$ and $Z \leq 42$.
This formulation of QED is somewhat unusual because it means that the
electron Hilbert space is inextricably linked to the photon Fock space.
But such a linkage appears to better describe the real world of photons
and electrons. (joint work with Michael Loss)

\b

\n{\bf 34 \h J. R. Mercado-Escalante: Inverse problem for the similarity exponents}

\b

\n The inverse problem of diffusion is solved by fractal methods and the groups analysis
 of the differential equations. In the multidimensional case, the direct problem of the Boussinesq's equation 
for the porous medium will be solved. The infinitesimal generator of the groups of the equation differential 
are found. The mutual relation between the infinitesimals and the dimension space is analysed. The group 
analysis of auxiliary and especial equations for the porous medium is made, regarding ordinary differential
 equations. The reduce equation is obtained. The inverse problem for the exponents of the similarity variables
 is solved. The similarity variables are defined and we find the self-similar solutions. An alternative 
presentation is conformed, with a condition of normalization to introduce an initial condition. 
The results to
 the mathematical models are applied the horizontal drains of agriculture soil, the 
vertical drains soil and the irrigation by drops.

\b

\n{\bf 35 \h B. Mityagin: Counting Lemma for  $1D$ periodic Dirac Operator }

\b


\newtheorem{Theorem}{Theorem}
\newtheorem{Lemma}[Theorem]{Lemma}
 
\newtheorem{Proposition}[Theorem]{Proposition}
 
 
 
\n   Let $ L= i J \frac{d}{dx} + V ,$
$\displaystyle
J =  \left (
\begin{array}{cc}
 1   & 0 \\
 0  &  -1
\end{array}
\right ) , $
$\displaystyle
V =  \left (
\begin{array}{cc}
 0   & p \\
 q  &  0
\end{array}
\right ) , $
be Dirac operator with periodic $L^2$-potential $V,$
$V(x+1) = V(x) .$ For any weight sequence  $B(k), $
$B(k) = B(-k) > 0, $
$B(k) \uparrow \infty .$
We define the space of potentials
$$ H(V) =  \{ V: \; \|V\|^2_B = \sum ( |p_k |^2 + |q_k |^2 )
B^2 (k) < \infty \} ,$$
where $p_k $ and $q_k $ are the Fourier coefficients of
the functions $p $ and $q. $
 
We consider boundary conditions (bc) of three types:
$Per^+ : F(1) = F(0);$
$Per^- : F(1) =- F(0);$
$ Dir :  \; f(0) = g(0), \; f(1) = g(0), $
and denote by $ L_{bc} $ the closure of $L_{|H^1_{bc}} $ in $L^2 .$
 
Put
$$ \Pi (X,Y) = \{ z \in {\bf C}^1 | \; |Re\,z | \leq X, \;
|Im \, z | \leq Y \} $$
and
$$D(a, \delta ) = \{ z \in {\bf C}^1 | \; |z-a| < \delta \} .$$
$L$ is well-defined on vector functions
$F = \left (
\begin{array}{c}
 f \\ g
\end{array}
\right ) , \;\; f,g \in H^1 . $
 
\begin{Theorem} (Counting Lemma)
There exists an absolute
constant $K,$
and dependent on $(B_k ) $
integer valued function $f(u), $
and a sequence $ \delta = (\delta_n ) ,$
$ \delta_n \downarrow 0, $ such that
$$ \sigma (L_{bc}) \subset \Pi (X, Y) \cup
\bigcup_{|n| > N} D(\pi n ; \delta_n \|V\|_{H(B)} ) ,$$
where
$$ X= K(1 + \|V\|_2 )^2 , \quad Y = \pi (N+1/2), \quad
N =  f(\| V\|_{H(B)}).$$
\end{Theorem}
 
\begin{Theorem}
Normalized eigen-functions of $L_{bc} $ are uniformly
$L^\infty $-bounded. Moreover, the same is true for all normalized
functions in
Riesz-subspaces $E_n =Im \, P_n ,$
$$ P_n = \frac{1}{2\pi i } \int_{|z-\pi n |= \delta_n }
(z- L_{bc})^{-1} dz , \quad |n| > N . $$
\end{Theorem}
 
These results remain valid if potential $V$ is in
the Banach-Orlicz space $L \log^2 L $
and $H(B) $ is substituted by a rearrangement-invariant space $E$ of
>$L^1 $-functions with compact imbedding $E \to L\log^2 L .$
 
 

\b
\n{\bf 36 \h P. I. Naumkin:Large time asymptotics for quadratic nonlinear Schr\"odinger equations 
in one space dimension}

 \b
\noindent We consider the Cauchy problem for the nonlinear Schr�dinger equations with quadratic nonlinearities of derivative type in one space dimension. We are interested in the global existence and large time asymptotic behavior of solutions of the Cauchy problem. Heuristically the quadratic type nonlinearities should be considered as sub critical from the point of view of large time behavior, since the quadratic nonlinearities decay in the uniform norm more slowly than the linear terms in the Schr�dinger equation. However as we show, due to the special oscillating structure of the nonlinear term the asymptotics behavior of solutions have a modified character. Our approach is similar to the method of normal forms of Shatah.


\b
\n{\bf 37 \h F. Nicoleau: A scattering inverse problem with the Aharonov-Bohm effect} 

\b
\n We study an inverse scattering problem for Schr�dinger operators with a convex obstacle and with an 
electromagnetic field in order to study the Aharonov-Bohm effect. In dimension greater than 3, we show that
 the electric and the magnetic field are uniquely determined. In the two dimensional case, some obstruction 
appears, based on a quantification of the magnetic flux. The same approach can be used to study an inverse 
scattering problem for a pair of Hamiltonians $(H(h) , H_0 (h))$, where $H_0 (h) = -h^2 \Delta$ and $H (h)=
 H_0 (h) +V$,$V$ is a short-range potential. We show that in dimension $n \geq 3$, the scatterings 
operators $S(h)$, $h \rightarrow 0$, which are localized near a fixed energy $\lambda >0$ determine 
the asymptotics of the potential $V$ at infinity. 

\b

\n{\bf 38 \h   R. H. Parmar: Super symmetric quantum mechanics with vector super potential }


  \def\n{\noindent}              \parindent 0in
  \def\dis{\displaystyle}          \parskip .15in
\def\ho{ harmonic oscillator }  \def\lo{ ladder operator }
\def\btd{\vec\bigtriangledown} \def\btds{\bigtriangledown ^2 }
   \def\p0{\psi_{0}}            \def \rh{\varrho}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
  \def\be{\begin{equation}}       \def\ee{\end{equation}}


\b


\n With P. C. Vinodkumar. SUSYQM deals usually with one-dimensional systems [1]. Here we define the 
SUSY charge   operators in three dimensional vector notations and define 
their scalar products to represent the Hamiltonian. It can then  easily be 
generlized for other higher dimensions.

We define here a general SUSY vector potential  $ \vec{\Phi } $ as $ - \btd 
\  \p0 / \p0 $ \ such that the SUSY charge operators can be written as,

\bea
\vec{Q^{+}} = \frac{1}{\sqrt{2}}    \ \left( -\btd - \frac{\btd \p0}{\p0}
\right)  \  ;   \  \vec{Q^{-}} = \frac{1}{\sqrt{2}} \left( \btd - \frac{\btd 
\p0}{\p0} \right)
\eea

\n Their scalar products according to SUSYQM [1] provide
\bea
\vec {Q^{+}} \cdot \vec{Q^{-}}   = H^{0}   =      \frac{1}{2} \left[ - \btds 
    +  \btd \cdot \left(\frac {\btd \ \p0}{\p0}  \right) +    \left(\frac 
{\btd \ \p0}{\p0} \right)^2    \right]
\eea
\n and
\bea
\vec {Q^{-}} \cdot \vec{Q^{+}}   = H^{1}   =      \frac{1}{2} \left[ - \btds 
  +    \left(\frac {\btd \ \p0}{\p0} \right)^2   -  \btd \cdot \left(\frac 
{\btd \ \p0}{\p0}  \right) \right]
\eea
\n Then the  two pair potentials $ V^0 $ and $ V^1 $ correspond to $H^0$ and 
$H^1$ are
\bea
V^{0} (\vec{r})  =  \frac{1}{2} \left[ \vec{\Phi } \cdot \vec{\Phi }    -  
\btd \cdot \vec{\Phi }
\right]
\eea
and
\bea
V^{1} (\vec{r})  =   \frac{1}{2} \left[ \vec{\Phi } \cdot \vec{\Phi }    +  
\btd \cdot \vec{\Phi }
\right]
\eea

\n For a solenoidal SUSY potential  $ \vec{\Phi } $,  $ H^1  =  H^0 $. This 
condition becomes a special case of the SUSYQM with vector potential.

\n Expressing the Hamiltonian in terms of Spherical polar co-ordinates, we 
get
\bea
H^{0}  =  \frac{1}{2} \left( -\btds + \ \Phi_{r}^{2} \ + \ \Phi_{\theta 
}^{2} \ + \ \Phi_{\varphi }^{2} \ - \ \Phi_{r}^{'} \ - \ \frac{2}{r} 
\Phi_{r} \ - \  \frac{\Phi_{\theta }^{'}}{r} \ - \ \frac{\Phi_{\theta }}{r \ 
tan \theta } \ - \  \frac{\Phi_{\varphi }^{'}}{r \ sin\theta } \right)
\eea
\n and
\bea
H^{1}  =  \frac{1}{2} \left( -\btds + \ \Phi_{r}^{2} \ + \ \Phi_{\theta 
}^{2} \ + \ \Phi_{\varphi }^{2} \ + \ \Phi_{r}^{'} \ + \ \frac{2}{r} 
\Phi_{r} \ + \  \frac{\Phi_{\theta }^{'}}{r} \ + \ \frac{\Phi_{\theta }}{r \ 
tan \theta } \ + \  \frac{\Phi_{\varphi }^{'}}{r \ sin\theta } \right)
\eea
\n Now, if  $ \vec{\Phi}$ is independent of $ \theta $ and $ \varphi $ and 
depend only on $r$,
the Hamiltonian become,
\bea
H^{0}  =  \frac{1}{2} \left( -\btds + \ \Phi_{r}^{2} \ - \ \Phi_{r}^{'} \ - 
\ \frac{2}{r} \Phi_{r} \right);
H^{1}  =  \frac{1}{2} \left( -\btds + \ \Phi_{r}^{2} \ + \ \Phi_{r}^{'} \ + 
\ \frac{2}{r} \Phi_{r} \right)
\eea
\n The advantage here is that the super potential $ \phi_r $ does not depend 
on the angular momentum unlike in the regular SUSYQM for spherical symmetric 
cases [2]. An additional term of
$ \pm \ 2 \Phi_r /r $  also appear in the Hamiltonian. For a choice of $ 
\vec{\Phi } = \vec{r} $ \ leads to the usual harmonic oscillator problem in 
SUSYQM. Other choices of $ \vec{\Phi }$ and their solutions will be 
discussed.

\medskip 
\noindent {\bf References.} 


\n 1. \  V A Kostelecky, D K Campbell Supersymmetry in Physics (North 
Hollend, 1985).

\n 2. \ Avinash Khare, Pramana Jnl. Phys {\bf 49} (19997) 41.


\b
\n{\bf 39 \h D. Pearson: Recent developments in value distribution theory for 
Schr\"odinger operators}

\b
\n The theory of value distribution for boundary values of Herglotz functions is closely linked to
 spectral properties of Schr\"odinger operators on the half line.The theory is used to provide a 
large distance asymptotics for solutions of the Schr\"odinger equation, and throws light on such questions
 as the existence of singular and absolutely continuous spectra and estimates of the Weyl-Titchmarsh 
m-function for sparse potentials. Some recent developments of this work are presented and lead to a 
characterization of asymptotics for the support of ac measures.

\b

\n{\bf 40 \h G. S. Pogosyan:  Coulomb-oscillator duality in spaces of constant curvature}

\b

With E. G. Kalnins, G.  and W. Miller, Jr. We have constructed a series of mappings 
$S_{2C} \rightarrow S_{2},
S_{4C}\rightarrow S_{3}$ and $S_{8C}\rightarrow S_5$, that
are generalize those well known from the  Euclidean space
Levi-Civita, Kustaanheimo-Stiefel and Hurwitz transformations.
It have shown, that as in case of flat space, these transformations
permit one to establish the {\it correspondence} between the
Kepler-Coulomb and oscillator problems in classical and quantum
mechanics for the respective dimensions.  Using these transformations
the quantum Coulomb system on the two-, three- and five-dimensional 
sphere are completely solved, including eigenfunctions with correct
normalization constant and energy spectrum.

\b
\n{\bf 41 \h R. Quezada: A class of non-conservative quantum dynamical semigroups}

\b
\n Quantum dynamical semigroups (qds) appear as solutions of the so called master equation which is a generalization of the Schrodinger equation in the Heisenberg representation with noise terms; that is the reason why a qds describes the irreversible (dissipative, nonunitary) evolution of a quantum system. In the theory of stochastic processes the master equation is a noncommutative extension of the backward Kolmogorov equation. The conservativity (Markovianity or unitality) of a qds is related with the non-explosion of a stochastic process and it is useful to assure the uniqueness of the minimal solution of a master equation. This property has been intensively studied in the recent years. Much less attention has received the class of non-conservative qds, nevertheless its study is important from both the mathematical point of view as well as for applications. Our main aim in this work is to study a class of non-conservative qds that naturally appear when a necessary condition for conservativity is not satisfied.


\b
\n{\bf 42 \h V. Rabinovich: Pseudodifferential operators with analytic symbols and estimates for  eigenfunctions of Schr\"{o}dinger operators}

\b
\n We study the behaviour of eigenfunctions of \ the Schr\"{o}dinger
operator $%
-\Delta +v(x)$ with potentials having power, exponential or
super-exponential growth at infinity and discontinuities on manifolds in
$%
\mathbf{R}^{n}$. We use a connection between the domain of
analyticity of the main symbol $\left( \left| \xi \right|
^{2}+v(x)\right) ^{-1}$ of the parametrix $-\Delta +v(x)$ at
infinity or near singularities of $v(x)$ and the behaviour of
eigenfunctions at infinity or near singularities of potentials.
Our approach is based on a general calculus of pseudodifferential
operators with analytic symbols.

\b


\n{\bf 43 \h R. del R\'{\i}o : Inverse spectral results for Dirac systems} 
\bigskip


\noindent
We generalize results of Hochstadt and Lieberman for Dirac Systems. From partial knowledge of the 
potential and partial knowledge of the spectra it is shown that the rest of the potential function is uniquely determined. An important ingredient in our strategy is the link between the rate of growth of an entire function and the distribution of its zeros. This is joint work with Benoit Grebert.

\b

\n {\bf 44 \h D. Robert: Long time propagation  for quantum observables and coherent states in the 
semiclassical r\'egime}

\b
\n In this lecture  we review some recent results obtained for solutions of time dependent Schr\"odinger equations in the semiclassical  r\'egime ($\hbar \rightarrow 0$). We consider as well propagation of quantum observables (Heisenberg picture)  and propagation  of Gaussian coherent states. The main problems discussed here concern the validity of the asymptotic expansions in $\hbar$  for large times. For general systems (in particular for  classical chaotic systems)  we prove that the Ehrenfest time $T_\hbar = c\log
\left(\frac{1}{\hbar}\right)$,
where $c>0$ depends on unstability Lyapounov exponents,  is a true time limitation for the semiclassical r\'egime.
On the opposite side, for integrable systems this time is much  longer (at leat o($\hbar^{-1/2}$)).\\
Furthermore for analytic or Gevrey smooth Hamiltonians we get analytic or Gevrey estimates for the corresponding semiclassical
  asymptotic expansions and consequently exponentially small remainder estimates.

\b

\n{\bf 45 \h Gert Roepstorff: On a Class of anomaly free gauge theories}

\b
\n This is a report on a calculation of the anomaly coefficients for the reducible representation 
$\theta$ of a Lie algebra $\mbox{Lie}\,G$ on
$ \bigwedge {\bf C}^n$. Assuming that $G\subset U(n)$ where $n\ge2$, the representation $\theta$ is obtained from lifting the action of $U(n)$
on  $  {\bf C}^n$ to the exterior algebra.
The coefficients vanish provided $G\subset SU(n)$ and $n\ne3$.
The singular role of the group $SU(3)$ is emphasized.

\b




\n{\bf 46 \h M. B. Ruskai: Comment on efficiency of adiabatic quantum algorithms}
 \bigskip

\b
\n It is argued that a proposed algorithm for adiabatic quantum computation can not solve an NP-�complete problem faster than Grover's algorithm, which requires $O(2^{n/2})$ time. 
There has been considerable interest in a proposed scheme for adiabatic 
quantum computation [1, 2] and speculation that it may provide a mechanism 
for effcient solution of hard problems. However, the latter is based on extrap� 
olations of numerical simulations for model problems, particularly the ``exact 
cover'' problem, with $n<20$. 
The adiabatic quantum algorithm is designed to take the ground state of 
an initial Hamiltonian H0 to that of a final Hamiltonian H1 using a linear 
interpolating Hamiltonian of the form H(t)=(1- t)H0  + tH1 . The efficiency 
of the adiabatic approximation has been shown to be $O(1/g 2 min )$ where g min 
denotes the minimum energy gap g(t)=E 1 (t) -E 0 (t) between the ground and 
first excited states of H(t). 
The ``computational basis'' 
|k 1...k n  consists of products of eigenfunctions 
of the Pauli matrix  z . Fahri, et al use the initial Hamiltonian H 0 =  j  x (j) 
whose eigenstates can be obtained by applying Hadamard transforms to the 
states 
|k 1...k n , yielding states of the form 2 -n/2  k 1 ...k n �1 
|k 1...k n  (with all 
signs +1 for the ground state). Fahri, et al define a non�negative final Hamil� 
tonian H 1 which is diagonal in the computational basis and has the solution of 
the Exact Cover problem as its ground state, i.e., they encode the Exact Cover 
problem in the computational basis. 
Grover [3, 4] showed how to encode searching an unordered list into the 
computational basis via the construction of a unitary operator G whose effect is 
simply to multiply the target state |j 1 ...j n  by 1 and all others by +1. (The 
state |j 1 ...j n  need not be known in advance, only a method of identifying it.) 
If we now let H 2 = GH 1 , we obtain a Hamiltonian [5] identical to H 1 except 
that the eigenvalue of the target state |j 1 ...j n  is multiplied by 1. Because H 1 
is non�negative, H 2 has exactly one negative eigenvalue so that its ground state is 
now the target state |j 1 ...j n . Applying the adiabatic algorithm to the modified 
interpolating Hamiltonian H 2 (t)=(1 t)H 0 +tH 2 should take the ground state 
of H 0 to the target state. Moreover, the only effect on the final Hamiltonian is to 
move one excited state below the previous ground state, without decreasing the 
final energy gap g(1). There is little reason to expect this to significantly affect 
g min (t); therefore, one would expect an adiabatic search to be as effcient as the 
solution of the problem encoded in H 1 . Thus, if Fahri, et al's projection of an 
efficiency of O(n 2 ) is correct, one would expect an efficiency of O([log N ] 2 ) for 
the adiabatic search of a list of N =2 n items described above. More generally, 
an adiabatic efficiency of O(n p ) would imply an adiabatic search efficiency for 
N =2 n items of O([log N ] p ). However, this contradicts earlier arguments [6, 4] 
that a quantum computer could not search an unordered list in better than 
O(  N) time. 
The validity of the adiabatic approximation depends on the existence of an 
eigenvalue gap (which should follow from the uniqueness of the ground state 
[7]), but the efficiency of the proposed algorithm, also requires that the gap 
decrease sufficiently slowly as n increases. This has not been established. In 
standard applications of the adiabatic approximation, it is reasonable to expect 
that perturbations of the form �(H 1 H 0 ) mix only a few low�lying states. 
However, the proposed algorithm transforms the initial state to a final (product) 
state which is an evenly weighted superposition of all 2 n excited states of H 0 . 
This suggests that a perturbation expansion will converge slowly which implies 
that g(t) must decrease rapidly or, alternatively, that many small time steps are 
needed to mix the required 2 n states. 
 

\n Acknowledgement: It is a pleasure to thank Professors Charles Bennett, David 
DiVicenzio, George Hagedorn and Christopher King for helpful discussions and 
comments. An expanded discussion of some of the issues raised here will be 
posted on the quant�ph preprint server. This research was partially supported 
by the National Security Agency and Advanced Research and Development 
Activity under Army Research Office contract number DAAG55�98�1�0374 and 
by the National Science Foundation under Grant number DMS�0074566. 

\medskip 
\noindent {\bf References.} 

\n 1.  E. Fahri, J. Goldstone, S. Gutmann and M. Spiser ``Quantum Computation 
by Adiabatic Evolution'' quant�ph/0001106
 
\n 2.   E. Fahri, et al ``A Quantum Adiabatic Evolution Algorithm Applied to 
Random Instances of an NP�complete Problem'' Science 292, 472--474 (20 
April 2001); see also quant�ph/0104129 

\n 3.  L. Grover in Proceedings of 28th Annual ACM 
Symposium on the Theory of 
Computation, pp. 212�219 (ACM Press, 1996)
 
\n 4. M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Infor� 
mation (Cambridge University Press, 2000).
 
\n 5. Because G and H 1 are both diagonal in the computational basis, they com� 
mute and H 2 is self�adjoint. 

\n. 6 C.H. Bennett, E. Bernstein, G. Brassard and U. Vazirani ``Strengths and 
Weaknesses of Quantum Computing'' SIAM J. Comput. 26(5), 1510�1523 
(1997). 

\n 7. As will be discussed in a paper to be posted at arXiv.org/quant�ph, there 
are some serious questions about Fahri, et al's use of the non�crossing rule 
to establish uniqueness of the ground state. However, for the Hamiltonians 
in question, this can be done by a standard Perron�Frobenius argument. 


\b
\n{\bf 47 \h A. Rybkin: On the generalized spectral shift function for one-dimensional 
Schrodinger operators with slowly decaying potentials and trace formulas} 

\b

\n In our talk we are concerned with self-adjoint Schrodinger operators 
on the line. It is well-known that if the potential is summable then 
Krein's spectral shift function exists (as well as the scattering 
matrix) and a variety of trace formulas can be described. 
The situation is different if the potential is decaying but slowly. We 
concentrate on the case when the potential is squire integrable. 
We intend to discuss the properties of a certain generalized 
spectral shift function and describe analogs of trace formulas in 
this setting.


\b

\n{\bf 48 \h A. Sachetti:  Tunneling instability for a double well Schroedinger operator with a non-linear perturbation}
 	
\b
 \n I consider a non-linear perturbation of a symmetric double-well potential as a model for molecular localization. In the semiclassical limit, I prove the existence of a critical value of the perturbation parameter giving the destruction of the beating effect.		

\n{\bf 49 \h R. Schrader: Statistical ensembles and density of states}

\b
\n  In our talk we will give a brief exposition of joint approach with 
V.Kostrykin to the
statistical theory of small systems using the concept of integrated 
density of states familiar from the theory of random Schr\"{o}dinger operators.
The idea to consider the thermodynamics for systems with a small number of 
particles is about 30 years old with one of the first articles due to 
Thirring. It has found applications in nuclear physics and in
astrophysics.  
The present approach covers both the classical and the quantum case and 
thus allows for a comparison. A microcanonical ensemble may be 
defined provided the integrated density of states is twice
differentiable. 
The structure of the van Hove singularities in the case of one particle
moving in a periodic potential in 3 dimensions implies that the
corresponding integrated density of states for n noninteracting
particles is [(3n-1)/2] times differentiable. The case of random
potentials and of an additional interaction between the
particles remains an open problem.
In analogy a canonical ensemble may be defined and may be compared 
with the microcanonical ensemble in a way familiar from the standard
formulation.
The classical case allows for a new example of a system giving rise to 
a negative specific heat. In the quantum case and for the 
microcanonical ensemble the relation between the integrated density of 
states and the scattering phase shift 
density - obtained in collaboration with V. Kostrykin -
combined with a conjectured high enery behavior of the phase shift  
gives rise to an explicit form of the (mean) energy-temperature 
relation in this high energy regime.
  
\b
\n{\bf 50 \h B. Schumayer: Painlev{\'e} test of coupled nonlinear Schr{\"{o}}dinger
       equations  and Bose-Einstein condensates}

\b
\n Nonlinear dynamical evolution equations are often employed in
physics, especially in field theories. Historically, the detailed
examination of the KdV equation shows that this type of equations
possesses special solutions, e.g., solitary waves, solitons.
Furthermore, these so called totally integrable equations, like
the KdV itself, or the nonlinear Schr{\"{o}}dinger equation, or
its variant, the Gross-Pitaevskii equation, can be solved by the
method of inverse scattering transformation. In 1978 Ablowitz,
Ramani and Segur [1] made a conjecture: if an
equation passes the Painlev{\'{e}}-test it can be solved by
inverse scattering transformation.

In our work we apply [2] the Painlev{\'{e}} test
to the coupled nonlinear Gross-Pitaevskii equations describing a
pile of weekly interacting atomic particles belonging to the same
Bose-Einstein condensed state. We show that the system of
equations becomes integrable for special external potentials
depending on system parameters such as masses and interaction
strengths.  We derive a formula for classifying the various cases
in terms of a single integer constant $m$.  Our example with $m=2$
turns out to agree with an earlier investigation carried out by
Clarkson [3] for the uncoupled case.  Finally, we
derive [4] also a transformation between the
solutions of the coupled nonlinear Schr{\"{o}}dinger equations and
those of the coupled Gross-Pitaevskii equations. Our
transformation ansatz and general formulas enable direct use of
the highly evolved soliton solutions of the coupled nonlinear
Schr{\"{o}}dinger equations in recent two-component Bose-Einstein
condensate experiments.

\medskip 
\noindent {\bf References.} 
 \begin{enumerate}


\item  M. J. Ablowitz and A. Ramani and H. Segur,
    Lett.\,Nuovo.\,Cim., \textbf{23}(9), 333--338 (1978)

\item D. Schumayer and B. Apagyi, J. Phys. A. {\textbf{34}}, 4969--4981 (2001)

\item  Peter A. Clarkson, Proc.\,Roy.\,Soc.\,Edin.\,A., \textbf{109A}, 109--126 (1988)

\item  D. Schumayer and B. Apagyi, Phys. Rev. A., submitted

\end{enumerate}

\b

 

\b
\n{\bf 51 \h R. Seiringer: Gross-Pitaevskii Theory of the Rotating Bose Gas}


\b
\n We study the Gross-Pitaevskii functional for a rotating
two-dimensional Bose gas in a trap. We prove that there is a
breaking of the rotational symmetry in the ground state; more
precisely, for any value of the angular velocity and for large
enough values of the interaction strength, the ground state of the
functional is not an eigenfunction of the angular momentum. This
has interesting consequences on the Bose gas with spin; in
particular, the ground state energy depends non-trivially on the
number of spin components, and the different components do not
have the same wave function. For the special case of a harmonic
trap potential, we give explicit upper and lower bounds on the
critical coupling constant for symmetry breaking.

\b 

\n{\bf 52 \h E Sere: A ground state for the Dirac-Fock model}
 
\b
\n The Dirac-Fock equations are the relativistic analogue of the well known Hartree-Fock equations. They are used in quantum chemistry, and yield results on the inner-shell electrons of heavy atoms. We give some existence results for the solutions of the Dirac-Fock equations, and we study the convergence of these solutions in the nonrelativistic limit. As a byproduct, we define a notion of ground state for the Dirac-Fock model, valid when the speed of light is large enough. This is joint work with M.J. Esteban.


\n{\bf 53 \h H. Siedentop: The Energy of relativistic one-electron atoms}

\b
\n Joint work with Edgardo Stockmeyer and Raymond Brummelhuis.
Relativistic particles have been described by Hamiltonians that are
the quantization of the relativistic symbol $\sqrt{p^2+m^2}-
Z\alpha/|x|$. This Hamiltonian has been investigated by Herbst and
Weder. The critical coupling constant, however, turns out to be
$\alpha Z \leq 2/\pi$ which does not cover all elements. In the
present talk we will present a model going back to Douglas and Kroll,
and Jansen and He{\ss}. We will show that the energy of this
Hamiltonian used in quantum chemistry which describes all known
elements numerically to high accuracy is bounded from below for
$\alpha Z \leq \gamma_c$ where $\gamma_c$ slightly exceeds 1. This
fact will allow to construct a Hamiltonian which is self-adjoint.

\b

\n{\bf 54\h F. Sobieckzky: When do digital images fragment into infinitely many segments}

\b
\n The image segmentation problem is the question of how to divide the domain of mapping into subsets which fulfill some homogeneity criterion. An analytical image segmentation method is proposed by which such criteria are investigated as to whether they separate digital images into infinitely many segments, as the dimension of the images' representation (and thereby the degree of detail) is increased to infinity. In order to describe the quantized structure of the discrete (digital) image data, the dynamics given by a Schroedinger Operator of a quantum spin system is considered. Using von Neuman's ergodic theorem, the procedure aimes at finding the orthogonal projectors onto characteristic subsets of a lattice, which contain a single given lattice point. Segments are identified with the invariant subsets of the underlying non-ergodic dynamical system. Key-words: Quantized Region Growing, Stability of Quantum Spin Systems, Hilbert Space methods and Image Segmentation, finite Magnetization.

\b


\n{\bf 55 \h S. Sontz: �New results in reverse inequalities}

\b
\n Based on earlier work of Carlen on reverse hypercontractivity and by the author on reverse log-Sobolev inequalities (both in the Segal-Bargmann space), we have recently obtained generalizations of these results to a certain class of complex manifolds. This is joint work with F. Galaz-Fontes and L. Gross. 

\b


\n{\bf 56  \h P. Stovicek: Weakly regular Floquet Hamiltonians with pure point spectrum}


\b 
\n We study the Floquet Hamiltonian 
$-i\partial _{t}+H+V(\omega t)$, acting in 
$L^{2}([\,0,T\,],\mathcal{H},dt)$, as depending
on the parameter $\omega =2\pi /T$. We assume that 
the spectrum of $H$ in $\mathcal{H}$ is discrete,
$\mathrm{Spec}(H)=\{h_{m}\}_{m=1}^{\infty }$,
but possibly degenerate, and that 
$t\mapsto V(t)\in \mathcal{B}(\mathcal{H})$
is a $2\pi$-periodic function with values in the space 
of Hermitian operators on $\mathcal{H} $. Let $J>0$ and 
set $\Omega _{0}=[\,\frac{8}{9}J,\frac{9}{8}J\,]$.
Suppose that for some $\sigma >0$ it holds true 
that $\sum_{h_{m}>h_{n}}\mu_{mn}(h_{m}-h_{n})^{-\sigma }<\infty$
where $\mu _{mn}=(\min \{M_{m},M_{n}\})^{1/2}M_{m}M_{n}$ 
and $M_{m}$ is the multiplicity of $h_{m}$. We show that in
that case there exist a suitable norm to measure the regularity 
of $V$, denoted $\epsilon _{V}$, and positive constants,
$\epsilon_{\star }$ and $\delta _{\star }$, with the property: 
if $\epsilon _{V}<\epsilon_{\star }$ then there exists 
a measurable subset $\Omega _{\infty }\subset \Omega _{0}$
such that its Lebesgue measure fulfills
$|\Omega_{\infty}|\geq |\Omega_{0}|-\delta_{\star}\epsilon_{V}$
and the Floquet Hamiltonian has a pure point spectrum for all 
$\omega \in\Omega_{\infty}$.

\b
\n{\bf 57 \h M. Tater: Scattering by a slab}




\b
\n The model consists of an infinite number of point interactions distributed periodically 
on a finite number of parallel planes. Exact formulae for the transmission coefficients are given. 
Numerical behaviour is presented, first in the case the incident plane wave has a wave vector perpendicular
 to the planes and then for some tilted directions. For some values of the interaction parameter the 
reflection coefficient is close to one for a relatively large set of incident wavelengths. 


\b

\n{\bf 58 \h A. Tip: Maxwell with a touch of Schr\"{o}dinger }

\b
\n We consider the macroscopic Maxwell's equations for a linear, in
general absorptive, dielectric. Such systems are of interest for
optical devices, in particular photonic crystals, dielectrics with
a spatial periodicity.
Without absorption and dispersion, i.e. if the electric permeability $%
\varepsilon ({\bf x)}$ is a function of position only, it is
straightforward to set up a unitary time evolution scheme and the
spectral structure of the generator then gives information about a
number of the system's properties.
In particular for a photonic crystal, where $\varepsilon ({\bf x+a)=}%
\varepsilon ({\bf x)}$, ${\bf a}$ being the lattice vector, a band
structure emerges and there may be band gaps. The latter can
become of great physical interest for the construction of photonic
devices, such as solar cells, solid state lasers and, possibly,
quantum computers. The reason is that excited states of atoms,
embedded in such crystals, can become stable if their transition
frequency falls in a band gap. Then there are no field modes
available to transport away the atomic excitation energy.

However, it is difficult to construct such photonic crystals with
gaps in the optical region and this has led to the study of
absorptive devices, which seem to be more promising. But now there
is a difficulty since ${\bf D}$ in Maxwell's equations
\begin{equation}
\partial _{t}{\bf D}{(}{\bf x},t)=\partial _{{\bf x}}\times {\bf B}{(}{\bf x}%
,t),\quad \partial _{t}{\bf B}{(}{\bf x},t)=-\partial _{{\bf x}}\times {\bf E%
}{(}{\bf x},t),
\end{equation}
is related to ${\bf E}$ through a time convolution
\begin{equation}
{\bf D}{(}{\bf x},t)={\bf E}{(}{\bf x},t)+\int_{-\infty }^{t}ds\chi ({\bf x}%
,t-s){\bf E}{(}{\bf x},s),
\end{equation}
and (1) does not define a unitary time evolution.

However, it is possible to introduce two auxiliary fields ${\bf F}_{2,4}{(}%
{\bf x},\lambda ,t)$, $\lambda \in [0,\infty )$, such that
\begin{equation}
{\bf F}(t)=\left(
\begin{array}{l}
{\bf F}_{1}(t) \\
{\bf F}_{2}(t) \\
{\bf F}_{3}(t) \\
{\bf F}_{4}(t)
\end{array}
\right) =\left(
\begin{array}{l}
{\bf E}(t) \\
{\bf F}_{2}(t) \\
{\bf B}(t) \\
{\bf F}_{4}(t)
\end{array}
\right) ,
\end{equation}
satisfies a unitary time evolution
\begin{equation}
\partial _{t}{\bf F}(t)=-i{\sf K}{\bf F}(t)
\end{equation}
in an appropriate Hilbert space of which the norm is proportional
to the conserved energy [1], the latter being an extension
of the conserved electromagnetic energy
\begin{equation}
{\cal E=}\frac{1}{2}\int d{\bf x\{E}{(}{\bf x},t)^{2}+{\bf B}{(}{\bf x}%
,t)^{2}\}.
\end{equation}

In an absorptive photonic crystal we have $\chi ({\bf x+a},t)=\chi ({\bf x}%
,t)$, leading to a discrete translational invariance of the
selfadjoint generator {\sf K}, thus allowing a Bloch decomposition
as in the Schr\"{o}dinger case. But in the zero order case, where
${\bf F}_{2,4}$ are decoupled from the physical fields ${\bf E}$
and ${\bf B}$, the generator for the subset
\begin{equation}
\left(
\begin{array}{l}
{\bf F}_{2} \\
{\bf F}_{4}
\end{array}
\right)
\end{equation}
has the full positive real axis as its spectrum and this excludes
the possibility of {\sf K} having spectral gaps. But we are only
interested in the physical fields and it is sufficient that their
Fourier transforms relative to the time variable, ${\bf
\tilde{F}}_{1,3}({\bf x},\omega )$, have gaps.

We shall present a precise definition of a band gap along these
lines and discuss the spectral properties of {\sf K} for the
photonic crystal case. Using a dilatation transformation on the
$\lambda $-variable, above, we find that the complex-dilated {\sf
K}$(\zeta )$ has spectrum consisting of areas in the lower complex
plane. The analysis, which uses the usual relative compactness
arguments, is a good deal more complicated than in the
Schr\"{o}dinger case, due to the presence of an
infinite-dimensional null space. In general band gaps no longer
occur [2]. We shall also show the results of calculations
of the dilated spectrum for a two-dimensional case, confirming
these results [3].

\medskip 
\noindent {\bf References.} 


\begin{enumerate}
\item  A. Tip,{\em \ Linear absorptive dielectrics}, Phys. Rev. A {\bf 57} (1998), 4818.

\item  A. Tip, A. Moroz and J.-M. Combes,{\em \ Band structure of
absorptive photonic crystals}, J. Phys. A {\bf 33} (2000), 6223.

\item  H. van der Lem, A. Moroz and A. Tip, in preparation.
\end{enumerate}


\b
\n{\bf 59 \h J. Toloza: Exponentially Small Error Estimates of Quasiclassical Eigenvalues}

\b

\n We study the behavior of truncated Rayleigh-Schr\"odinger series for the 
low-lying eigenvalues of the time-independent Schr\"odinger 
equation, in the semiclassical limit $\hbar\searrow 0$. We prove that, under 
certain hypotheses on the potential $V(x)$ and for small $\hbar>0$, the 
perturbation series for the eigenvalues admits an exponentially accurate 
truncation. That is, the difference between the truncated series and the actual
eigenvalue can be made smaller than $\exp(-C/\hbar)$ for a positive constant 
$C$. An analogous statement is shown for the corresponding eigenfuctions. 

\b

\n{\bf 60 \h A. Turbiner: Canonical Discretization (discretization as canonical transformation)}

\b
 
\n A new realization of the Heisenberg algebra in terms of finite-difference (translation covariant) and discrete (dilatation covariant) operators is presented. It is a realization of deformation type and can be treated as a quantum analogue of canonical trasformations. Fock space formalism which allows a notion of spectral problem is introduced. The formalism provides a natural isospectral connection of differential and difference equations which also leaves polynomial eigenfunctions invariant. (An)harmonic oscillator as well as the Hahn equation are considered in details as examples.

\b

\n{\bf 61 \h G. Uhlmann: On the local Dirichlet-to-Neumann map}

\b
\n We will discuss some recent results on the problem of determining a
conductivity or a potential by measuring the Dirichlet-to-Neumann
map associated to the conductivity equation or the Schroedinger
equation by making measurements on part of the boundary.


\b
\n{\bf 62 \h C. van der Mee: The Inverse generalized Regge problem} 
  \newcommand{\R}{{\bf R}} 
%\newcommand{\R}{\mathbb{R}} 
\newcommand{\za}{\alpha} 
\newcommand{\zb}{\beta} 
\newcommand{\zl}{\lambda}

\b 
 With V. Pivovarchik.  Small transversal vibrations of a nonhomogeneous string are described by a boundary value problem which by applying a Liouville transformation is transformed into the Sturm-Liouville problem \begin{eqnarray} y^{\prime\prime}(\zl,x)+\zl^2y(\zl,x)-q(x)y(\zl,x)&=&0,\label{eq:1}\\ y(\zl,0)&=&0,\label{eq:2}\\ y^\prime(\zl,a)+(i\za\zl+\zb)y(\zl,a)&=&0,\label{eq:3} \end{eqnarray} where $\za>0$, $\zb\in\R$, and $q\in L_2(0,a)$ is real-valued. For $\za=1$ and $\zb=0$ we obtain the so-called Regge problem. We provide a method of constructing the interval length $a$, the potential $q$ and the two constants $\za$ and $\zb$ from its eigenvalues. We give necessary and sufficient conditions on sequences of complex numbers to be the eigenvalues of Eqs. (\ref{eq:1})-(\ref{eq:3}) in the separate cases $0<\za1$. First, the eigenvalues are shown to be the zeros of an entire function belonging to a generalization of the Hermite-Biehler class and their asymptotic properties are derived. To solve the inverse problem, we employ the given sequence of complex 
numbers to construct entire functions which are then used to construct a scattering function. The potential $q$ then follows with the help of the usual Marchenko method. 

\b

\n{\bf 63 \h I. Veselic: Wegner estimate for alloy type models with non negative single site potential of small support}

\b

\n We study spectral properties of Schr\"odinger operators with a random potential of
Anderson or alloy type on $L^2(R^d)$ and their restrictions to finite boxes. We extend and unify W. Kirsch's result on  Wegner estimates for non-negative single site potentials with small support. Our estimate is valid for all bounded energy intervals. Subsequently we use recent results on the spectral shift function and establish the H\"older continuity of the integrated density of states on the whole energy axis. Wegner estimates play  a key role  in an existence proof of pure point spectrum of the considered random Schr\"odinger operators.\\


\medskip

{\bf References}

Combes, J. M.; Hislop, P. D.; Nakamura, Shu. The $L\sp p$-theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators. Comm. Math. Phys. 218 (2001), no. 1, 113--130.

Kirsch, W. Wegner estimates and Anderson localization for alloy-type potentials. Math. Z. 221 (1996), no. 3, 507--512.

\b
\n{\bf 64 \h C. Villegas Blas: The Bargmann transform, canonical transformations and the Kepler problem} 

\b

\n In this talk we describe a Bargmann transform when the configuration space is ${\cal S}^2$ or ${\cal S}^3$ (the unit spheres in the Euclidean spaces of three and four dimensions respectively). That is, a unitary transformation of $L^2({\cal S}^2)$ or $L^2({\cal S}^3)$ onto a Hilbert space of analytical functions. We show that in the kernel of such a transform, we have a generating function of a canonical transformation which is the classical analog of the Bargmann transform. This canonical transformation involves a complex quadric and it is also related to the regularization of the Kepler problem. We also study a space of analytical functions on the quadric and relate our study with results of V. Bargmann and V. Guillemin. Finally, we identify our Bargmann transform as a coherent states transform. 

\b
\n{ \bf 65 \h V. Vougalter: Pauli operator and Aharonov-Casher theorem for measure valued magnetic fields}

\b

\n  We define the two dimensional Pauli operator and identify its core for magnetic fields that are Borel measures. The magnetic field is generated by a scalar potential hence we bypass the usual ${\bf A}\in L^2_{loc}$ condition on the vector potential which does not allow to consider 
such singular fields.  We extend Aharonov-Casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case  of infinite total variation. One of the key technical tools is a weighted $L^2$ estimate on a singular integral operator.		

\b
\n{\bf 66\h K. B. Wolf: Finite Quantum/Optical Systems}     

\b
\n  Hamiltonian systems that satisfy the oscillator axioms
with a compact dynamical algebra, will have a finite number of energy
levels, and the range of positions will have a finite number of values.
Such systems satisfy Schrodinger difference equations, and their
eigenfunctions involve finite polynomials (Kravchuk, Meixner, Hahn)
times the square root of their orthonormality measure. These systems
exhibit coherent states, mixed states, a properly defined compact 
phase space, and a covariant Wigner quasiprobability distribution.
Finite Hamiltonian systems are useful in signal processing (finite optics)
since they define fractional Fourier-Kravchuk and Hankel-Hahn 
transforms on finite data sets, and on pixellated square or round screens.
These are a sequence of finite unitary approximants to the Fourier and
Hankel (Fourier-Bessel) transforms. 

\b

\n{\bf 67 \h K. Yajima: The local smoothing property and Strichartz inequality 
for Schr\"odinger equations 
with potentials superquadratic at infinity} 
 

\def\tr{{\bigtriangledown}}
\def\R{{\bf R}}
\def\D{{\Delta}}
\def\ep{{\varepsilon}}
\def\a{{\alpha}}
\def\c{{\gamma}}
\def\p{{\partial}}
\def\H{{\cal H}}
\def\th{{\theta}}
\def\lap{{\bigtriangleup}}
\def\la{{\langle}}
\def\ra{{\rangle}}
\def\ax{{\la x \ra}}
\def\ds{\displaystyle}

\def\bq{\begin{equation}}
\def\eq{\end{equation}}
\def\br{\begin{array}}
\def\er{\end{array}}

\def\lbeq(#1){\label{eqn:#1}}
\def\refeq(#1){{\rm (\ref{eqn:#1})}}
\def\lbth(#1){\label{th:#1}}
\def\refth(#1){{\rm Theorem \ref{th:#1}}}
\def\lbass(#1){\label{ass:#1}}
\def\refass(#1){{\rm Assumption \ref{ass:#1}}}

\def\bgdf{\begin{definition}}
\def\eddf{\end{definition}}
\def\bgth{\begin{theorem}}
\def\edth{\end{theorem}}
\def\bgass{\begin{assumption}}
\def\edass{\end{assumption}}
\renewcommand{\theequation}{\arabic{equation}}
\makeatother
\def\ben{\begin{enumerate}}
\def\een{\end{enumerate}}

\def\qed{\hbox {\hskip 1pt \vrule width 4pt height 6pt depth 1.5pt
\hskip 1pt}\\}% cqfd
\def\qed{\hbox {\hskip 1pt \vrule width 4pt height 6pt depth 1.5pt
         \hskip 1pt}\\}% cqfd



\newtheorem{theorem}{Theorem}%[section]
\newtheorem{assumption}[theorem]{Assumption}
\n With G. Zang. We consider $n$-dimensional Schr\"odinger equations
with potentials which grow super-quadratically at infinity:
\bq
i\frac{\p u}{\p t}= -(1/2)\lap u + V(x)u,\ \ x \in \R^n, \ t \in \R ; \quad
u(0,x)=u_0(x), \ \ x \in \R^n .
\lbeq(0)
\eq
\bgass $V(x)$ is real valued and is of $C^\infty$-class.
There exist $m>2$ and $R>0$ such that:
\ben
\item For $|x| \geq R$, $D_1 \ax^m \leq V(x) \leq D_2 \ax^m$,
where $D_1 \leq D_2$ are positive constants.
\item For $|\a|\geq 2$, ${\ds |\p^\a_x V(x)| \leq C_\a \ax^{m-|\a|}}$.
\een
\lbass(A)
\edass
Under the assumption, the operator
$L: u \mapsto -(1/2)\lap u + V(x)u$ defined on $C_0^\infty(\R^n)$ is
essentially selfadjoint in $L^2(\R^n)$ and the solution in $L^2(\R^n)$
of the initial value problem \refeq(0) is given by $u(t, \cdot) = U(t)u_0$
via the unitary group
$U(t)=e^{-itH}$ generated by the unique selfadjoint extension $H$ of $L$.
We show that the solution $u(t,\cdot)$, nonetheless, is
much smoother than $u_0$ and $1/m$ times differentiable at almost all time $t\not=0$. More precisely,
we prove the following theorem.  We write
$\la A \ra =(1+ |A|^2)^{\frac{1}{2}}$ for a self-adjoint operator $A$
and $D=(D_1, \ldots, D_n)$, $D_j=-i\p/\p x_j$.
$\|\cdot\|_p$ is the norm of Lebesgue space $L^p(\R^n)$ and
$\|\cdot\|=\|\cdot\|_2$, $1\leq p \leq \infty$.


\bgth Let $V$ satisfy {\rm \refass(A)}. Let $T>0$ and
$\Psi \in C_0^\infty(\R^n)$. Then, there exists a constant $C>0$ such that
\bq
\left(\int^T_{-T} \|\Psi(x) \la D \ra^{\frac1{m}}e^{-itH}u_0\|_2^2 dt
\right)^{\frac1{2}}
\leq C \| u_0\|, \quad u_0 \in L^2(\R^n).
\lbeq(impr)
\eq
\lbth(impr)
\edth
\refth(impr) is sharp in the sense that the exponent
$1/m$ in \refeq(impr) cannot in general be replaced by any larger number.
This can be seen by taking the potential
$V(x)=\la x_1\ra^{m}+ \cdots + \la x_n\ra^{m}$ and the initial state
$u_0(x)=e_{i_1}(x_1) \cdots e_{i_n}(x_n)$, where $e_j(x)$ is the $j$-th
eigenfunction of the one dimensional Schr\"odinger operator
$-(1/2)(d^2/dx^2)+\la x\ra^{m}$, and by using the well
known result on the asymptotic behavior as $j\to \infty$ of $e_j(x)$ for
$x$ in a compact set.


The method for proving \refth(impr) produces the following
Strichartz type inequality as a byproduct which we believe, however,
is much weaker than best possible.
\bgth Let $V$ satisfy {\rm \refass(A)}. Let $T>0$ and let
$p\in [2,\infty)$, $\th\in (2,\infty]$ be such that
$ 0 \leq {\ds \frac{2}{\th}= n\left(\frac1{2}-\frac1{p}\right)}<1$.
Then, for any ${\ds \gamma>\frac1{\th}\left(\frac1{2}-\frac1{m}\right)}$
there exists a constant $C>0$ such that
\bq
\left(\int^T_{-T} \|e^{-itH}u_0\|^\th_p dt \right)^{\frac1{\th}}
\leq C \| \la H \ra^{\gamma}u_0 \|,
\quad u_0 \in L^2(\R^n).
\lbeq(St-0)
\eq
\lbth(Str)
\edth

We shall first review the proof of \refth(impr) and \refth(Str) for the 
case
$V=0$ and the case $V$ is at most quadratic at infinity, in which case
\refth(impr)
and \refth(Str) hold with $m=2$ and $\gamma=0$ respectively, and explain
what are the
physical contents of these estimates. We then outline the proofs of
\refth(impr) and \refth(Str)
for superquadratic potentials following the idea which is suggested by the
physical
meaning of these estimates.

\b

\n{\bf 68 \h K. Yoshitomi: Band spectrum of the Laplacian on a slab with the Dirichlet 
boundary condition on a grid}

\b
\n We study the band spectrum of the Laplace operator on the slab $\Lambda={\bf R}\times ]-1/2,1/2[$ with the Dirichlet boundary condition on the grid $\partial\Lambda\cup (a{\bf Z}\times (]-1/2,1/2[\backslash [-\epsilon,\epsilon]))$. We obtain the asymptotic form of the first band function as $\epsilon$ tends to zero. We further investigate the Laplace operator on ${\bf R}^{2}$ with the Dirichlet boundary condition on a hexagonal grid.

\b

\n{\bf 69 \h G. Zhislin: Peculiarities of the structure of the spectrum of many particle hamiltonians of  charged systems in a homogeneous magnetic field}

\b 
 \n Let $H$ be the hamiltonian of the charged system $Z_1$ of $n$ quantum particles with finite masses in a homogeneous magnetic field with the direction of $z$-axes; $H$ is written after the separation of center-of-mass motion (c.m.m.) of the system in the direction of the field. Since the separation of c.m.m. in $(x,y)$-plane is impossible, in order to study the spectrum of the relative motion $Z_1$ Avron, Herbst and Simon suggested to study the spectrum of the restriction $H(P)$ of the operator $H$ of one component of pseudomomentum $Z_1$ eigenspace, corresponding to arbitrary fixed eigenvalue $P$ of this component.
They discovered the essential spectrum $s_e\Big(H(P)\Big)$ of $H(P)$, but the structure of the discrete spectrum $s_d\Big(H(P)\Big)$ of the operator $H(P)$ was not studied and we knew nothing on the spectrum $s(H)$ of the operator $H$.
 In this talk we prove the following assertions: 

\begin{enumerate}
\item
 For any positive atomic ions discrete spectrum $s_d\Big(H(P)\Big)$   is infinite.
\item
 Discrete spectrum $s_d\Big(H(P)\Big)$ may be infinite even for the systems $Z_1$
   with short-range interactions between the particles.
\item
 The spectrum $s\Big(H(P)\Big)$ of the operator $H(P)$ does not depend on $P$ 
\item 
 The spectrum $(H)$ coincides with the spectrum $s\Big(H(P)\Big)$, 
  but $s(H)$ has an additional infinite degeneration in comparing with 
   $s\Big(H(P)\Big)$; so for any positive atomic ions $s(H)$ consists of the
   infinite number of infinitely degenerated isolated eigenvalues, which 
   go to some point $b$ and of the all points of half-line $[b,+\infty)$ 
\end{enumerate}

\noindent {\bf Remark}. Assertion 2 is not connected with Efimov effect. It is based on the infinite degeneration of lower bound of $s_e\Big(H(P)\Big)$ as the eigenvalue of the hamiltonian $H(Z_2;P)$, corresponding to some decomposition $Z_2$ of $Z_1$ into 2 non-interacting clusters.

\b



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