Abstracts of Talks
Workshop on Spectral and Inverse Spectral Problems for
Schrödinger Operators
Goa, India
14-20 December 2000

***********************

Resonances for PDE

Walter Craig
McMaster University
Hamilton, Ontario, Canada

The talk discusses the Lyaponuv Center Theorem (1907), and its recent version (1973, 1976), and an analogy with the KAM theory for resonant tori for both finite degree of freedom dynamical systems, and for Hamiltonian PDE. Classically, resonances are problematic in KAM theory, and near-resonances give rise to the small small-divisor phenomena. I described the applications of the ideas of Fröhlich and Spencer to the analysis of this class of problems.

Scattering by Rotating Bodies

Völker Enss
Aachen, Germany
joint work with Vadim Kostrykin and Robert Schrader

For the time-evolution generated by the time-dependent Hamiltonian $H(t) = H_0 + V ( R^{-1} ( \omega t) x)$, with rotating potential with sufficiently fast decay, we prove asymptotic completeness and asymptotic conservation of $H_0 - \omega L$ under scattering.

In addition, the kinetic energy remains bounded so that the tools of geometric scattering theory are applicable. We study the transfer of energy from the rotating body to the scattered particles for various configurations yielding an effective acceleration of the particles in a thin gas.

Commutation, the Darboux Transform, and
Nonlinear Evolution Equations

H. Bhate
joint work with M. V. Prabhakar

If $A$ and $B$ are bounded operators on a Hilbert space, $\sigma (A B) \setminus \{ 0 \} = \sigma ( B A ) \setminus \{ 0 \}$. This was extended by Deift to the case when $A$ is a densely defined, closed operator on $H$, with $B = A ^*$, and then applied to the Strum Liouville operator $- {\displaystyle \frac{ d^2 }{ d x^2 }} + q (x)$, and to the KdV equation $u_t - u_{ x x x } + 6 u u_x = 0$.

We give an extension of this idea of factorization and commutation to a large class of nonlinear evolution equations. The corresponding eigenvalue problems include the AKNS system for $n \times n$ matrices, the Boussinesque equation, the KP equations, $(2 + 1)$-dimensional Davey Stewartson equations, and discrete analogues, including the 2-dimensional Toda Lattice.

Spectral Approximation by
Truncations and some related Problems

M. N. N. Namboodiri
Cochin, India

A linear algebraic method for analysing the spectra of bounded self-adjoint operators on separable Hilbert spaces is presented. There are some improved version of some results of A. Böttcher, A. V. Chitra, and the author himself. The self-adjoint operators are assumed to come from the well-known Arveson class. It is also observed that there is some hope in predicting gaps in the essential spectrum.

$\boldmath\mbox{$L^p$}$ Estimates on the Spectral Shift Function
and the Wegner Estimate

J. M. Combes
Toulon, and Marseille, France

The connection between the spectral shift function and the local density of states is established with the use of the Birman-Solomyak formula. We develop the $L^p$ theory of the spectral shift function applicable to pairs of self-adjoint operators whose difference is in the trace ideal ${\cal I}_p$ for $0 < p \leq 1$. Then we show how these $L^p$ estimates lead to so-called Wegner estimates on the density of states with the correct volume dependence; they also imply local Hölder continuity of the integrated density of states. Finally we describe some models of random operators to which these estimates apply. This is a joint work with P. Hislop and S. Nakamura.

Scattering Theory for the Dirac Operator
with a Long-Range Electromagnetic Potential

D. Yafaev
Rennes, France

We consider the Dirac operator with a long-range potential $V(x)$. Scalar, pseudo-scalar, and vector components of $V(x)$ may have arbitrary power-like decay at infinity. We introduce wave operators whose symbols are, roughly speaking, constructed in terms of approximate eigenfunctions of the stationary problem. We derive and solve eikonal and transport equations for the corresponding phase and amplitude functions. >From the analytical point of view, our proof of the existence and completeness of the wave operators relies on the limiting absorption principle and radiation estimates established in the paper.

Lifshitz Tail for Schödinger Operator
with Random Magnetic Field

Shu Nakamura
Graduate School of Mathematical Sciences
University of Tokyo
Tokyo, Japan

In this talk, we discuss the Lifshitz singularity of the integrated density of states for Schrödinger operators with random magnetic field. We consider 2-cases:

(1) Discrete Schrödinger operator on $Z \hspace{-.17cm} Z^2$ with Anderson type random magnetic field;

(2) Schrödinger operator on $I \hspace{-.17cm} R ^d   ( d \geq 2 )$ with metrically transitive random magnetic field.

In both cases, we show the usual Lifshitz singularity holds, in the same manner as for the Schrödinger operator with random potentials. (Here we consider Schrödinger operator without scalar potentials.) The key estimates are local energy estimates, or lower bounds of the Hamiltonian. The formulation and the proof of the local energy estimates are discussed in detail.

Recent Results in the Theory
of One-dimensional Random Operators

Günter Stolz
Birmingham, AL, USA
joint work with D. Buschmann, D. Damanik, and R. Sims

We discuss random Schrödinger operators in $d = 1$ of the type

$$H_\omega = - \frac{ d^2 }{ d x^2 } + V_{ per } (x) + \sum_{ ... ...hspace{-.17cm} Z } \: q_n (\omega) f ( x - n - d_n (\omega)) .$$

Here it is assumed that $V_{ per }$ is a 1-periodic background potential and that adjacent single site potentials $f ( \cdot - n - d_n )$ do not overlap. We are particularly interested in proofs of localization for the following situations:

(i) $q_n \equiv 1$, i.e. $H_\omega$ represents a (non-monotonic) displacement model;

(ii) $d_n \equiv 0$, and $q_n$ has singular distribution, in particular, the Bernoulli-Anderson model where $q_n$ assumes only two values.

Using the fact that the reflection coefficient of the single site potential $f \neq 0$ cannot vanish identically (in energy), we obtain exponential and dynamical localization at all energies for both models. In the case of dynamic localization a discrete set of energies at which the Lyapunov exponent vanishes has to be excluded. Similar methods allow us to prove localization for one-dimensional Poisson models and random operators of the type

$$- \frac{ d }{ d x } \: a_\omega (x) \frac{ d }{ d x } .$$

Spectral Theory for Sparse Random Potentials

Werner Kirsch
Bochum, Germany
joint work with Dirk Hundertmark, based on joint work
with M. Krishna and Jörg Obermeit

While the theory works in some generality, let us consider a specific example of sparse random potentials:

$$V_\omega (x) = \sum_{ i \in Z \hspace{-.17cm} Z ^d } \: \xi_i (\omega ) q_i (\omega) f ( x - i )$$

where $f \leq 0 ,   q_i$ and $\xi _i$ are independent random variables, $q_i$ are $iid$, and $\xi _i \in \{ 0,1 \}$, with $p_i = P ( \xi _i = 1 )$. So, if $p_i \rightarrow 0$ as $\vert i \vert \rightarrow \infty$ then the points $i$ where $V_\omega ( i ) \neq 0$ become more and more sparse as $\vert i \vert$ grows. We prove that $\sigma _{ a c } ( H_0 + V_\omega ) \supset [ 0 , \infty )$ if $p_i \leq C ( 1 + \vert i \vert )^{ - 2 - \epsilon }$. Moreover, under weak assumptions on the distribution of the $q_i$, the spectrum of $H_0 + V_\omega$ below zero is pure point. In particular, there is essential spectrum below zero iff ${\displaystyle \sum_{ i \in Z \hspace{-.17cm} Z ^d }} \: p_i = \infty$.

Homogenization of a Parabolic Equation

M. Rajesh
Indian Institute of Sciences
Bangalore, India
joint work with A. K. Nandakumar

We study the homogenization of the non-linear parabolic equation

$$\partial_t b \left( \frac{ x }{ \epsilon } , u_{ \epsilon } \... ...silon } , u_{ \epsilon } , \nabla u_{ \epsilon } \right) = f ,$$

as $\epsilon \rightarrow 0$, in a bounded domain $\Omega \subset I \hspace{-.17cm} R ^n$ assuming Dirichlet boundary conditions and with prescribed initial data, $u_0$. The functions $b ( \cdot ,   )$ and $a ( \cdot ,   ,   )$ are assumed to be $\epsilon$-periodic, and to satisfy suitable monotonicity conditions. The homogenized equation is obtained and we obtain corrector results to improve the weak convergence of $\nabla u_{ \epsilon }$ to strong convergence.

Probabilistic Approach to the Closure
of the Reflectionless Potentials

Shinichi Kotani
Osaka University, Japan

An ordinary reflectionless potential $q$ is given by

$$q (x) = - 2 \frac{ d^2 }{ d x^2 } \: \log \: \mbox{det} \: ( I + A (x) ) ,    A (x) = ( a_{ i j } (x) )$$

$$a_{ i j } (x) = \frac{ \sqrt{ w_i \cdot w_j }}{ \xi _i + \xi _j } e^{ - ( \xi _i + \xi _j ) x } ,     1 \leq i,j \leq n .$$

We introduce, for $\lambda _0 > 0$, the set

$$\Omega ( [ - \lambda _0 , 0 ] )$$

$$= \left\{ q :   q   \mbox{is a limit of reflectionless potentials such that } \; \mbox{sp} \; (L(q)) \subset [ - \lambda _0 , \infty ) \right\} ,$$ (1)

where $L (q) = - \frac{ d^2 }{ d x^2 } + q$. The limit means uniform limit on each compact set of $I \hspace{-.17cm} R$. Theorem (Marchenko-Lundina).

$$\Omega ( [ - \lambda_0 , 0 ] ) \cong \Sigma \: ( [ - \lambda_0 , 0 ] ),$$

where

$$\Sigma ( [ - \lambda_0 , 0 ] ) =$$

$$\{ ( \sigma _+ , \sigma_- )   :    \sigma_\pm    \mbox{are m... ...ma _+ + \sigma _- ) ( d \xi ) }{ 2 \sqrt{ - \xi }} \leq 1 . \}$$

By using the Sate-Segal-Wilson theorem one can construct KdV-flow on $\Omega ( [ - \lambda_0 , 0 ] )$. For simplicity set $\lambda _0 = 1$, and $H_+ = \{ f :   f   \mbox{holomorphic on}   \{ \vert z \vert \leq 1 \}   \mbox{... ...style \int_0^{ 2 \pi }} \vert f ( e^{ i \theta } ) \vert^2 d \theta < \infty \}$, the Hardy space on $\{ \vert x \vert = 1 \}$. Let $g (z)$ be an entire function $g (z) \neq 0 , z \in C$. Choose $f \in H_+$, define $F ( z^2 ) = \frac{ f ( z ) - f ( - z ) }{ 2 z }$. Solve $B \in H_+$

$$\begin{array}{r} B (z) - \int_{ 0 }^1 \: \frac{ B ( \xi ) - B... ...{ - 2 \zeta } ,   \zeta < 0 . \end{array} \right. \end{array}$$

Set

$$K_g f (z) = \int_{ 1 }^{ 1 } \: \frac{ 1 }{ \xi - z } \left ... ...( \xi ) }{ q (z) } - 1 \right ) B ( \xi^2 ) \sigma ( d \xi ) .$$

Then for $g_{ t , x } (z) = e^{ z x + 4 z^3 t }$

$$u ( t , x ) = - 2 \frac{ d^2 }{ d x^2 } \: \log \: \mbox{det} \: ( I + K_{ g_{ t , x } } )$$

solves KdV equation. And this defines a KdV-flow on $\Omega ( [ - 1 ,,0 , 0 ] )$.

Smoothing Property for Schrödinger Equations
with Potential Superquadratic at Infinity

Kenji Yajima
University of Tokyo, Japan
joint work with Guoping Zhang

We consider the initial value problem for the Schrödinger equation

$$(1)    \frac{ \partial u }{ \partial t } = - \frac{ 1 }{ 2 }... ..., x ) = \psi (x)    \mbox{in}   L^2 ( I \hspace{-.17cm} R^n ).$$

The smoothing property is the property that for all initial condition $\psi \in L^2 ( I \hspace{-.17cm} R ^n )$, the solution $u ( t , x )$ of (1) is smoother than $\psi (x)$, for a.e. $t \in I \hspace{-.17cm} R$. This has been proved for many dispensive equations with constant coefficients or with coefficients that approach constants at spatial infinity. For Schrödinger equations it can be stated in the form

$$(2)    \Vert e^{ - i t H _0 } u_0 \Vert _{ L^{ \theta } ( I ... ...hspace{-.17cm} R _x ^n ) ) } \leq C \Vert u_0 \Vert _{ L^2 } ,$$

for $0 < \frac{ 2 }{ \theta } = \left ( \frac{ 1 }{ 2 } - \frac{ 1 }{ p } \right ) < 1$, or

$$(3)    \Vert \Phi ( 1 - \Delta )^{ \alpha / 2 } e^{ - i t H_... ....17cm} R_x^n ) ) } \leq c \vert\vert u_0 \vert\vert _{ L^2 } ,$$

with $0 < \frac{ 2 }{ \theta } = 2 \alpha + n \left ( \frac{ 1 }{ 2 } - \frac{ 1 }{ p } \right ) < 1$ , where $H_0 = - \frac{ 1 }{ 2 } \Delta$. The property (2) and (3) are extended to the case $H = - \frac{ 1 }{ 2 } \Delta + V$, with $V (x) = {\cal O} ( \Vert x\Vert^2 )$ as $\Vert x\Vert \rightarrow \infty$. The purpose of this talk is to extend this property when $V (x) \geq C \Vert x \Vert^{ 2 + \epsilon }$ in which case the propagator $e^{ - i H t }$ has integral kernel which is nowhere smooth, and give some applications to non-linear Schrödinger equations.

On the Density of States for the
Periodic Schrödinger Operator

Y. Karpeshina
Birmingham, AL, USA

An asymptotic formula for the density of states of the operator $( - \Delta )^l + V$ in $R^n ,    n \geq 2 ,   l \geq \frac{ 1 }{ 2 }$ is obtained. Special consideration is given to the case of the Schrödinger equation $n = 3 ,   l = 1$, where the second term of the asymptotics is found.

Smoothness of the Density of States for
Random Decaying Interaction

M. Krishna
Chennai, India

We consider the random decaying model

$$H^\omega = \Delta + V^\omega ,$$

with $V_\omega (n) = a_n q^\omega (n)$, where $q^\omega (n)$ are independent, identically distributed random variables with probability measure $\mu$, and $a_n$ either growing or decaying. We consider the average total spectral measure $\sigma$ associated with $H^\omega$ and show that it is absolutely continuous with differentiable density whenever $a_n$, and $\mu$ satisfy some conditions with respect to each other.

The Wegner Estimate, Integrated

Density of States and Localization

for Nonsign Definite Potentials

P. D. Hislop
Lexington, KY, USA
joint work with F. Klopp

We prove a good Wegner estimate for various types of random operators with no sign condition on the single-site potential. The proof yield a volume dependence like $\vert \Lambda \vert$ so a consequence is the Hölder continuity of the integrated density of states at energies in the unperturbed spectral gap. The method combines a vector field construction due to Klopp and the $L^p$-estimate on the spectral shift function due to Combes, Hislop, and Nakamura. Under the condition that the negative part of the single site potential is suitably small, we prove band edge localization.

Long-Range Scattering of
Three-Body Quantum Systems

Erik Skibsted
Aarhus, Denmark

We give an account of some very recent results on asymptotic completeness (AC) for three-body systems with pair potentials obeying

$$- C \vert x^a \vert^{ - \mu } \leq V^a ( x^a ) \leq - C \vert x^a \vert^{ - \mu } ;    \vert x^a \vert \geq R ,$$

with $\mu \in ( 0 , \sqrt{ 3 } - 1 ]$. For $\mbox{dim} \: X^a = 1 ,    A C$ holds under the additional concavity assumption

$${V^{ a }} '' ( x^a ) \leq 0 ;    \vert x^a \vert \geq R .$$

The proof is naturally divided into the two cases $\mu \in \left ( 0 , \frac{ 1 }{ 2 } \right ]$ and $\mu \in \left. \left ( \frac{ 1 }{ 2 } , \sqrt{ 3 } - 1 \right. \right ]$. In higher dimension $A C$ holds for $\mu \in \left ( \frac{ 1 }{ 2 } , \sqrt{ 3 } - 1 \right ]$ under the additional assumption of spherical symmetry.

Internal Lifshitz Tails for Random Schrödinger Operators

F. Klopp
Villetaneuse, France

The material presented in this talk was mainly obtained on joint work with S. Wolff to whose memory this talk is dedicated.

We consider the continuous random Anderson model in $I \hspace{-.17cm} R ^d :   H = H_0 + V_\omega$, where $H_0 = - \Delta + W$, $W$ being $Z \hspace{-.17cm} Z^d$ periodic and $V_\omega ( \cdot ) = {\displaystyle \sum_{ \gamma \in Z \hspace{-.17cm} Z ^d }} \omega_{ \gamma } V ( \cdot - \gamma ) ,   ( \omega_{ \gamma } )_{ \gamma }$ independent, identically distributed random variables, and $V \geq 0    V \neq 0$ continuous, with compact support. Let us assume that $\mbox{supp} \: \omega_0 = [ 0 , \omega^+ ]$. Let $\sigma$ be the spectrum of $H_0$ and $\Sigma$ be the a.s. spectrum of $H_\omega$. Let $E_0$ be the upper edge of an open gap of $\Sigma$. Then, we know that the spectra are locally in the following relative position: the spectra $\sigma$ and $\Sigma$ have the same lower edge $E_0$ and to the left of that there is a gap in both spectra. Let $N (E)$ be the integrated density states of $H_\omega$. Then we prove

Theorem 1.

$$\limsup_{E \rightarrow E_0^+} \; {\displaystyle \frac{ \log ... ...og \: ( N (E) - N ( E_0 )) \vert }{ \log \: ( E - E_0 ) }} < 0$$

If we moreover assume that the edge $E_0$ is simple for $\sigma$, we prove

Theorem 2. For $d=2$,

$$\lim_{E \rightarrow E_0^+} \; {\displaystyle \frac{ \log \: \... ...E) - N ( E_0 )) \vert }{ \log \: ( E - E_0 ) }} = - \alpha < 0$$

and we compute $\alpha$ in terms of the Newton polygon associated to the Floquet eigenvalues reaching the value $E_0$ at the points i.e. Floquet parameter where this value is attained.

Schrödinger Operators and Geometry

K. B. Sinha
Indian Statistical Institute, Calcutta, India

In Conne's program of Non-Commutative Geometry, the Dixmier trace $\mbox{Tr}_\omega$ (a kind of non-normal trace on ${\cal B} ( {\cal H })$ vanishing on trace-class operators) plays an important role. In a compact Riemannian manifold $( M , g )$ of dimension $d$, the asymptotics as $t \rightarrow 0 +$ of the trace of the associated heat semigroup $T_t = \mbox{exp} \: ( t \Delta )$ (where $\Delta$ is the Laplace-Beltrami operator) gives various geometric parameters of the manifold. In fact, $\mbox{vol} \: (M) = {\displaystyle \lim_{ t \rightarrow 0 + }} \: t^{ d/2 } \: \mbox{Tr} \: ( T_t )$, which by a Tauberian theorem, equals $\mbox{Tr}_\omega [ ( - \hat{ \Delta } )^{ - 1 } P ]$, where $P = \mbox{Projection on}   {\cal N} ( \Delta ) ^{ \perp }$ and $\hat{ \Delta } = \Delta \vert { P {\cal H} }$. It turns out that the volume form on $M$, $v(f)$, is given by $v (f) = \mbox{Tr}_\omega [ f ( - \Delta + \epsilon )^{-d} ]$, for $\epsilon > 0$ and $f \in C (M)$. One establishes an identical formula for $I \hspace{-.17cm} R^{ 2 d }$, a non-compact flat manifold, with $\Delta$ the $2 d$-dim Laplacian, $f \in C_c^{ \infty } ( I \hspace{-.17cm} R^{ 2 d } )$. A more symmetric form of this is that $\mbox{Tr}_\omega [ \bar{f} ( - \Delta + \epsilon )^{ - 1 } f ] = {\displaystyle \int } \: \vert f(x) \vert^2 \: d x$. This implies that $\mu _n$, the eigenvalue of the operator $[ \bar{ f } ( - \Delta + \epsilon ) ^{-1} f ]$, is ${\cal O} ( n ^{-1} )$, and then leads to a connection with a family of Schrödinger operators $- \Delta - \mu _n ^{-1} \vert f \vert^2$.

Embedded Eigenvalues of Laplacians on Riemann Surfaces

E. Balslev
Aarhus, Denmark
joint work with A. Venkov

We consider the Laplacian $L_N = - y^2 \left ( {\displaystyle \frac{ \partial ^2 }{ \partial x^2 }} + {\displaystyle \frac{ \partial ^2 }{ \partial y^2 }} \right )$ on $\Gamma_0 (N)$-automorphic functions $f$ with primitive character $\chi$. Here $N = 4 N_1 ,   N_1 \equiv 2 \; \mbox{mod} \; 4$ or $N_1 \equiv 3 \; \mbox{mod} \; 4$, and $N_1 = p \cdots p_k$ is a product of different primes. The group $\Gamma_0 (N) = \left \{ \left ( \begin{array}{cc} a & b \ c & d \end{array} \... ...  c = N c_1 ,   a,b,c_1,d \in Z \hspace{-.17cm} Z ,    a d - b c = 1 \right \}$. We give the results for $\Gamma_0 (8)$, other values of $N$ are similar. The character $\chi ( \gamma ) = \chi_D (d)$, where $\chi_D (d) = 1$, for $d \equiv \pm 1 \: \mbox{mod} \: 8 ,   \chi_D (d) = - 1$, for $d \equiv \pm 3 \: \mbox{mod} \: 8$, and the function $f \in D ( L_N )$ if

(i) $f ( \gamma z ) = \chi ( \gamma ) f (z) ,   \gamma \in \Gamma_0(8) ,   z \in F_8$, the fundamental domain of $\Gamma_0 (8)$.
(ii) $f ,   L_N f \in L^2 ( F_8 ; d \mu ) ,   d \mu = y^{-2} d x d y$.

The operator $L_N$ is self-adjoint on $D ( L_N )$, and $L_N$ has an infinite sequence of embedded eigenvalues $\frac{1}{4} \leq \lambda _1 < \lambda _2 < \cdots < \lambda _n < \cdots$, satisfying a Weyl law. The perturbation of $L_N$ is defined by the group of characters

$$\chi_{ \alpha } ( \gamma ) = e^{ 2 \pi i \alpha \: \mbox{\tin... ...\displaystyle \int_{ z_0 }^{ \gamma z_0 }} \omega (t) \: d t }$$

where $\omega$ is the holomorphic modular form of weight $2$, for $\Gamma_0 (N)$ defined by

$$\omega (z) = P (z) - 7 P ( 2 z ) + 14 P ( 4z ) - 8 P ( 8 z ),$$

and

$$P (z) = 1 - 24 \: \sum_{ n = 1 }^{ \infty } \: \sigma (n) e^{ 2 \pi i nz },$$

where $E_2 (z) = P ( z ) + \frac{ 3 }{ \pi y }$ being the fundamental modular form of weight $2$.

Then $L_\chi f = - y^2 \left ( {\displaystyle \frac{ \partial^2 }{ \partial x^2 }} + {\displaystyle \frac{ \partial^2 }{ \partial y^2 }} \right ) f$, for $f ( \gamma z ) = \chi (\gamma) \chi_{ \alpha } ( \gamma ) f (z)$.

The unitary transformation $f \rightarrow \Omega_{ \alpha } f ,    \Omega _{ \alpha } (z) = e^{ 2 \pi i \alpha \: \mbox{\tiny Re} \: {\displaystyle \int_{ z_0 }^2 \omega (t) \: d t } }$ brings $L_\chi$ to the form $L (\chi) = L_N + \alpha M + \alpha ^2 N$ on $\Gamma_0 (N)$-automorphic functions where $M = - 4 \pi i y^2 \left ( \omega_1 {\displaystyle \frac{ \partial }{ \partial ... ...\right ) ,    N = \omega_1^2 + \omega_2^2 ,    \omega = \omega_1 + i \omega_2$.

Theorem. For every odd eigenfunction $v$ of $L_N$, which is also a common eigenfunction of all Hecke operators, with eigenvalue $\lambda > \frac{ 1 }{ 4 } ,    {\displaystyle \int_{ F_8 }} \: ( M v ) \bar{E}_{ \infty } \: d \mu ( z ) \neq 0$, where $E_{ \infty }$ is the Eisenstein series for the cusp at infinity. From every eigenspace contain ing odd eigenfunctions with eigenvalues $> 1/4$, at least one eigenfunction $v$ of $L \left ( \frac{ 1 }{2} \right )$ becomes a resonance function of $L (\chi)$.

Partial Differential Operators without the
Singularly Continuous Spectrum

M. Demuth
Clausthal, Germany

Let $H_1$ be a self-adjoint operator, bounded below, acting in $L^2 ( I \hspace{-.17cm} R ^d )$, given by a smooth increasing function of the Laplacian. Let $H_2$ be a further, self-adjoint, bounded below operator. Assume that

$$( e^{ - H_2 } - e^{ - H_1 } ) M_{ \gamma } \in {\cal B} ( L^2 ( I \hspace{-.17cm} R ^d ) ) ,    \gamma > 1 ,$$

where $M_{ \gamma }$ is the multiplication operator with the factor $( 1 + \vert x \vert^2 )^{ \gamma / 2 }$. Moreover, it is assumed that $( H_2 + a ) ^{-1} - ( H_1 + a ) ^{-1}$ is compact, for some $a$.

Under these conditions, the singular continuous spectrum of $H_2$ is empty.

This abstract result is applied to generators of $\alpha$-stable processes, $H_1 = ( - \Delta )^{ \alpha } ,    \alpha \in ( 0 , 1 )$. Potential perturbations and perturbations by imposing boundary conditions are considered.

This work was due in collaboration with K. B. Sinha (Delhi), M. Bovo (Clausthal), E. Gieve (Clausthal).

On the Norm Convergence of the Trotter-Kato
Product Formula with Error Bound

Takashi Ichinose

Department of Mathematics

Kanazawa University

Kanazawa, 920-1192, Japan

The norm convergence of the Trotter-Kato product formula with optimal error bound is shown for the self-adjoint semigroup generated by that operator sum of two nonnegative self-adjoint operators which is self-adjoint.

Namely, let $A$ and $B$ be nonnegative self-adjoint operators in a Hilbert space. Assume that the sum $C : = A + B$ is self-adjoint on $D [C] = D [A] \cap D [B]$. Then it holds that

$$\Vert \left( e^{ - \frac{ t B }{ 2 n } } \: e^{ - \frac{ t A... ...\right)^n - e^{ - t c } \Vert = \mbox{\large$O$} ( n ^{-1} ) ,$$

$$\Vert \left( e^{ - \frac{tA}{n} } e^{ - \frac{ t B }{ n } }... ...ert = \mbox{\large$O$} ( n ^{-1} ) ,    n \rightarrow \infty .$$

The convergence is uniform on each compact $t$-interval in the closed half-line $[ 0 , \infty )$, and further, if $C$ is strictly positive, uniform on the whole closed half line $[ 0 , \infty )$.

This is based on very recent works done jointly, with Hideo Tamura (Okayama) on one hand, and with Hideo Tamura, Hiroshi Tamuya (Kanazawa), and Valentin Zaghebnov (Marseille) on the other hand.

Universal Eigenvalue Inequalities for
Domains in Riemannian Manifolds

Mark Ashbaugh
University of Missouri
Columbia, MO 65211 USA

In 1955 and 1956 Payne, Pólya, and Weinberger presented universal inequalities for the eigenvalues of the Dirichlet Laplacian on bounded domanins in Euclidean space. These bounds are ``universal'' in the sense that they hold for an essentially arbitrary domain, with not even geometric data (aside from the dimension of space, $n$) needing to be supplied. The prototypical such PPW inequality is the bound

$$\lambda _{ m + 1 } - \lambda _m \leq \frac{ 4 }{ n m } \: \sum_{ i = 1 }^m \: \lambda _i ,$$

which applies for any bounded domain $\Omega$ in $I \hspace{-.17cm} R ^n$. This bound was subsequently improved and refined by various authors, most notably by Hile and Protter (1980) and by H. C. Yang (1991 preprint; revised significantly in 1995, but seemingly not published as yet). Beginning with S.-Y. Cheng (1975) various extensions of this bound to domains in certain Riemannian manifolds, and in particular spaces of constant curvature, have been made. We present the latest refinements of these bounds, with special emphasis on the forms obtained for domains in the sphere $S^n$ and on how these modified forms accomodate the behavior of the eigenvalues as the domain approaches all of $S^n$. This discussion is especially relevant for the low eigenvalues.

Finally, we discuss refined forms of the basic universal inequalities that apply to bounded domains in hyperbolic space $I \hspace{-.17cm} H^n$. Again, these are especially relevant for low eigenvalues: in particular, we have the result $\lambda _2 / \lambda _1 < 5$ for all bounded domains in $I \hspace{-.17cm} H^n$, improving (in two ways) upon the result $\lambda _2 / \lambda _1 \leq 17$ of Harrell and Michel for bounded domains in $I \hspace{-.17cm} H^2$. The bound $\lambda _2 / \lambda _1 < 5$ derives from an even stronger and more general inequality which we present. One of the interesting features of the stronger bound is that it is in a certain sense sharp for domains which are very large (think of geodesic balls with radius going to infinity), while continuing to be quite good even down to very small domains (where the the Euclidean behavior should predominate).

The Extraordinary Spectral Properties of Radially Periodic Schrödinger Operators

Andreas M. Hinz

Since it became clear in [1] that the band structure of the spectrum of periodic Sturm-Liouville operators $t = - \frac{ d^2 }{ d r^2 } + q(r)$ does not survive a spherically symmetric extension to Schrödinger operators $T = - \Delta + V$ with $V(x) = q ( \vert x \vert )$ for $x \in I \hspace{-.17cm} R ^d ,   d \in I \hspace{-.17cm} N \setminus \{ 1 \}$, a lot of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum $[ \mu_0 , \infty ]$ of $T$ with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, proved in [2] by spherical separation into Sturm-Liouville operators $t_c = t + \frac{ c }{ r^2 }$. Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues of $T$ more closely. A Welsh eigenvalue was discovered below the essential spectrum in the case $d=2$ (cf. [3]), and it turned out that there are in fact infinitely many, accumulating at $\mu_0$. Moreover, a method based on oscillation theory made it possible to count eigenvalues of $t_c$ contributing to an interval of dense point spectrum of $T$ (cf. [4]). We gained evidence that an asymptotic formula, valid for $c \rightarrow \infty$, does in fact produce correct numbers even for small values of the coupling constant, such that a rather precise picture of the spectrum of radially periodic Schrödinger operators has now been obtained.

References

[1] Hempel, R., Hinz, A. M., Kalf, H., On the Essential Spectrum of Schrödinger Operators with Spherically Symmetric Potentials, Math. Ann. 277 (1987), 197-208.

[2] Hempel, R., Herbst, I., Hinz, A. M., Kalf, H., Intervals of dense point spectrum for spherically symmetric Schrödinger operators of the type $- \Delta + \cos \vert x\vert$, J. London Math. Soc. (2) 43 (1991), 295-304.

[3] Brown, B. M., Eastham, M. S. P., Hinz, A. M., Kriecherbauer, T., McCormack, D. K. R., Schmidt, K. M., Welsh Eigenvalues of Radially Periodic Schrödinger Operators, J. Math. Anal. Appl. 225 (1998), 347-357.

[4] Brown, B. M., Eastham, M. S. P., Hinz, A. M., Schmidt, K. M., Distribution of eigenvalues in gaps of the essential spectrum of Sturm-Liouville operators - a numerical approach, submitted to J. Comput. Anal. Appl.

Spectral Problems in Superconductivity:
Magnetic Bottles

Bernard Helffer
Univerité Paris-Sud, FRANCE

When analyzing minimizers of the Ginzburg-Landau functional associated to a superconductive material submitted to an intense exterior magnetic field, we meet naturally the problem of analyzing the ground state and the corresponding ground state energy of the Neumann realization in an open set $\Omega$ of the Schrödinger operator with magnetic field $P_{ h , A } = \sum_{ j } \: ( h \partial_{ \partial x_j } - i A_j )^2$. We here consider the semi-classical problem $( h \rightarrow 0 )$ and we analyze two connected questions

(1) Find an expansion in powers of $h^{ 1/2 }$ of the lowest eigenvalue $P^{ (1) } (h)$ (with accurate remainder estimate)

(2) Localize the ground eigenfunction as $h \rightarrow 0$.

The results presented by the author are obtained in collaboration with A. Morame and extend previous works by us, by Baumann-Phillips-Tang, Berkoff-Sternberg, Del Pino-Föllmer-Sternberg, Lu-Pan.

In dimension 2, the main result is obtained in the case when the magnetic field is constant. We show that

$$P^{ (1) } (h) = h \Theta_0 \: \vert B \vert - C_1 h^{ 3/2 } K_{ \mbox{\tiny max}} + \mbox{\large$0$} ( h^{ 7/4 } )$$

where $\Theta_0 \in ] 0 , 1 [ , C_1 > 0$, and $K_{ \mbox{\tiny max}}$ is the maximum of the curvature. We obtain that the ground state is exponentially localized inside $\partial \Omega$ and near the points of maximal curvature showing that the ground state is, in the case of a constant magnetic field $\vec{B} = ( b_{ 2 3 } , b_{ i 3 } , b_{ i 2 } ) = \mbox{curl} \: A$, exponentially localized as $h \rightarrow 0$ at the points of the boundary where $\vec{B}$ is parallel to the tangent space.

This last result is based on the following ``exercise'': Analyze as a function of $\theta \in ] 0, \pi / 2[$, the bottom of the spectrum of the Neumann realization in $I \hspace{-.17cm} R _+^2$ of the Schrödinger operator:

$$D_{ x_1 }^2 + D_{ x_2 }^2 + ( x_1 \: \mbox{cos} \: \theta + x_2 \: \mbox{sin} \: \theta )^2.$$

Perturbation of Eigenvalues Embedded at a Threshold

Arne Jensen

Aalborg University, Denmark

and University of Tokyo, Japan

The work presented is joint work with Michael Melgaard. In many cases the spectrum of a Schrödinger operator has the following structure

$$\begin{array}{l} \hspace{.65in} \mbox{\small thresholds}   \\... ...all eigenvalues} \hspace{.62in} \sigma_{ a c } (H) \end{array}$$

Eigenvalues in the discrete case and embedded in $\sigma _{ a c } ( H )$ have been studied extensively. Here we present results on perturbations of eigenvalues embedded at a threshold. We consider the following model problem: ${\cal H} = {\cal H}_a \oplus {\cal H}_b,    H_a , H_b$ self-adjoint on ${\cal H}_a , {\cal H}_b ,    V_{ a b } = V_{ ba }^* \in {\cal B} ( {\cal H}_b , {\cal H}_a )$ and

$$H (g) = \left ( \begin{array}{cc} H_a & 0 \ 0 & H_b \end{arr... ...array}{cc} 0 & V_{ a b } \ V_{ b a } & 0 \end{array} \right )$$

We have $\sigma ( H(0) ) = \sigma ( H_a ) \cup \sigma ( H_b )$, thus eigenvalues at thresholds are easily obtained. Generally, the results show that an embedded eigenvalue leaves the threshold and becomes a discrete eigenvalue.

A Typical Result:

Assumptions:

(i) $\lambda \in \sigma ( H_a ) ,    {\cal K}_a \mapsto {\cal H}_a$ density and continuously such that

$$R_a (\zeta ) = ( H_a - \zeta )^{-1} = G_0 + i ( \zeta - \lamb... ... - ( \zeta - \lambda ) G_2 + o ( \vert \zeta - \lambda \vert )$$

as $\zeta \rightarrow \lambda$, with $\mbox{Im} \: \zeta > 0$, in ${\cal B} ( {\cal K}_a , {\cal K}_a ^* )$.

(ii) $V_{ a b } \in {\cal B} ( {\cal H}_b , {\cal K}_a )$

(iii) $\lambda$ an isolated eigenvalue of $H_b$ with eigenprojection $P_b$. Assume that $P_b V_{ b a } G_0 V_{ a b } P_b$ is strictly positive and invertible in ${\cal B} ( P_b {\cal H}_b )$.

Results:

There exist $\eta_0 > 0 ,    \delta _0 > 0$, and a function $\delta _l (g)$ satisfying $c g^2 \leq \delta_l (g ) \leq C g^2$, for some $c ,   C > 0$, such that

$$( \lambda - \delta_l( g) , \lambda + \delta _0 ) \cap \sigma_{ p p } ( H(g) ) = \emptyset$$

for all $g$ with $0 < \vert g \vert < \eta_0$.

The proof is based on the Feshbach formula.