Seminar in spectral theory and related topics

Patrick Gerard, Université Paris-Saclay

Spectral theory of first order operators with Toeplitz coefficients on the circle and applications to the Benjamin-Ono equation

I will discuss recent results on the spectral theory of selfadjoint operators of the form on the Hardy space of the circle, being a Toeplitz operator with a real symbol . Then, using a Lax pair structure, I will present applications to the Benjamin-Ono equation, in particular the existence of non smooth solutions which are both periodic in time and space. This is based on a recent joint work with Thomas Kappeler and Peter Topalov.

Sergey Khrushchev, Satbayev University

Uniformly convergent Fourier series with universal power parts on closed subsets of measure zero

TBA

Alexei Poltoratski, University of Wisconsin

TBA

TBA

Luis Silva, Universidad Nacional Autónoma de México

TBA

TBA

Andrew Comech, Texas A&M University and IITP RAS

Virtual levels and virtual states of Schrodinger operators

Virtual levels admit several equivalent characterizations:
(1) there are corresponding eigenstates from L^2 or a space “slightly weaker” than L^2;
(2) there is no limiting absorption principle in the vicinity of a virtual level (e.g. no weights such that the “sandwiched” resolvent remains uniformly bounded);
(3) an arbitrarily small perturbation can produce an eigenvalue.

We study virtual levels in the context of Schrodinger operators, with nonselfadjoint potentials and in all dimensions. In particular, we derive the “missing” limiting absorption principle — the estimates on the resolvent — near the threshold in two dimensions in the case when the threshold is not a virtual level.

This is a joint work with Nabile Boussaid based on the preprint arXiv:2101.11979

Maxim Zinchenko, University of New Mexico

TBA

TBA

Peter Kuchment, Texas A&M University

The nodal mysteries

Nodal patterns of oscillating membranes have been known for hundreds of years. Leonardo da Vinci, Galileo Galilei, and Robert Hooke have observed them. By the nineteenth century they acquired the name of Chladni figures. Mathematically, they represent zero sets of eigenfunctions of the Laplace (or a more general) operator. In spite of such long history, many mysteries about these patterns (even in domains
of Euclidean spaces, and even more on manifolds) still abound and attract recent attention of leading researchers working in physics, mathematics (including PDEs, math physics, and number theory) and even
medical imaging. The talk will survey these issues, with concentration on some recent results. No prior knowledge is assumed.

Leonid Parnovski, University College London

Floating mats and sloping beaches: spectral asymptotics of the Steklov problem on polygons

I will discuss asymptotic behaviour of the eigenvalues of the Steklov problem (aka Dirichlet-to-Neumann operator) on curvilinear polygons. The answer is completely unexpected and depends on the arithmetic properties of the angles of the polygon.

Jari Taskinen, University of Helsinki

Spectra of the Robin-Laplace- and Steklov-problems in bounded, cuspidal domains

It is well-known by works of several authors that the spectrum of the Neumann-Laplace operator may  be non-discrete even in bounded domains, if the boundary of the domain has some irregularities. In the same direction, in a paper in 2008 with S.A. Nazarov we considered the Steklov spectral problem in a bounded domain , , with a peak and showed that the spectrum may be discrete or continuous depending on the sharpness of the peak. Later, we proved that the spectrum of the Robin Laplacian in non-Lipschitz domains may be quite pathological since, in addition to countably many eigenvalues,  the residual spectrum may cover the whole complex plain. 

We have recently complemented this study in two papers, where we consider the spectral Robin-Laplace- and Steklov-problems in a bounded domain with a peak and also in a family of domains blunted at the small distance from the peak tip. The blunted domains are Lipschitz and the spectra of the corresponding problems on are discrete. We  study the  behaviour of the discrete spectra as  and their relations with the spectrum of case with . In particular we find various subfamilies of eigenvalues which behave in different ways (e.g. “blinking” and “stable” families”) and we describe a mechanism how the discrete spectra turn into the continuous one in this process. 

 The work is a co-operation with Sergei A. Nazarov (St. Petersburg) and Nicolas Popoff (Bordeaux).

Peter Yuditskii, Johannes Kepler Universität Linz

Reflectionless canonical systems: almost periodicity and character-automorphic Fourier transforms

We develop a comprehensive theory of reflectionless canonical systems with an arbitrary Dirichlet-regular Widom spectrum with the Direct Cauchy Theorem property. This generalizes, to an infinite gap setting, the constructions of finite gap quasiperiodic (algebro-geometric) solutions of stationary integrable hierarchies. Instead of theta functions on a compact Riemann surface, the construction is based on reproducing kernels of character-automorphic Hardy spaces in Widom domains with respect to Martin measure. We also construct unitary character-automorphic Fourier transforms which generalize the Paley-Wiener theorem. Finally, we find the correct notion of almost periodicity which holds for canonical system parameters in Arov gauge, and we prove generically optimal results for almost periodicity for Potapov-de Branges gauge, and Dirac operators. Based on joint work with Roman Bessonov and Milivoje Lukic.

Christophe Hazard, Institut Polytechnique de Paris

Curiosities about the spectrum of a cavity containing a negative material

In electromagnetism, a negative material is a dispersive material for which the real parts of the electric permittivity and/or the magnetic permeability become negative in some frequency range(s). In the last decades, the extraordinary properties of these materials have generated a great effervescence among the communities of physicists and mathematicians. The aim of this talk is to focus on their spectral properties. Using a simple scalar two-dimensional model, we will show that negative material are responsible for various unusual resonance phenomena which are related to various components of an essential spectrum. This is a common work with Sandrine Paolantoni.

Wencai Liu, Texas A&M University

Irreducibility of the Fermi variety for discrete periodic Schrödinger operators

Let be a discrete periodic Schrödinger operator on :

where is the discrete Laplacian and is periodic. We prove that for any , the Fermi variety at every energy level is irreducible (modulo periodicity). For , we prove that the Fermi variety at every energy level except for the average of the potential is irreducible (modulo periodicity) and the Fermi variety at the average of the potential has at most two irreducible components (modulo periodicity). This is sharp since for and a constant potential , the Fermi variety at -level has exactly two irreducible components (modulo periodicity). In particular, we show that the Bloch variety is irreducible (modulo periodicity) for any .

Svetlana Jitomirskaya, University of California

Spectral properties of the unbounded GPS model

We discuss spectral properties of the unbounded GPS model: a family of discrete 1D Schrodinger operators with unbounded potential and exact mobility edge. Based on papers in progress joint with Xu, You (Nankai) and Zhao (UCI).

Alexander Sobolev, University College London

On spectral properties of the one-particle density matrix

The one-particle density matrix is one of the key objects in the quantum-mechanical approximation schemes. The self-adjoint operator with the kernel is trace class but a sharp estimate on the decay of its eigenvalues was unknown. In this talk I will present a sharp bound and an asymptotic formula for the eigenvalues of .

A. Logunov, Princeton University

Nodal sets, quasiconformal mappings and how to apply them to Landis’ conjecture

A while ago Nadirashvili proposed a beautiful idea how to attack problems on zero sets of Laplace eigenfunctions using quasiconformal mappings, aiming to estimate the length of nodal sets (zero sets of eigenfunctions) on closed two-dimensional surfaces. The idea have not yet worked out as it was planned. However it appears to be useful for Landis’ Conjecture. We will explain how to apply the combination of quasiconformal mappings and zero sets to quantitative properties of solutions to on the plane, where is a real, bounded function. The method reduces some questions about solutions to Shrodinger equation on the plane to questions about harmonic functions. Based on a joint work with E.Malinnikova, N.Nadirashvili and F. Nazarov.

F. Klopp, Institut de Mathématiques de Jussieu – Paris Rive Gauche

A new look at localization

The talk is devoted to new, improved bounds for the eigenfunctions of random operators in the localized regime. We prove that, in the localized regime with good probability, each eigenfunction is exponentially decaying outside a ball of a certain radius, which we call the “localization onset length.” We count the number of eigenfunctions having onset length larger than, say, and find it to be smaller than times the total number of eigenfunctions in the system (for some positive constant ). Thus, most eigenfunctions localize on finite size balls independent of the system size.

We apply our techniques to obtain decay estimates for the -particles density matrices of eigenstates of non interacting fermionic quantum particles subjected to the random potential in a large box.

A. Khrabustovskyi, University of Hradec Kralove  

Homogenization of the Robin Laplacian in a domain with small holes: operator estimates

In the talk we revisit the problem of homogenization of the Robin Laplacian in a domain with a lot of tiny holes.

Let be a small parameter, be an open set in with , and be a perforated domain obtained by removing from a family of tiny identical balls of the radius () distributed periodically with a period . We denote by the Laplacian on subject to the Dirichlet condition on the external boundary of and the Robin conditions on the boundary of the balls:

where is an outward-facing unit normal. By we denote the Dirichlet Laplacian on . It is known (Kaizu (1985, 1989), Berlyand & Goncharenko (1990), Goncharenko (1997), Shaposhnikova et al. (2018)) that converges in a strong resolvent sense either to zero (solidifying holes), to (fading holes) or to the operator with a constant potential (critical case) as . The form of the limiting operator depends on certain relations between , and .

We will discuss our recent improvements of these results. Namely, for all three cases we show the norm resolvent convergence of the above operators and derive estimates in terms of operator norms. As an application we establish the Hausdorff convergence of spectra.

Anne-Sophie Bonnet-Ben Dhia, Institut Polytechnique de Paris

A new complex frequency spectrum for the analysis of transmission efficiency in waveguide-like geometries

We consider a waveguide, with one inlet and one outlet, and some arbitrary perturbation in between. In general, an ingoing wave in the inlet will produce a reflected wave, due to interaction with the perturbation. Our objective is to give an answer to the following important questions: what are the frequencies at which the transmission is the best one? And in particular, do they exist frequencies for which the transmission is perfect, in the sense that nothing is propagating back in the inlet?

Our approach relies on a simple idea, which consists in using a complex scaling in an original manner: while  the same stretching parameter is classically used in the inlet and the outlet, here we take them as two complex conjugated parameters. As a result, we select ingoing waves in the inlet and outgoing waves in the outlet, which is exactly what arises when the transmission is perfect. This simple idea works very well, and provides useful information on the transmission qualities of the system, much faster than any traditional approach. More precisely, we define a new complex spectrum which contains as real eigenvalues both the frequencies where perfect transmission occurs and the frequencies corresponding to trapped modes (also known as bound states in the continuum). In addition, we also obtain complex eigenfrequencies which can be exploited to predict frequency ranges of good transmission. Let us finally mention that this new spectral problem is PT -symmetric for systems with mirror symmetry.

Several illustrations performed with finite elements in several  simple 2D cases will be shown.

It is a common work with Lucas Chesnel (INRIA) and Vincent Pagneux (CNRS).

A. Fedotov, St. Peterburg State University

On Hierarchical Behavior of Solutions to the Maryland Equation in the Semiclassical Approximation

We describe a multiscale selfsimilar struture of solutions to one of the most popular models of the almost periodic operator theory, the difference Schroedinger equation with a potential of the form a , where , and are constants, and is an integer variable. The talk is based on a joint work with F.Klopp.

K. Pankrashkin, Carl von Ossietzky University of Oldenburg

Some convergence results for Dirac operators with large parameters

We consider Euclidean Dirac operators with piecewise constant mass potentials and investigate their spectra in several asymptotic regimes in which the mass becomes large in some regions. If the mass jumps along a smooth interface, then it appears that the (low-lying) discrete spectrum of such an operator converges to the (low-lying) discrete spectrum of an effective operator acting either on or in the interior of the interface. The effective operators admit a simple geometric interpretation in terms of the spin geometry, and the results can be extended to a class of spin manifolds as well. Most questions remain open if the jump interface is non-smooth. Based on joint works with Brice Flamencourt, Markus Holzmann, Andrei Moroianu, and Thomas Ourmieres-Bonafos.

D. Borisov, Bashkir State Pedagogical University and Institute o Mathematics UFRC RAS

Accumulation of resonances and eigenvalues for operators with distant perturbations

We consider a model one-dimensional problem with distant perturbations, for which we study a phenomenon of emerging of infinitely many eigenvalues and resonances near the bottom of the essential spectrum. We show that they accumulate to a certain segment of the essential spectrum. Then we discuss possible generalization of this result to multi-dimensional models and various situations of resonances and eigenvalues distributions. 

P. Exner, Doppler Institute for Mathematical Physics and Applied Mathematics

Spectral properties of spiral quantum waveguides

We discuss properties of a particle confined to a spiral-shaped region with Dirichlet boundary. As a case study we analyze in detail the Archimedean spiral for which the spectrum above the continuum threshold is absolutely continuous away from the thresholds. The subtle difference between the radial and perpendicular width implies, however, that in contrast to ‘less curved’ waveguides, the discrete spectrum is empty in this case. We also discuss modifications such a multi-arm Archimedean spirals and spiral waveguides with a central cavity; in the latter case bound state already exist if the cavity exceeds a critical size. For more general spiral regions the spectral nature depends on whether they are ‘expanding’ or ‘shrinking’. The most interesting situation occurs in the asymptotically Archimedean case where the existence of bound states depends on the direction from which the asymptotics is reached.

I. Vergara, EIMI

Positive definite radial kernels on trees and products

A classical result in harmonic analysis on trees characterises positive definite radial kernels on a -homogeneous tree () as integrals on the interval of a certain family of polynomials related to the Laplace operator on the tree. This gives a one-to-one correspondence between such kernels and finite Borel measures on . I will present an extension of this result to finite products of homogeneous trees. The main tools used in the proof come from the theory of Schur multipliers and the Hamburger moment problem.

D. Papathanasiou, EIMI

Weighted shifts on directed trees

Weighted backward shifts (unilateral or bilateral) on sequence spaces have been extensively studied and their dynamical properties are well known. We will extend this class of operators by substituting the set of natural numbers or integers by a directed tree and letting the underlying space to be a sequence space on that tree. Motivated by the work of Jablonski, Jung and Stochel, we will define weighted shifts on such spaces and we will discuss when they are hypercyclic or chaotic. We will be interested in relating the corresponding dynamical property to the family of weights of the shift, and the geometry of the tree.

M. Lukic, Rice University

Stahl-Totik regularity for continuum Schrödinger operators

This talk describes joint work with Benjamin Eichinger: a theory of regularity for one-dimensional continuum Schrödinger operators. For any half-line Schrödinger operator with a bounded potential , we obtain universal thickness statements for the essential spectrum, in the language of potential theory and Martin functions (which will be defined in the talk). Namely, we prove that the essential spectrum is not polar, it obeys the Akhiezer-Levin condition, and moreover, the Martin function at infinity obeys the two-term asymptotic expansion as . The constant in its asymptotic expansion plays the role of a renormalized Robin constant and enters a universal inequality . This leads to a notion of regularity, with connections to the exponential growth rate of Dirichlet solutions and limiting eigenvalue distributions for finite restrictions of the operator, and applications to decaying and ergodic potentials.

Ch. Berg, University of Copenhagen

Indeterminate Hamburger moment problems

A Hamburger moment sequence is of the form

where is a positive measure on . It can be characterized by positive semidefiniteness of the infinite Hankel matrix . The sequence can be determinate or indeterminate depending if there is exactly one or several measures on the real line such that holds.

In the talk I will focus on various problems related to the indeterminate case:

• The behaviour of the smallest eigenvalue of .
• The order and type of the functions of the Nevanlinna matrix.
• Behaviour of the infinite matrix of coefficients in the reproducing kernel for the moment problem.

In certain indeterminate cases—but not all—the following infinite matrix equations hold: .

Some of the results presented are joint work with Ryszard Szwarc, Wrocław.

St.Petersburg Summer Meeting in Mathematical Analysis

E. Liflyand, Bar Ilan University

Wiener algebras and trigonometric series in a coordinated fashion

Let be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proven in the paper that, in particular, a trigonometric series is the Fourier series of an integrable function if and only if there exists a such that , . If , then the piecewise linear continuous function defined by , , belongs to as well. Moreover, . Similar relations are established for more advanced Wiener algebras. These results are supplemented by numerous applications. In particular, new necessary and sufficient conditions are proved for a trigonometric series to be a Fourier series and new properties of are established. This is a joint work with R. Trigub.

I. Baibulov, EIMI

Eigenfunction expansion for three-body one-dimensional scattering problem

We consider a Schrödinger operator for three particles with pairwise interaction. In the case of d-dimensional particles for d>1 the scattering theory (description of continuous spectrum eigenfunctions or wave operators) is well known, first results are famously due to Faddeev. However, for the case of d=1 the description of generalized eigenfunctions was not complete. We will investigate the one-dimensional case and show how to modify the higher-dimensional results of Faddeev for d=1. At the end of the talk we will address how that approach naturally arises from a microlocal standpoint.

V. Kapustin, St.Petersburg Department of Steklov Mathematical Institute

The Riemann zeta function and kernels of Toeplitz operators

We modify the Riemann zeta function so that the resulting function becomes an element of the Hardy space in the half-plane where the real part is greater than 1/2. Then we construct the Toeplitz operator, whose kernel contains this function. For the kernel we show how one can construct the Hitt–Sarason representation from their theorem about nearly invariant subspaces.

A. Pushnitski, King’s College London

The spectrum of some Hardy kernel matrices

A Hardy kernel is an integral kernel in two variables and which is homogeneous in of degree . Integral operators on the positive semi-axis with Hardy kernels are explicitly diagonalisable by the Mellin transform. It is however by no means clear how to diagonalise the infinite matrix which is obtained by restricting a Hardy kernel onto natural numbers . In the talk, I will describe one specific explicit one-parametric family of Hardy kernels k when the spectral analysis of can be carried out. This matrix appears in the analysis of composition operators on the Hardy space of Dirichlet series; I will explain this connection at the end of the talk. This is joint work with Ole Brevig (Trondheim) and Kalle Perfekt (Reading).

G. Rozenblum, Chalmers University of Technology and SPbU

Eigenvalue properties of zero order pseudodifferential operators and an application to a 3D elasticity problem.

We consider a zero order pseudodifferential operator on a compact manifold. Suppose that the operator is polynomially compact. Then there may exist sequences of eigenvalues converging to the points of the essential spectrum. We find their asymptotics, by means of reduction to some negative order operators. The motivating example is the Neumann-Poincare (double layer) operator in 3D elastostatics, where the properties of eigenvlues converging to the points of the essential spectrum are studued.  The general results enable us to separate the contribution of the elastic properties and of the geometry of the body to the eigenvalue asymptotics. Details can be found in https://arxiv.org/abs/2006.10568

A. Kostenko, University of Vienna

Kirchhoff Laplacians on metric graphs

Laplacians on metric graphs (often called Quantum Graphs) play an important role as an intermediate setting between Laplacians on Riemannian manifolds and Laplacians on graphs and share properties with both. In this talk, we plan to discuss basic spectral properties of Laplacians on metric graphs. The main emphasis will be on graphs with infinitely many vertices and edges, the case which is much less understood.

R. Romanov, St.Petersburg State University

Discreteness of the spectrum of singular canonical systems.

We are going to provide a criterion for discreteness of the spectrum of canonical systems with one singular end. This criterion simultaneously answers the question of L. de Branges on description of canonical systems corresponding to arbitrary Hermite–Biehler functions. This is a joint work with Harald Woracek (Vienna).

M. Mitkovski, Clemson University

Uncertainty principles of Paneah-Logvinenko-Sereda type

I will present several new forms of the harmonic analysis uncertainty principle. These new forms can be viewed as a sharpening of the classical Paneah-Logvinenko-Sereda uncertainty principle, in a sense that we impose similar restrictions (sometimes more, sometimes less) on the Fourier support and deduce similar kind of sampling inequalities. I will also present applications of our uncertainty principles to some control and damping problems in linear PDE’s. This is a joint work with W. Green and B. Jaye.

M. Dorodnyi, EIMI

Operator error estimates for homogenization of the nonstationary Schrödinger-type equations: sharpness of the results.

In , we consider a selfadjoint matrix strongly elliptic second order differential operator , with periodic coefficients depending on . We study the behavior of the operator for small . It is proved that, as , this operator converges to in the norm of operators acting from the Sobolev space (with a suitable ) to . Here is the effective operator with constant coefficients. We prove sharp-order error estimates and study the question about the sharpness of the results with respect to the norm type, as well as to the dependence of estimates on . The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation .

V. Peller, St.Petersburg State University

Двойные операторные интегралы и формула следов Лифшица – Крейна

В докладе будут определены двойные операторные интегралы и будут обсуждаться их свойства. Затем мы перейдём к формуле следов Лифшица—Крейна и обсудим, как её можно получить с помощью двойных операторных интегралов. При этом важную роль играет теорема Сёкефальви-Надя и Фойаша об унитарных дилатациях сжатий.

V. Sloushch, St.Petersburg State University

Asymptotics of the discrete spectrum appearing in spectral gaps of the discrete Schrodinger operator under decaying sign-definite perturbation

We consider a periodic Schrodinger operator on discrete periodic graph in . The operator is perturbed by a sign-definite decaying potential on the graph ; , , . The asymptotics with the large coupling constant of a discrete spectrum of the perturbed operator , is investigated.

T. Weinmann, St.Petersburg State University

Spectral averaging for rank-one perturbations of self-adjoint operators

Let be a self-adjoint operator on a Hilbert space and . We study the properties of the family of rank-one perturbations of given by , where . In particular, we prove that if , then for almost every (here denotes ) and that in this case the spectral averaging formula holds:

Yu. Belov, St.Petersburg State University

Localization of zeroes for Cauchy transforms and canonical systems

If discrete measure is sufficiently small, then the zeroes of Cauchy transform are localized near the . Moreover, it may happen that this holds for any such that . We have found a description of such measures and attraction sets (i.e. subsets of which attract zeroes). We have proved that all attraction sets are ordered by inclusion. 

Such measures appear naturaly in the theory of canonical systems of differential equations. They correspond to the canonical systems whose Hamiltonian consists of indivisible intervals accumulating only to the left. Moreover, this correspondence is one-to-one under some additional assumptions. This topic is connected to the problem of density of polynomials and other classical problems in harmonic analysis. 

This is joint work with A. Baranov and E. Abakumov.

E. Korotyaev, St.Petersburg State University

Inverse scattering on half line, new results

We solve inverse scattering problem for Schrödinger operators with compactly supported potentials on the half line. We discretize S-matrix: we take the value of the  S-matrix on some infinite sequence of positive real numbers. Using this sequence obtained from S-matrix we recover uniquely the potential by a new explicit formula, without the Gelfand-Levitan-Marchenko equation.