# Workshop Programme

## for period 24 - 28 July 2006

### Spectral Theory and its Applications

24 - 28 July 2006

Timetable

 Monday 24 July 08:30-10:00 Registration 10:00-10:50 Klopp, F (Universite de Paris Nod) The Lyapunov exponent for certain ergodic matrix cocycles and the spectrum of the associated Schr\"{o}dinger operator Sem 1 The aim of this work in progress is the study of the generalized eigenfuinctions associated to singular (in particular singular continuous) spectrum for a Schrödinger operator on the half-line. This study reduces to the study of an ergodic matrix cocycle for which we develop an exact renormalization analysis. In particular, we get a complete description of the set of ergodic parameters for which the Lyapunov exponent of the cocycle exists and does not exist. 11:00-11:30 Coffee 11:30-12:20 Shubin, M (Northeastern) KdV, mKdV and eigenfunctions of Schr\"{o}dinger operators Sem 1 We investigate the relation between the Korteweg - de Vries and modified Korteweg - de Vries equations (KdV and mKdV), and find a new algebro-analytic mechanism, similar to the Lax L-A pair, which involves a first-order operator Q instead of the third-order operator A. In our framework, eigenfunctions of the Schroedinger operator L, whose time-dependent potential solves the KdV equation, evolve according to a linear first-order partial differential equation, giving explicit control over their time evolution. As an application, we establish global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity. These classes may even include functions which tend to infinity with the space variable. We also give a new proof of the invariance of the spectrum of the 1-dimensional Schr\"odinger operator under the KdV flow. It is, in particular, applicable to the potentials in the classes of possibly unbounded functions described above. The talk will be based on a joint work of T.Kappeler, P.Perry, T.Topalov and the speaker, see http://arxiv.org/pdf/math.AP/0601237 12:30-13:30 Lunch at Wolfson Court 14:00-14:50 Fulling, S (Texas A and M University) Quantum vacuum energy as spectral theory Sem 1 Vacuum (Casimir) energy, a topic of great current interest in physics, has close connections with spectral asymptotics, semiclassical approximations, periodic orbits, stationary-phase approximations, quantum graphs, etc. The Casimir effect is observable as the mutual attraction of neutral conducting or dielectric bodies via quantum fluctuations of the electromagnetic field; it is speculated that some related effect is responsible for the observed cosmological "dark energy". I will describe some current progress and issues in this field that are of mathematical interest. For example, contributions to vacuum energy arise from the oscillations in spectral densities associated with periodic classical paths of the associated classical (or ray-optical) system, and the sign of such a contribution is influenced by the phase of the oscillation (the Maslov index or a generalization thereof). Related Links www.math.tamu.edu/~fulling - personal home pagewww.math.tamu.edu/research/workshops/semivac - record of Jan. 2005 workshop 15:00-15:30 Tea 15:30-16:20 Pushnitski, A (Kings College London) The differences of spectral projections and scattering matrix Sem 1 Let A and B be self-adjoint operators in a Hilbert space such that the difference B-A is a trace class operator. For an interval I in the real line, we consider the self-adjoint operator D, defined as the difference between the spectral projection of B associated with I and the spectral projection of A associated to I. In his famous 1953 paper on the spectral shift function theory, M.G.Krein gave a simple example which shows that the operator D may fail to belong to the trace class even if the difference between B and A has rank one. Further analysis shows that in Krein's example, D is not even compact. We address the question of the description of the spectrum of D. It appears that under very general assumptions, the essential spectrum and the absolutely continuous spectrum of D can be completely described in terms of the spectrum of the scattering matrix for the pair A, B, evaluated at the endpoints of the interval I. In particular, it follows that D is compact if and only if the scattering matrix coincides with the identity operator. 16:30-17:20 Grigor'yan, A (Universitaet Bielefeld) Heat kernels for Schr\"{o}dinger operators Sem 1 We present two-sided estimates for the heat kernels of certain elliptic Schrodinger operators. The class of potentials include those with the quadratic decay at infinity. The proof is based on a joint work with L.Saloff-Coste providing conditions for the stability of the parabolic Harnack inequality under a non-uniform change of measure on weighted Riemannian manifolds. 17:30-18:30 Wine Reception 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Tuesday 25 July 09:00-09:50 Helffer, B (Université Paris Sud) Semi-classical analysis for non selfadjoint problems and applications to hydrodynamic instability Sem 1 Our aim is to show how semi-classical mechanics can be useful in questions appearing in hydrodynamics.We will emphasize the motivating examples and see how these problems can be solved by either harmonic approximation techniques in semi-classical analysis of the Schroedinger operator or by new semi-classical estimates for operators of principal type (mainly subelliptic estimates). This work is in collaboration with O. Lafitte. 10:00-10:50 Frank, R (Royal Institute of Technology) On Lieb-Thirring-Hardy inequalities Sem 1 We show that the Lieb-Thirring inequality on moments of negative eigenvalues remains true, with a possibly different constant, when the critical Hardy-weight is subtracted from the Laplace operator. A similar statement is true for fractional powers of the Laplacian, in particular relativistic Schr\"odinger operators. The talk is based on joint works with T. Ekholm and with E. Lieb and R. Seiringer. 11:00-11:30 Coffee 11:30-12:20 Liskevich, V (Bristol) Positive solutions to second-order semilinear and quasilinear elliptic equations in unbounded domains Sem 1 We survey some recent results on the existence and nonexistence of positive super-solutions to semilinear and quasi-linear elliptic equations equation (*): Lu=up in G, where G is an unbounded domain in RN (N > 2), L is a second order uniformly elliptic operator in divergence form with measurable coefficients and p in R. We prove the existence of critical exponents p- < 1 and p+ >1 such that equation (*) has no positive (super) solution if and only if p in [p-, p+] and show that the values and the properties of p-=p-(L, G) and p+=p+(L, G) essentially depend both on the geometry of the domain G and the coefficients of the elliptic operator L. The focus will be mainly on exterior domains and cone-like domains. . We also examine the same problem for the non-divergence type operators L and for some classes of quasi-linear operators with critical potentials. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behaviour of super-harmonic functions associated to the operator, Phragmën--Lindelöf type comparison arguments and an improved version of Hardy's inequality. 12:30-13:30 Lunch at Wolfson Court 14:00-14:50 Ashbaugh, M (Missouri) Some Eigenvalue comparison results Sem 1 This talk will concern inequalities for the eigenvalues of certain differential operators, particularly those of the Laplacian under Dirichlet and/or Neumann boundary conditions, and of the eigenvalue problems arising in the study of the vibration and buckling of a clamped plate. Various results will be presented, including recent results of the speaker and F. Chiacchio for the Laplacian and of the speaker for the two aforementioned biharmonic problems. 15:00-15:30 Tea 15:30-16:20 Hoffman-Ostenhof, T (Vienna) Nodal domains and spectral minimal partitions Sem 1 Abstract: We consiser two-dimensional Schr\"odinger operators in bounded domains. We analyze relations between the nodal domains of eigengunctions, spectral minimal partitions and spectral properties of the corresponding operator. The main results concern the existence and regularity of the minimal partitions and the characterization of the minimal partitions associated with nodal sets as the nodal domains for the case that in Courant's nodal theorem there is equality. 16:30-17:20 Friedlander, L (Arizona) Determinants of zeroth order operators Sem 1 This is a joint work with Victor Guillemin. We compare two ways of regularizing the determinant of a pseudo-differential operator of zeroth order. The first way is based on a generalization of Szego's strong limit theorem; the second regularization uses a generalized zeta-function. Both regularizations of the determinant are non-local, and they give different answers. However, and this is our main result, the ratio of them is a local quantity. Related Links http://math.arizona.edu/~friedlan/zero_ord_det.pdf - preprint 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Wednesday 26 July 09:00-09:50 Coulhon, T (Cergy-Pontoise) Large time behavior of heat kernels on forms Sem 1 We derive large time upper bounds for heat kernels on vector bundles of differential forms on a class of non-compact Riemannian manifolds under certain curvature conditions. 10:00-10:50 Buslaev, V (St Petersburg State) Adiabatic linear equation whose generator has continuous spectrum Sem 1 We describe the asymptotic behavior of solutions of the Cauchy problem for the linear differential equation of the first order in the Banach space, containing a small paremeter before the derivative. Denote the main variable by t and the generator of the equation by -iA(t). It is supposed that the linear operator A(t) for any t on a given finite interval has continuous spectrum (plus, may be, some number of eigenvalues) and allows the reduction to a constant model operator by application of methods of the stationary scattering theory. For example, A(t) can be a Schroedinger operator with a quickly decreasing at infinity potential depending on t, such that the initial equation is a non-stationary Schroedinger equation. Then the asymptotic behavior of solutions of the problem can be described by formulas that are similar to the known formulas for the case of the generator with purely point simple (for any t) spectrum when the ideas of the "Quantum Adiabatic Theorem" can be used. In particular, that allows to describe the asymptotic behavior of the solution whose initial data at t=0 is the eigenfunction of the continuous spectrum of A(0). 11:00-11:30 Coffee 11:30-12:20 Rochon, F (Stony Brook) Eta invariant, boundaries and the determinant line bundle Sem 1 We will discuss the definition of the eta invariant on manifolds with boundary using cusp suspended operators. This will be used to show that the (exponentiated) eta invariant of a family of elliptic operators trivialzes the determinant bunle of the associated family of operators on the boundary, giving a pseudodifferential generalization of a result of Dai and Freed. 12:30-13:30 Lunch at Wolfson Court 14:00-14:50 Gordon, CS (Dartmouth College, Hanover) Inverse spectral results on line bundles over even dimensional tori Sem 1 Given a Hermitian line bundle L over a flat torus M, a connection on L, and a function Q on M, one associates a Schrodinger operator acting on sections of L. In the case of line bundles of Chern number one over two-dimensional tori for which both the connection and the potential are even, V. Guillemin showed under genericity conditions that the spectrum determines both the connection and the potential. We address the question in higher even dimensions of whether, fixing an invariant'' connection, the spectrum determines the potential. With a genericity condition, we show that the spectrum determines the even part of the potential. We also show that if we allow the connection to vary over the invariant connections but keep the potential Q fixed, then the resulting collection of spectra uniquely determine the potential, without any assumption of evenness. This collection of spectra is a natural analog in the case of line bundles to the classical Bloch spectrum of the torus. We also obtain counterexamples showing that the genericity conditions are needed even in the case of the two-dimensional tori studied by Guillemin. 15:00-15:30 Tea 15:30-16:20 Banuelos, R (Purdue) Finite dimensional distributions of Brownian motion and stable processes Sem 1 We investigate properties of finite dimensional distributions of Brownian motion and other Levy processes which arose from our efforts to answer the following very "simple" question: What is the lowest eigenvalue for the rotationally invariant stable process in the interval (-1, 1)? While we still don't know the answer to this question, its investigation has led to interesting applications of finite dimensional distributions (multiple integrals) to eigenvalues and eigenfunctions of the Laplacian and "fractional" Laplacian. 16:30-17:20 Jacobson, D (McGill) Estimates from below for the remainder in Weyl's law Sem 1 We obtain asymptotic lower bounds for the spectral function of the Laplacian on compact manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Our results can be considered pointwise versions (on a general manifold) of Hardy's lower bounds for the error term in the Gauss circle problem. Next, we obtain a lower bound for the remainder in Weyl's law on a negatively curved surface. On higher-dimensional negatively curved manifolds, we prove a similar bound for the oscillatory error term. This extends earlier results of Hejhal and Randol on surfaces of constant negative curvature proved by methods of analytic number theory. Our approach uses wave trace asymptotics, equidistribution of closed geodesics and small-scale microlocalization. Related Links http://www.ams.org/era/2005-11-09/S1079-6762-05-00149-6/home.html - ERA-AMS paperhttp://front.math.ucdavis.edu/math.SP/0505400 - arxiv preprinthttp://www.math.mcgill.ca/jakobson/papers/umd06.pdf - Slides 19:30-18:00 Conference Dinner at Magdalene College
Thursday 27 July
09:00-09:50 Davies, B (Kings College, London)
Non-self-adjoint operators and pseudospectra Sem 1
10:00-10:50 Rozenblioum, G (Chalmers Institute of Technology)
Spectral properties of the perturbed Landau Hamiltonian Sem 1

The Landau Hamiltonian describes the movement of a charged quantum particle confined to a plane, under the action of the constant magnetic field. The spectrum of this operator consists of the landau levels, infinitely degenerate eigenvalues placed at the points of an arithmetic progression. We describe what happens with the spectrum and the spectral subspaces under the perturbation of the operator by localyzed magnetic and electric fields.

11:00-11:30 Coffee
11:30-12:20 Lieb, E (Princeton)
Lieb-Thirring inequalitites for Schr\"{o}dinger operators with complex-valued potentials Sem 1

Inequalities are derived for power sums of the real-part and the mobulus of the eigenvalues of a Schrödinger operators with a complex-valued potential.

12:30-13:30 Lunch at Wolfson Court
14:00-14:50 Poster Session
15:00-15:30 Tea
15:30-16:20 Suslina, T (St Petersburg State)
Homogenization of periodic differential operators as a spectral threshold effect Sem 1

In L_2(R^d), we study matrix periodic elliptic second order differential operators of the form A=X^*X. Here X is a homogeneous first order differential operator. Many operators of mathematical physics can be written in this form. We study a homogenization problem for A. Namely, for the corresponding operator with rapidly oscillating coefficients, we study the behavior of the resolvent in the small period limit. We find approximation for this resolvent in the (L_2)-operator norm in terms of the resolvent of the effective operator. The error estimate is order-sharp, and the constant in this estimate is well controlled. Next, taking the so called corrector into account, we obtain more accurate approximations for the resolvent both in the (L_2)-operator norm and in the "energetic" norm. The error estimates are order-sharp. The obtained results are of new type in the homogenization theory.

The method is based on the abstract operator theory approach for selfadjoint operator families A(t)=X(t)^*X(t) depending on one-dimensional parameter t, where X(t)=X_0 +tX_1. It turns out that the homogenization procedure is determined by the spectral characteristics of the periodic operator A near the bottom of the spectrum. Therefore, a homogenization procedure can be treated as a spectral threshold effect.

General results are applied to particular periodic operators of mathematical physics: the acoustics operator, the operator of elasticity theory, the Maxwell operator, the Schrodinger operator, the twodimensional Pauli operator.

16:30-17:20 Kuchment, P (Texas A and M)
Liouville theorems and spectral edge behavior for periodic operators on Abelian coverings of compact manifolds Sem 1

The classical Liouville theorem says that any entire function in C^n that grows polynomially, is a polynomial. Thus, for a fixed rate of polynomial growth, the space of such functions is finite dimensional, and its dimension can be easily computed. There is also a classical version of this theorem that deals with harmonic functions in R^n. A program of extending this theorem to solutions of Laplace-Beltrami equations on more general Riemannian manifolds (or to more general elliptic operators in Euclidean space) was started by S. T Yau about 30 years ago. Major work on this has been done by Colding and Minicozzi and P. Li with his co-authors.

A new development since the beginning on 1990s was devoted to Liouville theorems for periodic elliptic operators in R^n, where amazingly explicit dimension formulas could be obtained. This was due to work by Avellaneda and Lin, Moser and Struwe, and later P. Li and Wang.

At the same time, similar questions about validity of Liouville type theorems have been asked in the settings of holomorphic functions on coverings of complex manifolds (e.g., work by P. Lin and A. Brudnyi) and discrete harmonic functions on graph coverings with compact bases (G. Margulis).

In this talk, we provide an overview of recent results by the authors that treat all these three situations simultaneously for periodic operators on abelian coverings of compact bases. This includes elliptic operators on Riemannian manifolds, Cauchy-Riemann operators on analytic manifolds, discrete equations on graphs, and differential operators on quantum graphs. One can give necessary and sufficient conditions for the validity of Liouville theorems, as well as explicit dimension formulas. One can also mention striking similarities with the formulas obtained in the work by Gromov and Shubin concerning Riemann-Roch theorems for elliptic operators.

Albeit the problem does not look like being related to spectral theory, the answers (as well as the tools) are given in terms of the structure of the dispersion relation near edges of the spectrum of the operator.

 Friday 28 July 09:00-09:50 Vasy, A (MIT) Diffraction by edges Sem 1 I will discuss the propagation of singularities for the wave equation on manifolds with corners, with special attention paid to the issue whether some part of the reflected waves is weaker than the incoming wave (in terms of Sobolev regularity). This is joint work with Richard Melrose and Jared Wunsch. 10:00-10:50 Melrose, R (MIT) Adiabatic and semiclassical limits Sem 1 11:00-11:30 Coffee 11:30-12:20 Yafaev, D (I.R.M.A.R) A balance of kinetic and potential energies in the semi-classical limit Sem 1 Consider a bound state of a quantum particle in a potential well $v(x)$. Suppose that the energy of a particle is higher than the minimum of the function $v(x)$. We show that in the semi-classical limit, as the Planck constant $h$ tends to zero, the ratio of kinetic and potential energies tends to a finite positive limit. 12:30-13:30 Lunch at Wolfson Court 14:00-14:50 Laptev, A (Royal Institute of Technology, Sweden) Some eigenvalue inequalities for Schr\"{o}dinger operators with positive potentials Sem 1 We shall discuss some estimates from below on the ratio of the $\lambda_k/\lambda_1$ where $\lambda_k$ are the eigenvalues of the Dirichlet boundary value problem for the operator $-\Delta +V$, $V\ge 0$, in $L^2(\Omega)$, $\Omega\subset{\Bbb R}^d$. Surprisingly some of such estimates are independent of $V$ or $\Omega$. This is my joint paper with R.Frank and S.Molchanov. 15:00-15:30 Tea 15:30-16:20 Fournais, S (Paris-Sud) Spectral confinement and current for atoms in strong magnetic fields Sem 1 We study confinement of the ground state of atoms in strong magnetic fields to different subspaces related to the lowest Landau band. The results obtained allow us to calculate the quantum current in the entire semiclassical region $B \ll Z^3$. 16:30-17:20 Kiselev, A (Wisconsin, Madison) Enhancement of dissipation by fast mixing Sem 1 We consider a dissipative evolution problem f_t = iALf - Gf. Here L, G are self-adjoint operators, G>0 has discrete spectrum, and A is a large parameter. The question we would like to ask is under what assumptions on L the dissipation speed is greatly enhanced as the parameter A becomes large. We provide a sharp characterization of operators L that lead to strong enhancement. The work lies at the interface of spectral theory, dynamical systems and PDE. In particular, the original motivation for this study comes from passive scalar equation, where enhancement of diffusion by fluid flow is a classical problem of interest. The key tools invlove estimates related to self adjoint quantum dynamics corresponding to continuous and point spectra. This is a joint work with P. Constantin, L. Ryzhik and A. Zlatos. Related Links http://www.math.wisc.edu/~kiselev/pub2.html 18:15-19:00 Dinner at Wolfson Court (Residents only)