Yuri A. Kordyukov

Semiclassical Spectral Asymptotics for a Two-Dimensional Magnetic Schrödinger Operator II: The Case of Degenerate Wells

We continue our study of a magnetic Schrödinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive. We analyze the case when the magnetic field has degenerate magnetic wells. The main result of the paper is an asymptotics of the groundstate energy of the operator in the semiclassical limit. The upper bounds are improved in the case when we have a localization by a miniwell effect of lowest order. These results are applied to prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.

Adiabatic Limits and the Spectrum of the Laplacian on Foliated Manifolds

We present some recent results on the behavior of the spectrum of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions normal to the leaves (in the adiabatic limit).

THE EGOROV THEOREM FOR TRANSVERSE DIRAC TYPE OPERATORS ON FOLIATED MANIFOLDS

Abstract Egorov’s theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first-order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel transport by the parallel transport determined by the transverse Levi-Civita connection. These results allow us to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.

Egorov’s Theorem for Transversally Elliptic Operators on Foliated Manifolds and Noncommutative Geodesic Flow

Abstract The main result of the paper is Egorov’s theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.

Spectral Gaps for Periodic Schrödinger Operators with Strong Magnetic Fields

We consider Schrödinger operators Hh=(ih d+A)*(ih d+A) with the periodic magnetic field B=dA on covering spaces of compact manifolds. Using methods of a paper by Kordyukov, Mathai and Shubin [14], we prove that, under some assumptions on B, there are in arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of the strong magnetic field h→0.

Index theory and non-commutative geometry on foliated manifolds

This paper gives a survey of the index theory of tangentially elliptic and transversally elliptic operators on foliated manifolds as well as of related notions and results in non-commutative geometry.

Eigenvalue estimates for a three-dimensional magnetic Schrödinger operator

We consider a magnetic Schrödinger operator H = (−ih∇− ~ A) with the Dirichlet boundary conditions in an open set Ω ⊂ R, where h > 0 is a small parameter. We suppose that the minimal value b0 of the module | ~ B| of the vector magnetic field ~ B is strictly positive, and there exists a unique minimum point of | ~ B|, which is non-degenerate. The main result of the paper is upper estimates for the low-lying eigenvalues of the operator H in the semiclassical limit. We also prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting. 1. Preliminaries and main results 1.1. Main assumptions. We would like to analyze the asymptotic behavior, in the semiclassical regime, of the low-lying eigenvalues of the Dirichlet realization of the magnetic Schrödinger operator in an open set Ω in R: H = (hDX1 −A1(X)) + (hDX2 −A2(X)) + (hDX3 −A3(X)) , where ~ A = (A1, A2, A3) ∈ C(Ω̄,R) is a magnetic potential and h > 0 is a small parameter. We will denote the coordinates in R as X = (X1, X2, X3) = (x, y, z) . Let ~ B = rot ~ A = (B1, B2, B3) be the corresponding vector magnetic field: B1 = ∂A3 ∂y − ∂A2 ∂z , B2 = ∂A1 ∂z − ∂A3 ∂x , B3 = ∂A2 ∂x − ∂A1 ∂y . Put b0 = min{| ~ B(X)| : X ∈ Ω}. We assume that there exist a (connected) bounded domain Ω1 ⊂⊂ Ω and a constant ǫ0 > 0 such that (1.1) | ~ B(X)| ≥ b0 + ǫ0, x 6∈ Ω1 . We also assume that: (1.2) b0 > 0 , and that there exists a unique minimum X0 ∈ Ω such that | ~ B(X0)| = b0, which is non-degenerate: in some neighborhood of X0 (1.3) C|X −X0| ≤ | ~ B(X)| − b0 ≤ C|X −X0| . 2000 Mathematics Subject Classification. 35P20, 35J10, 47F05, 81Q10.

Semiclassical asymptotics and gaps in the spectra of periodic Schrödinger operators with magnetic wells

We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schrodinger operator with magnetic wells on a noncompact Riemannian mani- fold M such that H 1 (M, R) = 0 equipped with a properly discon- nected, cocompact action of a finitely generated, discrete group of isometries has an arbitrarily large number of spectral gaps in the semi-classical limit.

Semiclassical Spectral Asymptotics for a Magnetic Schrödinger Operator with Non-vanishing Magnetic Field

We consider a magnetic Schrodinger operator Hh on a compact Riemannian manifold, depending on the semiclassical parameter h > 0. We assume that there is no electric field. We suppose that the minimal value b0 of the intensity of the magnetic field b is strictly positive. We give a survey of the results on asymptotic behavior of the eigenvalues of the operator Hh in the semiclassical limit.

Integer points in domains and adiabatic limits

We prove an asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space with smooth boundary, which remain unchanged along some linear subspace and stretch out in the directions, orthogonal to this subspace. A more precise estimate for the remainder is obtained in the case when the domains are strictly convex. Using these results, we improved the remainder estimate in the adiabatic limit formula (due to the first author) for the eigenvalue distribution function of the Laplace operator associated with a bundle-like metric on a compact manifold equipped with a Riemannian foliation in a particular case when the foliation is a linear foliation on the torus and the metric is the standard Euclidean metric on the torus.