Integer points in domains and adiabatic limits (2010)
Kordyukov, Yuri A., Yakovlev, Andrey A.
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...
Adiabatic limits and noncommutative Weyl formula (2010)
We discuss asymptotic behavior of the eigenvalue distribution of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions...
Helffer, Bernard, Kordyukov, Yuri A.
We consider a magnetic Schr\"odinger operator $H^h$, depending on the semiclassical parameter $h>0$, on a two-dimensional Riemannian manifold. We assume that there is no electric field. We suppose...
TWO-DIMENSIONAL MAGNETIC SCHRÖDINGER OPERATOR: THE CASE OF DISCRETE WELLS (2010)
Bernard Helffer, Yuri A. Kordyukov, Bernard Helffer, Yuri, A. Kordyukov
Dedicated to Misha Shubin on the occasion of his 65th birthday Abstract. We consider a magnetic Schrödinger operator H h, depending on the semiclassical parameter h> 0, on a two-dimensional...
Classical and quantum dynamics in transverse geometry of Riemannian foliations (2009)
First, we survey some results on classical and quantum dynamical systems associated with transverse Dirac operators on Riemannian foliations. Then we illustrate these results by two examples of...
Noncommutative Hamiltonian dynamics on foliated manifolds (2009)
First, we review the notion of a Poisson structure on a noncommutative algebra due to Block-Getzler and Xu and introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra....
Index theory and non-commutative geometry on foliated manifolds (2009)
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.
Semiclassical analysis of Schr\"odinger operators with magnetic wells (2008)
Helffer, Bernard, Kordyukov, Yuri A.
We give a survey of some results, mainly obtained by the authors and their collaborators, on spectral properties of the magnetic Schr\"odinger operators in the semiclassical limit. We focus our...
Transversality and Lefschetz numbers for foliation maps (2008)
López, Jesús A. Álvarez, Kordyukov, Yuri A.
Let $F$ be a smooth foliation on a closed Riemannian manifold $M$, and let $\Lambda$ be a transverse invariant measure of $F$. Suppose that $\Lambda$ is absolutely continuous with respect to the...
Spectral gaps for periodic Schr\"odinger operators with hypersurface magnetic wells (2008)
Helffer, Bernard, Kordyukov, Yuri A.
We consider a periodic magnetic Schr\"odinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M, \RR)=0$ endowed with a properly discontinuous cocompact isometric action of a discrete...
The Egorov theorem for transverse Dirac type operators on foliated manifolds (2007)
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...
Vanishing theorem for transverse Dirac operators on Riemannian foliations (2007)
We obtain a vanishing theorem for the half-kernel of a transverse ${\rm Spin}\sp c$ Dirac operator on a compact manifold endowed with a transversely almost complex Riemannian foliation twisted by a...
Adiabatic limits and the spectrum of the Laplacian on foliated manifolds (2007)
Kordyukov, Yuri A., Yakovlev, Andrey A.
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...
Lefschetz distribution of Lie foliations (2007)
Lopez, Jesus A. Alvarez, Kordyukov, Yuri A.
Let $\mathcal F$ be a Lie foliation on a closed manifold $M$ with structural Lie group $G$. Its transverse Lie structure can be considered as a transverse action $\Phi$ of $G$ on $(M,\mathcal F)$;...
The periodic magnetic Schr\"odinger operators: spectral gaps and tunneling effect (2007)
Helffer, Bernard, Kordyukov, Yuri A.
A periodic Schr\"odinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M, \mathbb R)=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group is...
Helffer, Bernard, Kordyukov, Yuri A.
We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schr\"odinger operator with magnetic wells on a noncompact Riemannian manifold $M$ such that...
Noncommutative geometry of foliations (2005)
We review basic notions and methods of noncommutative geometry and their applications to analysis and geometry on foliated manifolds.
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...
Spectral gaps for periodic Schr\"odinger operators with strong magnetic fields (2003)
We consider Schr\"odinger operators $H^h = (ih d+{\bf A})^* (ih d+{\bf A})$ with the periodic magnetic field ${\bf B}=d{\bf A}$ on covering spaces of compact manifolds. Under some assumptions on $\bf...
The trace formula for transversally elliptic operators on Riemannian foliations (2000)
We prove a Duistermaat-Guillemin trace formula for transversally elliptic operators on a compact foliated manifold.
Adiabatic limits and spectral sequences for Riemannian foliations (1999)
Lopez, Jesus A. Alvarez, Kordyukov, Yuri A.
For general Riemannian foliations, spectral asymptotics of the Laplacian is studied when the metric on the ambient manifold is blown up in directions normal to the leaves (adiabatic limit). The...
The transversal wave equation and the noncommutative geodesic flow in Riemannian foliations (1997)
We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between...
Noncommutative spectral geometry of Riemannian foliations (1997)
We construct spectral triples in a sense of noncommutative differential geometry, associated with a Riemannian foliation on a compact manifold, and describe its dimension spectrum.
Long time behavior of leafwise heat flow for Riemannian foliations (1996)
Lopez, Jesus A. Alvarez, Kordyukov, Yuri A.
For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This...
Adiabatic Limits and Spectral Geometry of Foliations (1995)
We study spectral asymptotics for the Laplace operator on differential forms on a Riemannian foliated manifold equipped with a bundle-like metric in the case when the metric is blown up in directions...