arXiv research

Paper introduces magnetic Steklov operator on differential forms and its properties.

Study magnetic Steklov eigenvalues on manifolds with boundary.

Study magnetic potentials on Anosov manifolds using spectral data.

The paper calculates spectral invariants of magnetic Steklov eigenvalues on Riemannian manifolds.

Study eigenvalues of magnetic Steklov problem on Riemannian annuli.

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…

Upper bounds for Steklov eigenvalues on curved submanifolds.

The paper studies magnetic field effects on surface eigenvalues and spectral properties.

We use layer potential to establish that the boundary biharmonic Steklov operators are elliptic pseudo-differential operators. Thus we are able to establish lower bounds on both the measure of boundary nodal sets and interior nodal sets for biharmonic Steklov eigenfunctions.

Upper bounds found for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.

Study Cheeger inequalities for Riemannian manifolds with boundary.

Study on biharmonic Steklov problem on differential forms.

Upper bounds for Steklov eigenvalues on manifolds with boundary.

We study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and compute the first few of them explicitly in terms of the scalar and mean curvatures. This is done by applying the Seeley …

Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.

We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. It is shown that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neuma…

Study on magnetic Dirac operators and their spectrum.

Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklo…

We prove Li-Yau-Kröger type bounds for Neumann-type eigenvalues of the poly-harmonic operator and of the biharmonic operator on bounded domains in a Euclidean space. We also prove sharp estimates for lower order eigenvalues of a biharmonic Steklov problem and of the Laplacian, which directly implies two sharp Reilly-ty…

The article studies eigenvalues and spectrum of magnetic Dirac operators.

Study magnetic Schrödinger operators in Euclidean space.

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of thes…

Guillarmou extends X-ray transform to magnetic and thermostat flows.

Upper bounds for Steklov eigenvalues of warped products are derived.

We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin …

The paper solves the Steklov spectral inverse problem for conformal metrics.

We consider a magnetic Schrödinger operator $H^h$, depending on a semiclassical parameter $h>0$, on a compact Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value $b_0$ of the intensity of the magnetic field $b$ is strictly positive. We give a survey of the results on asympt…

The paper studies magnetic curvature and proves the existence of closed orbits on low energy levels.

Study eigenvalues and shapes, proving sharp inequalities for Steklov eigenvalues.

The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometr…

Constructs smooth integrable magnetic systems on a two-torus.

Study Dirichlet-to-Neumann maps on manifolds, focusing on covering and total spaces.

Study sharp lower bounds on negative eigenvalues of magnetic Pauli operator.

Bounds on Steklov eigenvalues for manifolds with boundary.

The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian)…

In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or a Hadamard manifold, and of a biharmonic Steklov problem on bounded domains of a…

A quasiclassical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is not given by an exact 2-form. For this, the multidimensional WKB method in the form of Maslov canonical operator is applied. In this case, the canon…

We consider an open domain with a compact boundary in an Euclidean space and a Schroedinger operator with magnetic field on this domain. We give sufficient conditions on the rate of growth of the magnetic field near the boundary which guarantees essential self-adjointness of this operator. From the physical point of vi…

Paper introduces magnetic Hodge Laplacian for differential forms.

Magnetic Brunn-Minkowski inequalities on Riemannian manifolds

We investigate nodal sets of magnetic Schroedinger operators with zero magnetic field, acting on a non simply connected domain in $\r^2$. For the case of circulation 1/2 of the magnetic vector potential around each hole in the region, we obtain a charactisation of the nodal set, and use this to obtain bounds on the mul…

The known upper bounds for the multiplicities of the Laplace-Beltrami operator eigenvalues on the real projective plane are improved for the eigenvalues with even indexes. Upper bounds for Dirichlet, Neumann and Steklov eigenvalues on the real projective plane with holes are also provided.

We solved the Schr{ö}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / Λ is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice Λ. The lattice is not neces…

Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.

Paper finds how Steklov eigenvalues change on graphs and trees.

The paper compares Steklov and Laplacian eigenvalues on graphs.

Upper bound found for Steklov eigenvalues counting function.

Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.