The error estimates for the infinite element method for eigenvalue problems
The Essential Spectrum of Elliptic Systems of Mixed Order.
The essential spectrum of relativistic multi-particle operators
The essential spectrum of relativistic one-electron ions in the Jansen-Hess model.
The Existence of Wave Operators in Scattering Theory.
The first eigenvalue for the -Laplacian operator.
The first eigenvalue of -Laplacian systems with nonlinear boundary conditions.
The gaps in the spectrum of the Schrödinger operator
We obtain inequalities between the eigenvalues of the Schrödinger operator on a compact domain Ω of a submanifold M in with boundary ∂Ω, which generalize many existing inequalities for the Laplacian on a bounded domain of a Euclidean space. We also establish similar inequalities for a closed minimal submanifold in the unit sphere, which generalize and improve Yang-Yau’s result.
The generalized Morawetz problem for Lavrent'ev-Bitsadze equation with spectral parameter.
The integral equation methods for the perturbed Helmholtz eigenvalue problems.
The inverse -body problem. A geometrical approach
The Lane-Emden Function and Nonlinear Eigenvalues Problems
We consider a semilinear elliptic eigenvalues problem on a ball of and show that all the eigenfunctions and eigenvalues, can be obtained from the Lane-Emden function.
The Laplace-Beltrami operator in almost-Riemannian Geometry
We study the Laplace-Beltrami operator of generalized Riemannian structures on orientable surfaces for which a local orthonormal frame is given by a pair of vector fields that can become collinear.Under the assumption that the structure is 2-step Lie bracket generating, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence, a quantum particle cannot cross the singular set (i.e., the set where the vector fields become collinear) and the...
The laplacian operator on a Riemann surface
The laplacian operator on a Riemann surface, II
The laplacian operator on a Riemann surface, III
The Leading Singularity of Scattering Kernel for a Transparent Obstacle.
The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems
We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem where is a bounded domain, is a real number and , satisfy appropriate growth conditions. Moreover, the coefficient contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in . The main tool is the investigation of the associated homogeneous eigenvalue problem and an application...
The Leray measure of nodal sets for random eigenfunctions on the torus
We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity .The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant, equal...
The light in the neighborhood of a caustic