On the First Eigenvalue of Non-Uniformly Elliptic Boundary Value Problems.
On the generic spectrum of a riemannian cover
Let be a compact manifold let be a finite group acting freely on , and let be the (Fréchet) space of -invariant metric on . A natural conjecture is that, for a generic metric in , all eigenspaces of the Laplacian are irreducible (as orthogonal representations of ). In physics terminology, no “accidental degeneracies” occur generically. We will prove this conjecture when dim dim for all irreducibles of . As an application, we construct isospectral manifolds with simple eigenvalue...
On the heat trace of the magnetic Schrödinger operators on the hyperbolic plane.
On the homogenization of the Poisson equation in partially perforated domains with arbitrary density of cavities and mixed type conditions on their boundary
In this paper we study the behavior of solutions of the boundary value problem for the Poisson equation in a partially perforated domain with arbitrary density of cavities and mixed type conditions on their boundary. The corresponding spectral problem is also considered. A short communication of similar results can be found in [1].
On the Lax-Phillips scattering theory for transport equation
On the linearization of some singular, nonlinear elliptic problems and applications
On the long range scattering for Stark Hamiltonians.
On the lowest eigenvalue of the Laplacian for the intersection of two domains.
On the nodal line of the second eigenfunction of elliptic operators in two dimensions.
On the nodal set of the second eigenfunction of the laplacian in symmetric domains in
We present a simple proof of the fact that if is a bounded domain in , , which is convex and symmetric with respect to orthogonal directions, , then the nodal sets of the eigenfunctions of the laplacian corresponding to the eigenvalues must intersect the boundary. This result was proved by Payne in the case for the second eigenfunction, and by other authors in the case of convex domains in the plane, again for the second eigenfunction.
On the nonrelativistic limit of the Dirac hamiltonian
On the p-biharmonic operator with critical Sobolev exponent
We study the existence of solutions for a p-biharmonic problem with a critical Sobolev exponent and Navier boundary conditions, using variational arguments. We establish the existence of a precise interval of parameters for which our problem admits a nontrivial solution.
On the point spectrum of Schrödinger operators
On the poles of the scattering matrix for two convex obstacles
On the principal eigencurve of the p-Laplacian related to the Sobolev trace embedding
We prove that for any λ ∈ ℝ, there is an increasing sequence of eigenvalues μₙ(λ) for the nonlinear boundary value problem ⎧ in Ω, ⎨ ⎩ on crtial ∂Ω and we show that the first one μ₁(λ) is simple and isolated; we also prove some results about variations of the density ϱ and the continuity with respect to the parameter λ.
On the Principal Eigenvalue of a Second Order Linear Elliptic Problem with an Indefinite Weight Function.
On the principal eigenvalue of elliptic operators in and applications
Two generalizations of the notion of principal eigenvalue for elliptic operators in are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and “limit periodic” operators. These results apply to questions of existence and uniqueness for some semilinear problems in the whole space. We also indicate several outstanding open problems and formulate some conjectures.
On the Principal Eigenvalues of Indefinite Elliptic Problems.
On the quasi-classical asymptotics of the forward scattering amplitude and of the total scattering cross-section
On the quasi-classical limit of the total scattering cross-section in nonrelativistic quantum mechanics