Mathematics

In this paper, we give a new characterization of generalized Browder's theorem by considering equality between the generalized Drazin-meromorphic Weyl spectrum and the generalized Drazin-meromorphic spectrum. Also, we generalize Cline's formula to the case of generalized Drazin-meromorphic invertibility under the assumption that A k B k A k = A k+1 for some positive integer k .

In this work, we present a new characterization of symmetric H + -tensors. It is known that a symmetric tensor is an H + -tensor if and only if it is a generalized diagonally dominant tensor with nonnegative diagonal elements. By exploring the diagonal dominance property, we derive new necessary and sufficient conditions for a symmetric tensor to be an H + -tensor. Based on these conditions, we propose a new method that allows to check if a tensor is a symmetric H + -tensor in polynomial time. In particular, this allows to efficiently compute the minimum H -eigenvalue of tensors in the related and important class of M -tensors. Furthermore, we show how this result can be used to approximately solve polynomial optimization problems.

We show, in the same vein of Simon's Wonderland Theorem, that, typically in Baire's sense, the rates with whom the solutions of the Schrödinger equation escape, in time average, from every finite-dimensional subspace, depend on subsequences of time going to infinite.

In this note the three dimensional Dirac operator A m with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that A m is self-adjoint in L 2 (Ω; C 4 ) for any open set Ω⊂ R 3 and its spectrum is described explicitly in terms of the spectrum of the Dirichlet Laplacian in Ω . In particular, whenever the spectrum of the Dirichlet Laplacian is purely discrete, then also the spectrum of A m consists of discrete eigenvalues that accumulate at ±∞ and one additional eigenvalue of infinite multiplicity.

Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature -1. For each n∈N , let X n be a random degree- n cover of X sampled uniformly from all degree- n Riemannian covering spaces of X . An eigenvalue of X or X n is an eigenvalue of the associated Laplacian operator Δ X or Δ X n . We say that an eigenvalue of X n is new if it occurs with greater multiplicity than in X . We prove that for any ϵ>0 , with probability tending to 1 as n→∞ , there are no new eigenvalues of X n below 3 16 −ϵ . We conjecture that the same result holds with 3 16 replaced by 1 4 .

We consider the case of scattering of several obstacles in R d for d≥2 . In this setting the absolutely continuous part of the Laplace operator Δ with Dirichlet boundary conditions and the free Laplace operator Δ 0 are unitarily equivalent. For suitable functions that decay sufficiently fast we have that the difference g(Δ)−g( Δ 0 ) is a trace-class operator and its trace is described by the Krein spectral shift function. In this paper we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles we consider the Laplace operators Δ 1 and Δ 2 obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then g(Δ)−g( Δ 1 )−g( Δ 2 )+g( Δ 0 ) is a trace class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman-Krein formula. In case g(x)= x 1 2 the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using non-rigorous path integral derivations and our formula provides both a rigorous justification as well as a generalisation.

We study the trace class perturbations of the whole-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we refine the Lieb--Thirring inequality due to Hansmann--Katriel. The spectral enclosure for such operators is also discussed.

In this paper we construct a Birkhoff normal form for a semiclassical magnetic Schr{ö}dinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even dimensional riemannian manifold M. We use this normal form to get an expansion of the first eigenvalues in powers of h^{1/2}, and semiclassical Weyl asymptotics for this operator.

For the almost Mathieu operator with a small coupling constant, for a series of spectral gaps, we describe the asymptotic locations of the gaps and get lower bounds for their lengths. The results are obtained by analysing a monodromy matrix.

We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem