Narrow Results

This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schrödinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an asymptotic expansion of the number of eigenvalues below the essential spectrum (Weyl-type asymptotics). The methods of proof rely on results concerning the asymptotic behavior of the first eigenvalue obtained in a previous work [10].

The aim of the paper is to derive spectral estimates into several classes of magnetic systems. They include three-dimensional regions with Dirichlet boundary as well as a particle in ${ℝ}^3$ confined by a local change of the magnetic field. We establish two-dimensional Berezin–Li–Yau and Lieb–Thirring-type bounds in the presence of magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians.

We study the spectrum of the spin-boson Hamiltonian with two bosons for arbitrary coupling $\alpha >0$ in the case when the dispersion relation is a bounded function. We derive an explicit description of the essential spectrum which consists of the so-called two- and three-particle branches that can be separated by a gap if the coupling is sufficiently large. It turns out, that depending on the location of the coupling constant and the energy level of the atom (w.r.t. certain constants depending on the maximal and the minimal values of the boson energy) as well as the validity or the violation of the infrared regularity type conditions, the essential spectrum is either a single finite interval or a disjoint union of at most six finite intervals. The corresponding critical values of the coupling constant are determined explicitly and the asymptotic lengths of the possible gaps are given when $\alpha$ approaches to the respective critical value. Under minimal smoothness and regularity conditions on the boson dispersion relation and the coupling function, we show that discrete eigenvalues can never accumulate at the edges of the two-particle branch. Moreover, we show the absence of the discrete eigenvalue accumulation at the edges of the three-particle branch in the infrared regular case.

We present a review of recent results about a class of Schrödinger operators usually called soft quantum waveguides, with the focus on relations between their spectral properties and the geometry of the confinement. We also mention a number of open problems in this area.

We analyze the (discrete) spectrum of the semirelativistic "spinless-Salpeter" Hamiltonian

We generalize the semiclassical Bohr–Sommerfeld quantization rule to an exact, implicit spectral formula for linear, generalized Schrödinger equations admitting a discrete spectrum. Special cases include the position-dependent mass Schrödinger equation or the Schrödinger equation for weighted energy. Requiring knowledge of the potential and the solution associated with the lowest spectral value, our formula predicts the complete spectrum in its exact form.

The spectrum of masses of the colorless states of DW Hamiltonian operator of quantum SU(2) Yang–Mills field theory on ${ℝ}^D$ obtained via the precanonical quantization is shown to be purely discrete and bounded from below. The scale of the mass gap is estimated to be of the order of magnitude of the scale of the ultra-violet parameter $\varkappa$ which is introduced by precanonical quantization on dimensional grounds.

The concept of Isogeometric Analysis recently introduced by Hughes et al. [2005] is herein applied to the study of structural vibrations. In this framework, the favourable behaviour of the method is verified and compared with some classical finite element results. Numerical experiments are shown for structural one-, two- and three-dimensional problems in order to test the performance of this promising technique in the field of the analysis of natural frequencies and modes.

We provide a generalization and new proofs of the formulas of Collins et al. for the spectrum of polynomials in cyclic monotone elements. This is applied to Random Matrices with discrete spectrum.

A discrete eigenvalue E_n of a Schrödinger operator H = −Δ + νf(r) is given, as a function F_n(ν) of the coupling parameter ν ≥ ν_c. It is shown how the potential shape f(x) can be reconstructed from F_n(ν): A constructive inversion algorithm and a functional inversion sequence are both discussed.