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Absolute continuity for Jacobi matrices with power-like weights

Wojciech Motyka (2007)

Colloquium Mathematicae

This work deals with a class of Jacobi matrices with power-like weights. The main theme is spectral analysis of matrices with zero diagonal and weights λ : = n α ( 1 + Δ ) where α ∈ (0,1]. Asymptotic formulas for generalized eigenvectors are given and absolute continuity of the matrices considered is proved. The last section is devoted to spectral analysis of Jacobi matrices with qₙ = n + 1 + (-1)ⁿ and λ = ( q n - 1 q ) .

Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian

Soeren Fournais, Bernard Helffer (2006)

Annales de l’institut Fourier

Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, many papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schrödinger operator with magnetic field have appeared recently. Here we would like to mention the works by Bernoff-Sternberg, Lu-Pan, Del Pino-Felmer-Sternberg and Helffer-Morame and also Bauman-Phillips-Tang for the case of a disc. In the present paper we settle one important...

Approximation of eigenvalues for unbounded Jacobi matrices using finite submatrices

Anne Monvel, Lech Zielinski (2014)

Open Mathematics

We consider an infinite Jacobi matrix with off-diagonal entries dominated by the diagonal entries going to infinity. The corresponding self-adjoint operator J has discrete spectrum and our purpose is to present results on the approximation of eigenvalues of J by eigenvalues of its finite submatrices.

Bound states of a converging quantum waveguide

Giuseppe Cardone, Sergei A. Nazarov, Keijo Ruotsalainen (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider a two-dimensional quantum waveguide composed of two semi-strips of width 1 and 1 − ε, where ε > 0 is a small real parameter, i.e. the waveguide is gently converging. The width of the junction zone for the semi-strips is 1 + O(√ε). We will present a sufficient condition for the existence of a weakly coupled bound state below π2, the lower bound of the continuous spectrum. This eigenvalue in the discrete spectrum is unique and its asymptotics is constructed and justified when ε → 0+....

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