Spectral properties of Schrodinger operators defined on N-dimensional infinite trees

Details Spectral properties of Schrodinger operators defined on N-dimensional infinite trees Spectral properties of Schrodinger operators defined on N-dimensional infinite trees

2006-08-29

arxiv.org/abs/math/0608716v1

Mathematics

Spectral Theory

34 pages, 4 figures

Scientific paper

We study the discreteness of the spectrum of Schrodinger operators which are defined on N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their eigenvalues using operators on a one-dimensional tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N-dimensional tree. We show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero. Moreover, the projections to the one-dimensional tree of eigenfunctions of the N-dimensional Laplace operator converge to the corresponding eigenfunctions of the one-dimensional limit operator.

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