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

Yehuda Pinchover·Gershon Wolansky·Daphne Zelig
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Abstract

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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Yehuda Pinchover, Gershon Wolansky, Daphne Zelig. 2006-08-29. Spectral properties of Schrodinger operators defined on N-dimensional infinite trees. https://arxiv.org/abs/math/0608716

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