# Spectral properties of Schrödinger operators on radial *N*-dimensional infinite trees

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## Abstract

We study the discreteness of the spectrum of Schrödinger operators which are defined on a class of radial *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.

## Keywords

Weight Function Laplace Operator Discrete Spectrum Lipschitz Domain Counting Function## References

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