2006/08/29 by Pinchover, Yehuda, Wolansky, Gershon, Zelig, Daphne
#34B10 #34L15 #35J10 #35P15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.math/0608716
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.