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Schr\"odinger Operator on Homogeneous Metric Trees: Spectrum in Gaps by Sobolev, A. V.; Solomyak, M. is a scholarly article available to read on EtoBox.

What is Schr\"odinger Operator on Homogeneous Metric Trees: Spectrum in Gaps about?

The paper studies the spectral properties of the Schr\"odinger operator $A_{gV} = A_0 + gV$ on a homogeneous rooted metric tree, with a decaying real-valued potential $V$ and a coupling constant $g\ge 0$. The spectrum of the free Laplacian $A_0 = -\Delta$ has a band-gap structure with a single eigenvalue of infinite multiplicity in the middle of each finite gap. The perturbation $gV$ gives rise to extra eigenvalues in the gaps. These eigenvalues are monotone functions of $g$ if the potential $V$ has a fixed sign. Assuming that the latter condition is satisfied and that $V$ is symmetric, i.e. depends on the distance to the root of the tree, we carry out a detailed asymptotic analysis of the counting function of the discrete eigenvalues in the limit $g\to\infty$. Depending on the sign and decay of $V$, this asymptotics is either of the Weyl type or is completely determined by the behaviour of $V$ at infinity.

Author

Sobolev, A. V.; Solomyak, M.

Published

2001

Language

EN