Schrödinger operator on homogeneous metric trees: spectrum in gaps

Details Schrödinger operator on homogeneous metric trees: spectrum in gaps Schrödinger operator on homogeneous metric trees: spectrum in gaps

2001-09-04

arxiv.org/abs/math/0109016v1

Mathematics

Spectral Theory

AMS LaTex file, 47 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Schrödinger operator on homogeneous metric trees: spectrum in gaps does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Schrödinger operator on homogeneous metric trees: spectrum in gaps, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Schrödinger operator on homogeneous metric trees: spectrum in gaps will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-270412