An Agmon estimate for Schr\"odinger operators on Graphs

DOI10.1007/S11005-023-01635-5arXiv2206.09521MaRDI QIDQ6402541

Stefan Steinerberger

Publication date: 19 June 2022

Abstract: The Agmon estimate shows that eigenfunctions of Schr"odinger operators, $-Deltaphi+Vphi=Ephi$, decay exponentially in the `classically forbidden' region where the potential exceeds the energy level $leftx:V(x)>Eight$. Moreover, the size of $|phi(x)|$ is bounded in terms of a weighted (Agmon) distance between $x$ and the allowed region. We derive such a statement on graphs when $-Delta$ is replaced by the Graph Laplacian $L=D-A$: we identify an explicit Agmon metric and prove a pointwise decay estimate in terms of the Agmon distance.

Mathematics Subject Classification ID

Schrödinger operator, Schrödinger equation (35J10) PDEs on graphs and networks (ramified or polygonal spaces) (35R02)

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