Schr\"odinger operators with complex sparse potentials

TL;DR

This paper provides new bounds and characterizations for Schr"odinger operators with complex, sparse potentials, revealing eigenvalue behaviors and extending classical theorems to non-selfadjoint, multidimensional cases.

Contribution

It introduces quantitative bounds for eigenvalues, improves existing bounds for sparse potentials, and generalizes spectral theorems to non-selfadjoint multidimensional operators.

Findings

Eigenvalues can accumulate at every point of the essential spectrum.

Bounds for eigenvalues are sharp for certain sparse potentials.

One-dimensional imaginary step potentials have many eigenvalues, similar to resonances.

Abstract

We establish quantitative upper and lower bounds for Schr\"odinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S.\ Boegli (Comm. Math. Phys., 2017, 352, 629-639), of a Schr\"odinger operator with eigenvalues accumulating to every point of the essential spectrum. The second result shows that the eigenvalue bounds of Frank (Bull. Lond. Math. Soc., 2011, 43, 745-750 and Trans. Amer. Math. Soc., 2018, 370, 219-240) can be improved for sparse potentials. The third result generalizes a theorem of Klaus (Ann. Inst. H. Poincar\'e Sect. A (N.S.), 1983, 38, 7-13) on the characterization of the essential spectrum to the multidimensional non-selfadjoint case. The fourth result shows that, in one dimension, the purely imaginary (non-sparse) step potential has unexpectedly many eigenvalues, comparable to the number of resonances. Our examples show that several known upper bounds are sharp.