Uniform Spectral Asymptotics for a Schrödinger Operator on a Segment with Delta-Interaction

We consider a Schrödinger operator on the segment \((0,1)\) subject to the Dirichlet condition and perturb it by a delta-potential concentrated at the point \(x= \varepsilon \), where \( \varepsilon \) is a small positive parameter. We show that the perturbed operator converges to the unperturbed one in the norm resolvent sense and this also implies the convergence of the spectrum. However, the latter convergence is true only inside each compact set on the complex plane and it does not characterize the behavior of the total ensemble of the eigenvalues under the perturbation. Our main result is the spectral asymptotics for the eigenvalues of the perturbed operator with an estimate for the error term uniform in the small parameter. This asymptotics involves an additional nonstandard term, which allows us to describe a global behavior of the total ensemble of the eigenvalues under the perturbation.

DOI 10.1134/S1061920824020018