On the spectral instability of the Sturm-Liouville operator with a complex potential
For selfadjoint operators, the spectrum of a perturbation can be estimated by relative bounds of the perturbations. However, this is no longer true for non-selfadjoint operators, and it is therefore interesting to find necessary and sufficient conditions to preserve spectral asymptotics, or to find counterexamples. In this paper, the author is concerned with the case that the spectrum is located around a ray. In Theorem 1, sufficient conditions are given to ensure that if the spectra of the operator \(T_0\) and its perturbation \(T_0+V\) both are located around a ray, say \(\arg \lambda =\varphi_0 \) and \(\arg \lambda = \psi _0\), respectively, then \(\psi _0 =\varphi_0 \). This result is then applied to the anharmonic oscillator \(H_\theta \) in \(L^2(0,\infty )\) given by \(-y''+e^{i\theta }x^\alpha y\) (\(\alpha >0\), \(0<|\theta |<\pi\)) and the boundary condition \(y(0)=0\). \(H_\theta \) is perturbed by a potential of the form \(V_0+V\), where the assumptions on \(V_0\) imply that \(H_\theta \) and \(H_\theta +V_0\) have the same spectral asymptotics. However, if \(V\neq0\), then \(V\) is assumed to be of the form \(V(x)=(b-x)^mW(x)\), where \(m\geq0\), supp\,\(W=[0,b]\) and \(W(b-0)\neq0\) exists, and it follows that \(H_\theta +V_0+V\) has two eigenvalue sequences, one with the same asymptotics as \(H_\theta \) and the second one having the same asymptotics as the squares of the eigenvalues of the Regge problem on \([0,b]\). This shows that a small change \(W\) of the potential does not lead to a small change in the spectrum.
scientific article; zbMATH DE number 5611056
scientific article