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On complex perturbations of infinite band Schrödinger operators. (English) Zbl 1363.34317

Let \(H_0=-\frac{d^2}{dz^2}+V_0\) be an infinite band Schrödinger operator on \(L^2(\mathbb R)\) with a real valued nonnegative potential \(V_0\in L^\infty (\mathbb R)\). Denote by \(I\) the essential spectrum of \(H_0\). The authors study complex perturbations \(H=H_0+V\) and obtain the Lieb-Thirring type inequalities for the rate of convergence of eigenvalues of \(H\) to the set \(I\). It is assumed that \(V\in L^p(\mathbb R)\) with \(p\geq 2\). If \(\operatorname{Re} V\geq 0\), it is sufficient to assume \(p>1\).

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47A55 Perturbation theory of linear operators
47A75 Eigenvalue problems for linear operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators