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Essential m-sectoriality and essential spectrum of the Schrödinger operators with rapidly oscillating complex-valued potentials. (English) Zbl 1344.35016

Author’s abstract: Schrödinger operators \(T_0 = -\Delta + q(x)\) with rapidly oscillating complex-valued potentials \(q(x)\) are considered. Each of such operators is sectorial and hence has Friedrichs extension. We prove that \(T_0\) is essentially \(m\)-sectorial in the sense that the closure of \(T_0\) coincides with its Friedrichs extension \(T\). In particular, \(T_0\) is essentially self-adjoint if the rapidly oscillating potential \(q(x)\) is realvalued. Further, we prove \(\sigma_{\mathrm{ess}} (T) = [0, \infty)\) under somewhat stricter condition on the potentials \(q(x)\).

MSC:

35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs
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