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Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains. (English) Zbl 0951.35033
The authors investigate the Schrödinger operator on some connected infinite domain \(\Omega\subset \mathbb{R}^d\) with a smooth boundary and with Dirichlet boundary conditions on \(\partial\Omega\). The first main result shows that certain upper bounds on the rate of growth of \(L^r\) norms of generalized eigenfunctions over expanding balls imply certain minimal singularity of the spectral measures. From this follow new criteria for the existence of the absolutely continuous spectrum or the singular continuous spectrum of given dimensional characteristics.
The second major result establishes a fundamental relation between generalized eigenfunctions and the behavior of the time-averaged moments of the position operator \(X\) under the Schrödinger evolution.

MSC:
35J10 Schrödinger operator, Schrödinger equation
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35P05 General topics in linear spectral theory for PDEs
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