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On the absence of eigenvalues of a periodic matrix Schrödinger operator in a layer. (English) Zbl 1186.81059
Summary: We consider the matrix Schrödinger operator $$-D+V(x)$$ in the layer $$\mathbf R^{d-1}\times (0,T)$$, $$d\geq 2$$. The potential $$V$$ is assumed to be periodic along the layer. We supply the operator with appropriate boundary conditions. The Dirichlet and Neumann boundary conditions, the third boundary condition with periodic coefficients, and quasiperiodic conditions are allowed. The spectra of the corresponding operators are shown to be free of eigenvalues. In the self-adjoint case, the spectrum is absolutely continuous.

##### MSC:
 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35P05 General topics in linear spectral theory for PDEs 47A10 Spectrum, resolvent 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)