Construction of the Essential Spectrum for a Multidimensional Non-self-adjoint Schrodinger Operator via the Spectra of Operators with Periodic Potentials, I

摘要

We describe the essential spectrum α_e(H) of a multidimensional Schrodinger operator H with a complex-valued potential V(x) in terms of a family f Schrodinger operators {H~y}_(y∈R~m) with periodic potentials V_y(x) which approximate the potential V(x) at infinity in a sense. Under some conditions we prove that the set α_e(H) coincides with the set Γ{V_y} of such points λ ∈ C for which the family of norms {‖R_λ(H~y)‖}_(y∈IR) is unbounded at infinity. Sometimes the set Γ{V_y} coincides with the set Σ{V_y} of limit points of the spectra α(H~y) of the operators H~y for |y| → ∞. In this case we call the family {V_y(x)}_(x∈R~m) spectrally non-degenerate. We find some conditions of the spectral non-degeneracy. To this end we carry out an estimation of resolvents of the operators H~y with the help of generalized perturbation determinants for the corresponding cyclic boundary problems on the lattices of the periodicity.