The Morse and Maslov indices for Schrödinger operators
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We study the spectrum of Schrödinger operators with matrixvalued potentials, utilizing tools from infinite-dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrödinger operators with periodic potentials. For quasi-convex domains in Rn, we recast the results, connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.
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- [DK]Ju. Daleckii and M. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI, 1974.Google Scholar
- [GM08]F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, Amer. Math. Soc., Providence, RI, 2008, pp. 105–173.CrossRefGoogle Scholar
- [PW]A. Portaluri and N. Waterstraat, A Morse-Smale index theorem for indefinite elliptic systems and bifurcation, J. Differential Equations 258, 1715–1748.Google Scholar