The Morse and Maslov indices for Schrödinger operators
- 62 Downloads
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.
Unable to display preview. Download preview PDF.
- [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