The Morse and Maslov Indices for Schr\"odinger Operators

We study the spectrum of Schr\"odinger operators with matrix valued 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 Schr\"{o}dinger operators with quasi-periodic, Dirichlet and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for the multidimensional Schr\"odinger operators with periodic potentials. For quasi convex domains in $\mathbb{R}^n$ we recast the results connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.