V. S. Rabinovich

Pseudodifferential Operators with Operator Valued Symbols. Fredholm Theory and Exponential Estimates of Solutions

abstract:
We consider a class of pseudodifferential operators with operator-valued symbols $a=a(x,\xi )$ having power growth with respect to the variables $x$ and $\xi$. Moreover we consider the symbols analytically extended with respect to $\xi$ onto a tube domain in $\mathbb{C}^{n}$ with a base being a ball in $\mathbb{R}^{n}$ with a radius depending on the variable $x$.

The main results of the paper are the Fredholm theory of pseudodifferential operators with operator valued symbols and exponential estimates at infinity of solutions of pseudodifferential equations $Op(a)u=f$.

We apply these results to Schrödinger operators with operator-valued potentials and to the spectral properties of Schrödinger operators in quantum waveguides.

Mathematics Subject Classification: 35Sxx, 58Jxx, 81Q10

Key words and phrases: Pseudodifferential operators with operator-valued symbols, Fredholmness, exponential estimates of solutions, quantum waveguides