**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