Generalized Metaplectic Operators and the Schr\"odinger Equation with a Potential in the Sj\"ostrand Class

TL;DR

This paper extends the concept of metaplectic operators to include pseudodifferential operators in the Sj"ostrand class, showing they form a decay-preserving algebra and that solutions to the Schr"odinger equation with such potentials maintain phase-space concentration.

Contribution

It introduces a class of generalized metaplectic operators generated by Hamiltonians with Sj"ostrand class potentials, linking them to Fourier integral operators and phase-space preservation.

Findings

The algebra of generalized metaplectic operators exhibits decay properties similar to classical metaplectic operators.

The evolution group of the Schr"odinger equation with Sj"ostrand class potentials consists of these generalized operators.

Solutions preserve modulation space norms, indicating phase-space concentration is maintained.

Abstract

It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential opertators in a Sj\"ostrand class enjoys the same decay properties. We study the behavior of these generalized metaplectic operators and represent them by Fourier integral operators. Our main result shows that the one-parameter group generated by a Hamiltonian operator with a potential in the Sj\"ostrand class consists of generalized metaplectic operators. As a consequence, the Schr\"odinger equation preserves the phase-space concentration, as measured by modulation space norms.