Elliptic Z-Operators Associated with the Metaplectic Group

Abstract: In the paper, the ellipticity (Fredholm property) of operators of the form (Formula presented.) is investigated, where (Formula presented.) are pseudodifferential operators of Shubin type and (Formula presented.) is a metaplectic operator, in Sobolev-type spaces on (Formula presented.). Explicit conditions for the ellipticity of such operators are given for the case in which the symplectic matrix associated with the operator (Formula presented.) induces a topologically free action of the group (Formula presented.) on the sphere (Formula presented.). The results thus obtained are applied to the inverse problem for the Schrödinger equation describing the dynamics of two noninteracting quantum systems for which the values of given observations at the initial and final times are known. © 2021, Pleiades Publishing, Ltd.