Propagation of Exponential Phase Space Singularities for Schrödinger Equations with Quadratic Hamiltonians

Abstract

We study propagation of phase space singularities for the initial value Cauchy problem for a class of Schrödinger equations. The Hamiltonian is the Weyl quantization of a quadratic form whose real part is non-negative. The equations are studied in the framework of projective Gelfand-Shilov spaces and their distribution duals. The corresponding notion of singularities is called the Gelfand-Shilov wave front set and means the lack of exponential decay in open cones in phase space. Our main result shows that the propagation is determined by the singular space of the quadratic form, just as in the framework of the Schwartz space, where the notion of singularity is the Gabor wave front set.

Pseudo-differential operators in a Gelfand-Shilov setting

“…The assertion (1) in the previous theorem is essentially a special case of Theorem 32 in [22]. Independently by the authors, the case (2) was recently deduced by Carypis and Wahlberg in [5], using similar arguments as in [22]. In order to be self-contained we present a complete but different proof.…”

Schrödinger-type equations in Gelfand-Shilov spaces

“…[8,13,14]. More recently, some papers treating Schrödinger equations with real valued coefficients in these spaces appeared, see [16,21]. In the case of lower order terms with complex coefficients, the choice of Cauchy data in Gelfand-Shilov spaces intersects another field of investigation, namely the study of the smoothing effect produced by exponential decay of the data on the Gevrey regularity of the solution to the Schrödinger equation, see [28,31,34].…”

Pseudo-differential calculus in a Bargmann setting

“…The Gelfand-Shilov distribution spaces S 1 s pR d q and Σ 1 s pR d q are the dual spaces of S s pR d q and Σ s pR d q, respectively. 6 We have…”

“…Here we note that the operator e ixAD ξ ,Dxy is homeomorphic on Σ 1 pR 2d q and its dual (cf. [5,6,29]). For modulation spaces we have the following subresult of Proposition 2.8 in [28].…”

“…Then the condition on a in Theorem 3.8 is that }V φ 0 a¨ω 0 } L p,q ă 8. In view of [5,6,29] and Proposition 1.13 (2), the previous condition is the same as }V φ A a¨ω 0 } L p,q ă 8 because ω 0 is moderate. We observe that all weights in Theorem 3.8 are moderate, while there are no such assumptions or other restrictions on the involved weight functions in Theorem 3.3.…”