Continuity of the integral kernel of Schrodinger propagator

In this paper, we prove that the integral kernel of e(-it(-Delta+V)) is bounded and jointly continuous in (t, x, y) for a class of potentials which need not to be smooth. This result gives a partial affirmative answer for Simon's conjecture ([21, p. 482]). As a byproduct, we prove a short time asymptotic estimate for the integral kernel of e(-it(Delta+V)). Moreover, when V belongs to Schwartz function classes, we can write e(-it((-Delta)lambda/2) +V) as Fourier integral operators and give expansions of the symbols. (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this paper, the boundedness and joint continuity of the integral kernel of e(-it(-Delta+V)) is proven for a class of potentials that do not need to be smooth. This result partially affirms Simon's conjecture. A short time asymptotic estimate for the integral kernel of e(-it(Delta+V)) is also provided. Moreover, when V belongs to Schwartz function classes, e(-it((-Delta)lambda/2) +V) can be expressed as Fourier integral operators and expansions of the symbols are given.