Classical proofs of Kato type smoothing estimates for the Schrödinger equation with quadratic potential in \(\mathbb R^{n+1}\) with application. (English) Zbl 1240.35429

Chen, Xuwen

Differ. Integral Equ. 24, No. 3-4, 209-230 (2011).

The author studies the Schrödinger equation with quadratic potential which models behaviour of bosons in the magnetic field. He shows the Kato-type smoothing estimates without the use of pseudodifferential technique. It appears to be equivalent to the proof of uniform \(L^2(\mathbb {R}^N)\) boundedness result for a family of singularized Hermite projection kernels. As a consequence, the author also proves the \(\mathbb {R}^9\) collapsing variable-type Strichartz estimate.
The main tool of the proof are fine properties of Hermite functions.

Reviewer: Milan Pokorný (Praha)

MSC:

35Q41	Time-dependent Schrödinger equations and Dirac equations
35B45	A priori estimates in context of PDEs
35A23	Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
42C10	Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

Schrödinger equation; Hermite function; Strichartz estimate; Hermite projection kernels; Morawetz inequality