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On the local smoothing for the Schrödinger equation

Authors: Luis Vega and Nicola Visciglia
Journal: Proc. Amer. Math. Soc. 135 (2007), 119-128
MSC (2000): Primary 35-xx
Published electronically: June 28, 2006
MathSciNet review: 2280200
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Abstract: We prove a family of identities that involve the solution $ u$ to the following Cauchy problem:

$\displaystyle {\bf i} \partial_t u + \Delta u=0, u(0)=f(x), (t, x)\in {\mathbf R}_t\times {\mathbf R}^n_x, $

and the $ \dot H^\frac 12({\mathbf R}^n)$-norm of the initial datum $ f$. As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate proved by Constantin and Saut (1989), Sjölin (1987) and Vega (1988) and a uniqueness criterion for the solutions to the Schrödinger equation.

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Additional Information

Luis Vega
Affiliation: Departamento de Matemáticas, Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain

Nicola Visciglia
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy

Received by editor(s): July 21, 2005
Published electronically: June 28, 2006
Additional Notes: This research was supported by HYKE (HPRN-CT-2002-00282). The first author was also supported by a MAC grant (MTM 2004-03029) and the second author by an INDAM (Istituto Nazionale di Alta Matematica) fellowship
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2006 American Mathematical Society