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Kato Smoothing Effect for Schrödinger Operator

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Studies in Phase Space Analysis with Applications to PDEs

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 84))

Abstract

In this paper we give a survey of results on smoothing effect for Schrödinger operators. Several phenomena can be called smoothing effect. Here we limit us on the Kato or one half smoothing effect. We shall give the new and old results in different contexts, global in time and local in time in all spaces, in exterior domain. In this last case we shall give the recent result where the geometrical control condition is not satisfied and replaced by a damping condition. This kind of assumption originates in the control theory. We shall give also some references on the other smoothing effect, Strichartz estimate and related problem for wave equation, and KdV equation without exhaustiveness.

2010 Mathematics Subject Classification: 35Q41, 35B65.

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Notes

  1. 1.

    For a function on q(x, ξ), there are several manner to associate an operator. For instance if q(x, ξ) = a(x)ξ, the classical quantification of q is a(x)D j . Other possibility is D j a(x). The Weyl quantification is the mix of the previous one, i.e., \((1/2)(a(x)D_{j} + D_{j}a(x))\). For a general symbol, the formula is more complicated; see Hrmander [32, Chap. 18, 5].

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  1. Laboratoire de Mathématiques, Université de Versailles Saint-Quentin, 45, avenue des États-Unis, F-78035, Versailles Cedex, France

    Luc Robbiano

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  1. Luc Robbiano
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  1. , Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, Bologna, I-40126, Italy

    Massimo Cicognani

  2. , Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, Pisa, I-56127, Italy

    Ferruccio Colombini

  3. , Dipartimento di Matematica e Geoscienze, Università di Trieste, Via Valerio, 12/1 12/1, Trieste, I-34127, Italy

    Daniele Del Santo

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Robbiano, L. (2013). Kato Smoothing Effect for Schrödinger Operator. In: Cicognani, M., Colombini, F., Del Santo, D. (eds) Studies in Phase Space Analysis with Applications to PDEs. Progress in Nonlinear Differential Equations and Their Applications, vol 84. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6348-1_16

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