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Divergence-type Operators: Spectral Theory and Spacetime Estimates

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Evolution Equations of Hyperbolic and Schrödinger Type

Part of the book series: Progress in Mathematics ((PM,volume 301))

Abstract

The paper is concerned with various aspects of the spectral structure of the operator \( {H = - \Sigma^{n}_{j,k=1}\partial_{x_j}a_{j,k}(x)\partial{x_k}}.\) It is assumed to be formally self-adjoint in \( {L^2(\mathbb{R}^n),n \geq 2}.\) The real coefficients \( {a_{j,k}(x)= a_{k,j}(x)}\) are assumed to be bounded and H is assumed to be uniformly elliptic and to coincide with −∆ outside of a ball. A Limiting Absorption Principle (LAP) is proved in the framework of weighted Sobolev spaces. It is then used for (i) A general eigenfunction expansion theorem and (ii) Global spacetime estimates for the associated (inhomogeneous) generalized wave equation.

Mathematics Subject Classification. Primary 47A10; Secondary 35P05, 35L15, 47F05.

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  1. Institute of Mathematics, Hebrew University, Jerusalem, 91904, Israel

    Matania Ben-Artzi

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  1. Matania Ben-Artzi
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  1. , Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, United Kingdom

    Michael Ruzhansky

  2. , Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

    Mitsuru Sugimoto

  3. , Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany

    Jens Wirth

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Ben-Artzi, M. (2012). Divergence-type Operators: Spectral Theory and Spacetime Estimates. In: Ruzhansky, M., Sugimoto, M., Wirth, J. (eds) Evolution Equations of Hyperbolic and Schrödinger Type. Progress in Mathematics, vol 301. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0454-7_1

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