Fundamentals of scattering theory and resonances in quantum mechanics

• Peter D Hislop ^[1]
  1. [1] University of Kentucky

     University of Kentucky

     Estados Unidos

     
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 14, Nº. 3, 2012, págs. 1-39
• Idioma: inglés
• DOI: 10.4067/S0719-06462012000300001
• Enlaces
  • Texto completo
• Resumen
  • español

    Presentamos los conceptos básicos de la teoría de dispersión cuanto-mecánica de dos cuerpos y la teoría de resonancias cuánticas. El operador de ondas y la matriz S se construyen para perturbaciones del potencial suaves y de soporte compacto del Laplaciano. La continuación meromórfica de la resolvente truncada se prueba para la misma familia de operadores de Schrödinger. Las resonancias cuánticas se definen como los polos de la continuación meromórifca de la resolvente truncada. Se muestra que ellas son las mismas que los polos de la matriz S continuada meromórficamente. Los problemas básicos de la existencia de resonancias y las estimaciones de la función de conteo de la resonancia se describen y resultados recientes se presentan.

  • English

    ABSTRACT We present the basics of two-body quantum-mechanical scattering theory and the theory of quantum resonances. The wave operators and S-matrix are constructed for smooth, compactly-supported potential perturbations of the Laplacian. The meromorphic continuation of the cut-off resolvent is proved for the same family of Schrödinger operators. Quantum resonances are defined as the poles of the meromorphic continuation of the cut-off resolvent. These are shown to be the same as the poles of the meromorphically continued S-matrix. The basic problems of the existence of resonances and estimates on the resonance counting function are described and recent results are presented.

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