Fundamentals of scattering theory and resonances in quantum mechanics

Abstract

Peter D. HislopDepartment of Mathematics,University of Kentucky,Lexington, Kentucky 40506-0027, USAemail: hislop@ms.uky.eduABSTRACTWe present the basics of two-body quantum-mechanical scattering theory and the the-ory of quantum resonances. The wave operators and S-matrix are constructed forsmooth, compactly-supported potential perturbations of the Laplacian. The meromor-phic continuation of the cut-off resolvent is proved for the same family of Schr¨odingeroperators. Quantum resonances are defined as the poles of the meromorphic con-tinuation of the cut-off resolvent. These are shown to be the same as the poles of themeromorphically continued S-matrix. The basic problems of the existence of resonancesand estimates on the resonance counting function are described and recent results arepresented.RESUMENPresentamos los conceptos ba´sicos de la teor´ia de dispersi´on cuanto-mec´anica de doscuerpos y la teor´ia de resonancias cu´anticas. El operador de ondas y la matriz Sseconstruyen para perturbaciones del potencial suaves y de soporte compacto del Lapla-ciano. La continuaci´on meromorfica de la resolvente truncada se prueba para la mismafamilia de operadores de Schrdinger. Las resonancias cu´anticas se definen como los po-los de la continuaci´on meromorifca de la resolvente truncada. Se muestra que ellas sonlas mismas que los polos de la matriz Scontinuada meromorficamente. Los problemasba´sicos de la existencia de resonancias y las estimaciones de la funci´on de conteo de laresonancia se describen y resultados recientes se presentan.Keywords and Phrases: Scattering theory, resonances, Schr¨odinger equation, wave operators,quantum mechanics2010 AMS Mathematics Subject Classification: 35J10, 35P25, 35Q40,47A40,47A55, 81U05,81U20