Eingenvalue accumulation for the Schrödinger equation: a singular Sturm-Liouville problem in the spectral parameter

• Autor(a) principal: Borin, Daniel
• Data de Publicação: 2019
• Tipo de documento: Trabalho de conclusão de curso
• Idioma: eng
• Título da fonte: Repositório Institucional da UNESP
• Texto Completo: http://hdl.handle.net/11449/235239
• topic: Sturm-Liouville problem; Eigenvalue accumulation; Schrödinger equation; Problema de Sturm-Liouville; Acumulação dos autovalores; Equação de Schrödinger

In this work we will present the Sturm-Liouville classic theory for the regular problem, showing that several properties of the eigenvalues and eigenfunctions of this problem, for example, that the eigenvalues form an infinite increasing sequence and that the eigenfunctions forms an orthonormal basis for the CL2(r) [a;b] space. In this way, we define the singular Sturm-Liouville problem that it depends continuously of a parameter l and we prove that the eigenvalues accumulate into the endpoint n 2 LR under certain conditions. We also study the case where it non-accumulate. Lastly, we realize an application in Quantum Mechanics, where analyzing certain properties of the potential function in the system we may obtain information about the accumulation or non-accumulation of the energy.