Analytical Perturbation Theory for Linear Operators

Student thesis: Master Thesis and HD Thesis

  • Mette Kristensen
4. term, Mathematics, Master (Master Programme)
In quantum mechanics the energy of a system is described by means of the Hamiltonian of the particular system. A Hamiltonian is a linear self-adjoint operator and therefore the spectrum of the Hamiltonian is real. The physical interpretation of the spectrum of the Hamiltonian is that the spectrum is the energy levels of the system. A Hamiltonian can be a compact operator, but it can also be an unbounded operator. If it is a compact operator the spectrum consists only of eigenvalues and therefore the physical interpretation is that the system only have discrete energy levels and this correspond to a system only containing bounded states. If the operator is bounded the spectrum can consist of both discrete eigenvalues and a continuous spectrum which correspond to a system of both bounded states and scattered states. In order to find the eigenvalues and the eigenstates of the system it is necessary to solve the eigenvalue equation. This is often impossible to do explicitly for a Hamiltonian that reflects reality, and this is the reason why perturbation theory is used to approximate the eigenvalues and eigenstates. The main idea in perturbation theory is to observe how the eigenvalues and eigenstates of a known operator changes when a small potential in the form of a self-adjoint operator is added to the original operator. In this report only compact operators are studied, and it will be shown that when the solvable Hamiltonian has a nondegenerate eigenvalue the perturbed Hamiltonian has nondegenerate eigenvalue close to the known eigenvalue if the perturbation is small enough. The perturbed Hamiltonian will be written as a sum of a solvable Hamiltonian, and a perturbation in the form of a self-adjoint compact operator. In the report it will be proved that a nondegenerate eigenvalue of the perturbed Hamiltonian is analytic an analytic function. This means that the eigenvalue can be written as a Taylor series when the perturbation is small enough and through the report the first five coefficients of the Taylor series will be determined. This will be done by means of the Feshbach Formula and therefore this formula will be stated and proved.
Publication date2009
Number of pages71
Publishing institutionInstitut for Matematiske Fag
ID: 17644689