Aspects of Operator Theory in Quantum Mechanics: And a Study of Magnetic Many-body Quantum Systems in the Hartree-Fock Approximation

Thorn, Mikkel Hviid

4. term

Mathematics, Master

2025

2025-05-28

100 pages

This thesis explores aspects of operator theory relevant to quantum mechanics. We primarily focus on three different subjects: Self-adjoint unbounded operators, pseudodifferential calculus, and simple quantum systems of particles in Euclidean space. First we present some of the theory of self-adjoint unbounded operator covering basic definitions, variational operators, and spectral theory. While introducing the basic definitions we prove standard results such as criteria for self-adjointness and the KatoRellich Theorem. For variational operators we also include Friedrich’s Extension. As for spectral theory we prove the spectral theorem for bounded and unbounded self-adjoint operators, Stone’s Formula, and Helffer-Sjöstrand Formula. Secondly we study tempered distributions and pseudo-differential operators. The Schwartz space and space of tempered distributions are introduced, and results such as reflexivity of the spaces, Schwartz Kernel Theorem, and the Structure Theorem are proven. Afterwards we deal with quite general quantization schemes for pseudodifferential operators, mostly working with Hörmander classes of smooth symbols with decay controlled by a tempered weight. For these we establish a Calderón-Vaillancourt Theorem, a Moyal product, and for certain quantizations a Beal’s Commutator Criterion. To prove all these results we make use of modulated tight Gabor frame, and we characterize the different spaces by their coordinates or matrices in this frame. Lastly we analyze some one particle systems directly and a many-body particle system in the Hartree-Fock approximation, both under the influence of a regular magnetic field. As a start we omit the magnetic field and give classical results on the free Schrödinger operator and harmonic oscillator. Then we give elementary results on free magnetic Schrödinger operators and find the Landau spectrum. Afterwards we turn our attention toward the Hartree-Fock approximation of a many-body particle system under the influence of a constant magnetic field. Essentially the many-body particle system is approximated by a single particle Schrödinger operator with an added potential representing the particle cloud. This potential satisfies a fix-point equation, which we solve.

Operators ; Unbounded Operators ; Self-adjointness ; Spectral Theory ; Quantization ; Pseudo-differential Calculus ; Pseudo-differential Operator ; Moyal Product ; Schrödinger Operator ; Magnetic Schrödinger Operator ; Hartree-Fock

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