On Semiclassical Operators: Semiclassical Analysis
Translated title
On Semiclassical Operators
Authors
Kjøller, Kristian Lundgaard ; Johannesen, Tróndur Tummasarson
Term
4. term
Education
Publication year
2018
Submitted on
2018-06-07
Pages
42
Abstract
Dette speciale undersøger centrale værktøjer i semiklassisk analyse, som forbinder klassisk og kvantemekanisk beskrivelse gennem en lille parameter h. Først giver vi en klar definition af anti-Wick-kvantisering for klassiske pseudodifferentialoperatorer og udvider den til det semiklassiske regime. Med dette viser vi Gårdings ulighed, en grundlæggende nedre grænse for sådanne operatorer. Dernæst udleder vi en formel for Fourier-transformen af e^{i c x^m}, når m er lige; for m > 2 involverer formlen generaliserede hypergeometriske funktioner. Til sidst studerer vi oscillerende integraler og deres adfærd, når h går mod 0. Det fører til stationær fase-asymptotik under forskellige forudsætninger, hvor vi blandt andet forsøger at anvende den nævnte Fourier-transform. I højere dimensioner kræver analysen brug af Morse-lemmaet, som forenkler fasen nær ikke-degenererede kritiske punkter.
This thesis explores key tools in semiclassical analysis, which links classical and quantum descriptions via a small parameter h. We first give a clear definition of anti-Wick quantization for classical pseudodifferential operators and extend it to the semiclassical setting. Using this framework, we prove Gårding’s inequality, a basic lower bound for such operators. We then derive a formula for the Fourier transform of e^{i c x^m} when m is even; for m > 2 the expression involves generalized hypergeometric functions. Finally, we study oscillatory integrals and how they behave as h tends to 0. This leads to stationary phase asymptotics under various conditions, where we also attempt to use the above Fourier transform. In higher dimensions, the analysis requires the Morse Lemma, which reduces the phase near nondegenerate critical points to a quadratic form.
[This abstract was generated with the help of AI]
Keywords
Analysis ; Matematik ; Mathematics ; Maths ; Semiclassical
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