Spectral, scattering, and regularity properties related to various functional and differential equations

Abstract

Waves occur in many real world phenomena and much of the scientific literature concerns the study of waves. Different types of waves are described by different wave equations, for example Schrödinger's equation describing the time evolution of the wave function in a quantum mechanical system, or the Euler-Bernoulli equation describing bending waves in beams. This dissertation mainly considers these two types of waves and related topics. However, the dissertation also has a secondary focus unrelated to the topic of wave equations, which is the study of random variables with stationary digits. The dissertation consists of two parts. The first gives an introduction to the subjects listed above and an overview of the papers included in the second part. The second part consists of 6 papers, labeled A-F, concerning the aforementioned subjects.Paper A and B deal with topics related Schrödinger's equation. Specifically, in Paper A spectral results for the magnetic Weyl quantization of S 0 0,0 v male profil h bestemmes ved at anvende metoder fra variationsregningen, og det vises, at det optimale højdeprofil h er en generalisering af højdeprofilen h(x) = εx 2 , der har fået meget opmaerksomhed i litteraturen.Artikel E og F omhandler emner relateret til stokastiske variabler X ∈ [0, 1], hvis cifre {X n } n≥1 udgør en stationaer stokastisk proces. Bemaerk at dette emne ikke har nogen direkte sammenhaeng med de foregående emner relateret til bølgefaenomener. I Artikel E bevises en funktionalligning for den kumulative fordelingsfunktion F tilhørende X. Fra funktionalligningen bevises yderligere karakteriseringer af stationaritet af cifrene {X n } n≥1 ud fra egenskaberne ved funktionen F. I Artikel F betragtes bestemte stationaere stokastiske modeller for cifrene. Konkret betragtes stationaere Markovkaeder og stationaere fornyelsesprocesser. Det vises, at for sådanne processer vil F enten vaere givet ved F(x) = x for x ∈ [0, 1], eller vaere singulaer, dvs. F (x) = 0 for naesten alle x ∈ [0, 1]. Miksturer af stationaere Markovkaeder eller stationaere fornyelsesprocesser behandles ligeledes i Artikel F, og helt fundamentalt for denne analyse er egenskaberne ved normale tal.