Pointwise convergence problems and some sharp inequalities arising in quantum mechanics

• Material Type: 博士論文
• Title: Pointwise convergence problems and some sharp inequalities arising in quantum mechanics
• Author/Editor: 白木, 尚武
• Author Heading: 白木, 尚武
• Publication, Distribution, etc.: 埼玉大学大学院理工学研究科
• Publication Date: 2021-03
• Publication Date (W3CDTF): 2021-03
• Alternative Title: 量子力学に現れる各点収束問題と不等式の最良定数問題
• Periodical title: 博士論文（埼玉大学大学院理工学研究科（博士後期課程））
• Degree Grantor: 埼玉大学
• Date Granted: 2021-03-25
• Date Granted (W3CDTF): 2021-03-25
• Dissertation Number: 博理工甲第1203号
• Degree Type: 博士(理学)
• Text Language Code: eng
• Target Audience: 一般
• Note (General): 出版タイプ： VoR; vi, 69p; 1 A variety of pointwise convergence problems for Schrödinger-type equations 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 To the fractional Schrödinger equation . . . . . . . . . . . . . . . . 3 1.1.2 Carleson's problem and the divergence sets . . . . . . . . . . . . . 4 1.1.3 Path along lines generated by a fractal set . . . . . . . . . . . . . . 5 1.1.4 Path along a tangential curve . . . . . . . . . . . . . . . . . . . . . 7 1.1.5 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 First new result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 Proof of Theorem 1.2.1 . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Second new result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3.1 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3.2 Proof of (Theorem 1.3.2⇒ Corollary 1.3.4) . . . . . . . . . . . . 20 1.3.3 Proof of Theorem 1.3.1 . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.4 The necessary conditions regarding Theorem 1.3.3 . . . . . . . . . 232 Sharp bilinear estimates for the Klein-Gordon equation 27 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Some connections to recent results and corollaries . . . . . . . . . . . . . . 31 2.2.1 Wave regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.2 Non-wave regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 Proof of Theorem 2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 On estimate (2.15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.1 Estimate (2.15) with explicit constant . . . . . . . . . . . . . . . . 37 2.4.2 Threshold of our argument for βε (3-d/4 , 5-d/4) . . . . . . . . . . . . 40 2.4.3 Contributions of radial symmetry . . . . . . . . . . . . . . . . . . . 41 2.5 Sharpness of constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.1 Wave regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5.2 Non-wave regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.5.3 Non-existence of an extremiser . . . . . . . . . . . . . . . . . . . . 463 Hypercontractivity via Heat-flow method 48 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Markov semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.2 The closure property . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.3 Ornstein-Uhlenbeck semigroup . . . . . . . . . . . . . . . . . . . . 54 3.3 Proofs of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.1 Proof of Theorem 3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.2 Proof of Corollary 3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4.1 Reverse hypercontractivity . . . . . . . . . . . . . . . . . . . . . . 57 3.4.2 Related work and extensions . . . . . . . . . . . . . . . . . . . . . 58; 主指導教員 : Neal Bez 准教授; text; application/pdf
• DOI: 10.24561/00019569; https://doi.org/10.24561/00019569
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• Summary, etc.: We are concerned three different themes strongly related to quantum mechanics by employing harmonic analytic approaches. The first chapter pertains to...