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
• Publication Date (W3CDTF): 2021
• 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/type: 埼玉大学
• Date Granted: 2021-03-25
• Date Granted (W3CDTF): 2021-03-25
• Dissertation Number: 甲第1203号
• Degree Type: 博士(理学)
• Conferring No. (Dissertation): 甲第1203号
• Text Language Code: eng
• Target Audience: 一般
• Note (General): type:text; We are concerned three different themes strongly related to quantum mechanics by employing harmonic analytic approaches. The first chapter pertains to the pointwise convergence problem for the Schrödinger type operator, the so called Carleson's problem, initiated by the mathematical giant Lennart Carleson in 1980. He showed that some smoothness condition on the initial data is required for the solutions to the standard Schrödinger equation to converge to the initial data almost everywhere in R. While the one spatial dimensional case was completely understood in a very early stage, the higher dimensional case turns out to be extremely difficult. In 2016, Jean Bourgain finally provided a plausible necessary condition, then soon later, Xiumin Du, Larry Guth, Xiaochun Li and Ruixian Zhang proved that Bourgain's regularity threshold is essentially suffcient as well. Their proof contains state-of-the-art technologies in harmonic analysis, which also reflects the well-known connections among Carleson's problem and other major open problems in harmonic analysis, such as Stein's restriction conjecture and the Kakeya conjecture. Many variations of Carleson's problem are also concerned, for instance, convergence along generalized paths and refinements by measuring the corresponding divergence sets in a more precise sense than Lebesgue measure. Chu-hee Cho, Sanghyuk Lee and Ana Vargas considered the following two distinct generalized paths in one spatial dimensional case; (1) paths along lines generated by a given fractal set, and (2) path along a tangential line onto the hyperplane Rd⨉｛0｝. We extend their results from the standard Schrödinger equation to the fractional Schrödinger equation, which has been also studied actively because of its useful applications. By our novel approach, we prove that the Minkowski dimension of the given fractal set influences to the smoothness condition on the initial data for pointwise convergence in the situation of (1), and for (2), the Hölder exponent of the curve and the order of the fractional Schrödinger operator influences the smoothness condition of the initial data. We also consider the refined problem of estimating the size of the associated divergence sets in case (2). In the second part of the thesis, we consider the Strichartz estimate for the Klein-Gordon operator which can somehow be considered to be the hybrid of Schrödinger and wave operators. Strichartz estimates are one of the most important results in harmonic analysis since they have very useful applications in non-linear PDE theory and have connection with Stein's restriction conjecture. In 2007, Damiano Foschi obtained the sharp Strichartz estimate for wave equation in some special cases and a complete characterization of extremisers. Here, sharp estimate means the estimate with the optimal constant. The latest extension of this result is due to Neal Bez, Chirs Jeavons and Tohru Ozawa who further discussed this subject in the context of the so-called null-form estimates. René Quilodrán and soon later Jeavons, simultaneously, naturally extended Foschi's argument from wave to the Klein-Gordon equation and obtained analogous results. Jeavons further proved an improved Strichartz estimate in five spatial dimensions. In this chapter, we take the philosophy of Bez-Jeavons-Ozawa and extend results due to Quilodrán and Jeavons to two different directions, which we call the wave regime and the non-wave regime. In the non-wave regime, we also obtained an improved Strichartz estimate in four spatial dimensions. In the last chapter, Nelson's celebrated hypercontractivity inequality is concerned and a new perspective of supersolutions is provided. Jonathan Bennett and Bez have pursued a remarkable study of algebraic closure properties of supersolutions in their series of papers. For example, in 2009 they presented a new significantly simple proof of the sharp n-fold Young's convolution inequality and its inverse by combining the closure property and heat-flow monotonicity argument. The purpose of this chapter is to reprove the hypercontractivity inequality for the Ornstein-Uhlenbeck semigroup, another key object in quantum mechanics, by this technique and formally extend this result for far more abstract Markov semigroups which enjoy the diffusion property.; 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
• DOI: 10.24561/00019569; https://doi.org/10.24561/00019569
• Persistent ID (NDL): info:ndljp/pid/12313440; https://dl.ndl.go.jp/pid/12313440
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• Date Accepted (W3CDTF): 2022-08-08T06:02:54+09:00
• Date Created (W3CDTF): 2022-06-15
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