{"title":"具有无限多导数的线性方程和泽塔非局部方程的显式解","authors":"","doi":"10.1016/j.nuclphysb.2024.116680","DOIUrl":null,"url":null,"abstract":"<div><p>We summarize our theory on existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives of the form<span><span><span><math><mi>f</mi><mo>(</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo><mi>ϕ</mi><mo>=</mo><mi>J</mi><mo>(</mo><mi>t</mi><mo>)</mo><mspace></mspace><mo>,</mo><mspace></mspace><mi>t</mi><mo>≥</mo><mn>0</mn><mspace></mspace><mo>,</mo></math></span></span></span> where <em>f</em> is an analytic function such as the (analytic continuation of the) Riemann zeta function. We explain how to analyse initial value problems for these equations, and we prove rigorously that the function<span><span><span><math><mi>ϕ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></munder><mfrac><mrow><mi>μ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msup></mrow></mfrac><mi>J</mi><mo>(</mo><mi>t</mi><mo>−</mo><mi>ln</mi><mo></mo><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mspace></mspace><mo>,</mo></math></span></span></span> in which <em>μ</em> is the Möbius function and <em>J</em> satisfies some technical conditions to be specified in Section <span><span>4</span></span>, is the solution to the <em>zeta nonlocal equation</em><span><span><span><math><mi>ζ</mi><mo>(</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>h</mi><mo>)</mo><mi>ϕ</mi><mo>=</mo><mi>J</mi><mo>(</mo><mi>t</mi><mo>)</mo><mspace></mspace><mo>,</mo><mspace></mspace><mi>t</mi><mo>≥</mo><mn>0</mn><mspace></mspace><mo>,</mo></math></span></span></span> in which <em>ζ</em> is the Riemann zeta function and <span><math><mi>h</mi><mo>></mo><mn>1</mn></math></span>. We also present explicit examples of solutions to initial value problems for this equation. Our constructions can be interpreted as highlighting the importance of the <em>cosmological daemon functions</em> considered by Aref'eva and Volovich (2011) <span><span>[1]</span></span>. Our main technical tool is the Laplace transform as a unitary operator between the Lebesgue space <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and the Hardy space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":54712,"journal":{"name":"Nuclear Physics B","volume":null,"pages":null},"PeriodicalIF":2.5000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0550321324002463/pdfft?md5=3c6c94d585794459cbf36ea4d23f7705&pid=1-s2.0-S0550321324002463-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Linear equations with infinitely many derivatives and explicit solutions to zeta nonlocal equations\",\"authors\":\"\",\"doi\":\"10.1016/j.nuclphysb.2024.116680\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We summarize our theory on existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives of the form<span><span><span><math><mi>f</mi><mo>(</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo><mi>ϕ</mi><mo>=</mo><mi>J</mi><mo>(</mo><mi>t</mi><mo>)</mo><mspace></mspace><mo>,</mo><mspace></mspace><mi>t</mi><mo>≥</mo><mn>0</mn><mspace></mspace><mo>,</mo></math></span></span></span> where <em>f</em> is an analytic function such as the (analytic continuation of the) Riemann zeta function. We explain how to analyse initial value problems for these equations, and we prove rigorously that the function<span><span><span><math><mi>ϕ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></munder><mfrac><mrow><mi>μ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msup></mrow></mfrac><mi>J</mi><mo>(</mo><mi>t</mi><mo>−</mo><mi>ln</mi><mo></mo><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mspace></mspace><mo>,</mo></math></span></span></span> in which <em>μ</em> is the Möbius function and <em>J</em> satisfies some technical conditions to be specified in Section <span><span>4</span></span>, is the solution to the <em>zeta nonlocal equation</em><span><span><span><math><mi>ζ</mi><mo>(</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>h</mi><mo>)</mo><mi>ϕ</mi><mo>=</mo><mi>J</mi><mo>(</mo><mi>t</mi><mo>)</mo><mspace></mspace><mo>,</mo><mspace></mspace><mi>t</mi><mo>≥</mo><mn>0</mn><mspace></mspace><mo>,</mo></math></span></span></span> in which <em>ζ</em> is the Riemann zeta function and <span><math><mi>h</mi><mo>></mo><mn>1</mn></math></span>. We also present explicit examples of solutions to initial value problems for this equation. Our constructions can be interpreted as highlighting the importance of the <em>cosmological daemon functions</em> considered by Aref'eva and Volovich (2011) <span><span>[1]</span></span>. Our main technical tool is the Laplace transform as a unitary operator between the Lebesgue space <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and the Hardy space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</p></div>\",\"PeriodicalId\":54712,\"journal\":{\"name\":\"Nuclear Physics B\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2024-09-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0550321324002463/pdfft?md5=3c6c94d585794459cbf36ea4d23f7705&pid=1-s2.0-S0550321324002463-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nuclear Physics B\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0550321324002463\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, PARTICLES & FIELDS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nuclear Physics B","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0550321324002463","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, PARTICLES & FIELDS","Score":null,"Total":0}
Linear equations with infinitely many derivatives and explicit solutions to zeta nonlocal equations
We summarize our theory on existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives of the form where f is an analytic function such as the (analytic continuation of the) Riemann zeta function. We explain how to analyse initial value problems for these equations, and we prove rigorously that the function in which μ is the Möbius function and J satisfies some technical conditions to be specified in Section 4, is the solution to the zeta nonlocal equation in which ζ is the Riemann zeta function and . We also present explicit examples of solutions to initial value problems for this equation. Our constructions can be interpreted as highlighting the importance of the cosmological daemon functions considered by Aref'eva and Volovich (2011) [1]. Our main technical tool is the Laplace transform as a unitary operator between the Lebesgue space and the Hardy space .
期刊介绍:
Nuclear Physics B focuses on the domain of high energy physics, quantum field theory, statistical systems, and mathematical physics, and includes four main sections: high energy physics - phenomenology, high energy physics - theory, high energy physics - experiment, and quantum field theory, statistical systems, and mathematical physics. The emphasis is on original research papers (Frontiers Articles or Full Length Articles), but Review Articles are also welcome.