{"title":"同构逆问题","authors":"E. Korotyaev","doi":"10.1134/S1061920824601745","DOIUrl":null,"url":null,"abstract":"<p> Consider two inverse problems for Sturm–Liouville problems on the unit interval. This means that there are two corresponding mappings <span>\\(F, f\\)</span> from a Hilbert space of potentials <span>\\(H\\)</span> into their spectral data. They are called isomorphic if <span>\\(F\\)</span> is a composition of <span>\\(f\\)</span> and some isomorphism <span>\\(U\\)</span> of <span>\\(H\\)</span> onto itself. An isomorphic class is a collection of inverse problems isomorphic to each other. We consider basic Sturm–Liouville problems on the unit interval and on the circle and describe their isomorphic classes of inverse problems. For example, we prove that the inverse problems for the case of Dirichlet and Neumann boundary conditions are isomorphic. The proof is based on nonlinear analysis. </p><p> <b> DOI</b> 10.1134/S1061920824601745 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"32 2","pages":"314 - 340"},"PeriodicalIF":1.5000,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Isomorphic Inverse Problems\",\"authors\":\"E. Korotyaev\",\"doi\":\"10.1134/S1061920824601745\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p> Consider two inverse problems for Sturm–Liouville problems on the unit interval. This means that there are two corresponding mappings <span>\\\\(F, f\\\\)</span> from a Hilbert space of potentials <span>\\\\(H\\\\)</span> into their spectral data. They are called isomorphic if <span>\\\\(F\\\\)</span> is a composition of <span>\\\\(f\\\\)</span> and some isomorphism <span>\\\\(U\\\\)</span> of <span>\\\\(H\\\\)</span> onto itself. An isomorphic class is a collection of inverse problems isomorphic to each other. We consider basic Sturm–Liouville problems on the unit interval and on the circle and describe their isomorphic classes of inverse problems. For example, we prove that the inverse problems for the case of Dirichlet and Neumann boundary conditions are isomorphic. The proof is based on nonlinear analysis. </p><p> <b> DOI</b> 10.1134/S1061920824601745 </p>\",\"PeriodicalId\":763,\"journal\":{\"name\":\"Russian Journal of Mathematical Physics\",\"volume\":\"32 2\",\"pages\":\"314 - 340\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2025-07-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Russian Journal of Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1134/S1061920824601745\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S1061920824601745","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Consider two inverse problems for Sturm–Liouville problems on the unit interval. This means that there are two corresponding mappings \(F, f\) from a Hilbert space of potentials \(H\) into their spectral data. They are called isomorphic if \(F\) is a composition of \(f\) and some isomorphism \(U\) of \(H\) onto itself. An isomorphic class is a collection of inverse problems isomorphic to each other. We consider basic Sturm–Liouville problems on the unit interval and on the circle and describe their isomorphic classes of inverse problems. For example, we prove that the inverse problems for the case of Dirichlet and Neumann boundary conditions are isomorphic. The proof is based on nonlinear analysis.
期刊介绍：
Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.
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