{"title":"有界区间上Schrödinger算子的极大行列式","authors":"C. Aldana, Jean-Baptiste Caillau, P. Freitas","doi":"10.5802/jep.128","DOIUrl":null,"url":null,"abstract":"We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrodinger operator defined on a bounded interval with Dirichlet boundary conditions under an $L^q$-norm restriction ($q\\geq 1$). This is done by first extending the definition of the functional determinant to the case of $L^q$ potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all $q\\geq 1$, providing a complete characterization of extremal potentials in the case where $q$ is one (a pulse) and two (Weierstrass's $\\wp$ function).","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"7 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Maximal determinants of Schrödinger operators on bounded intervals\",\"authors\":\"C. Aldana, Jean-Baptiste Caillau, P. Freitas\",\"doi\":\"10.5802/jep.128\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrodinger operator defined on a bounded interval with Dirichlet boundary conditions under an $L^q$-norm restriction ($q\\\\geq 1$). This is done by first extending the definition of the functional determinant to the case of $L^q$ potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all $q\\\\geq 1$, providing a complete characterization of extremal potentials in the case where $q$ is one (a pulse) and two (Weierstrass's $\\\\wp$ function).\",\"PeriodicalId\":106406,\"journal\":{\"name\":\"Journal de l’École polytechnique — Mathématiques\",\"volume\":\"7 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal de l’École polytechnique — Mathématiques\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.5802/jep.128\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de l’École polytechnique — Mathématiques","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5802/jep.128","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Maximal determinants of Schrödinger operators on bounded intervals
We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrodinger operator defined on a bounded interval with Dirichlet boundary conditions under an $L^q$-norm restriction ($q\geq 1$). This is done by first extending the definition of the functional determinant to the case of $L^q$ potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all $q\geq 1$, providing a complete characterization of extremal potentials in the case where $q$ is one (a pulse) and two (Weierstrass's $\wp$ function).