{"title":"一维薛定谔算子共振的奇异性和渐近分布","authors":"T. J. Christiansen, T. Cunningham","doi":"10.3233/asy-241928","DOIUrl":null,"url":null,"abstract":"We obtain new results about the high-energy distribution of resonances for the one-dimensional Schrödinger operator. Our primary result is an upper bound on the density of resonances above any logarithmic curve in terms of the singular support of the potential. We also prove results about the distribution of resonances in sectors away from the real axis, and construct a class of potentials producing multiple sequences of resonances along distinct logarithmic curves, explicitly calculating the asymptotic location of these resonances. The results are unified by the use of an integral representation of the reflection coefficients, refining methods used in (J. Differential Equations 137(2) (1997) 251–272) and (J. Funct. Anal. 178(2) (2000) 396–420).","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Singularities and asymptotic distribution of resonances for Schrödinger operators in one dimension\",\"authors\":\"T. J. Christiansen, T. Cunningham\",\"doi\":\"10.3233/asy-241928\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We obtain new results about the high-energy distribution of resonances for the one-dimensional Schrödinger operator. Our primary result is an upper bound on the density of resonances above any logarithmic curve in terms of the singular support of the potential. We also prove results about the distribution of resonances in sectors away from the real axis, and construct a class of potentials producing multiple sequences of resonances along distinct logarithmic curves, explicitly calculating the asymptotic location of these resonances. The results are unified by the use of an integral representation of the reflection coefficients, refining methods used in (J. Differential Equations 137(2) (1997) 251–272) and (J. Funct. Anal. 178(2) (2000) 396–420).\",\"PeriodicalId\":55438,\"journal\":{\"name\":\"Asymptotic Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2024-08-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Asymptotic Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3233/asy-241928\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Asymptotic Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3233/asy-241928","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Singularities and asymptotic distribution of resonances for Schrödinger operators in one dimension
We obtain new results about the high-energy distribution of resonances for the one-dimensional Schrödinger operator. Our primary result is an upper bound on the density of resonances above any logarithmic curve in terms of the singular support of the potential. We also prove results about the distribution of resonances in sectors away from the real axis, and construct a class of potentials producing multiple sequences of resonances along distinct logarithmic curves, explicitly calculating the asymptotic location of these resonances. The results are unified by the use of an integral representation of the reflection coefficients, refining methods used in (J. Differential Equations 137(2) (1997) 251–272) and (J. Funct. Anal. 178(2) (2000) 396–420).
期刊介绍:
The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand. Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.