{"title":"一维薛定谔方程的显式求解","authors":"Peter C. Gibson","doi":"10.1137/22m1514441","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4466-4493, August 2024. <br/> Abstract. Evaluation of a product integral with values in the Lie group SU(1,1) yields the explicit solution to the impedance form of the Schrödinger equation. Explicit formulas for the transmission coefficient and [math]-matrix of the classical one-dimensional Schrödinger operator with arbitrary compactly supported potential are obtained as a consequence. The formulas involve operator theoretic analogues of the standard hyperbolic functions and provide new tools with which to analyze acoustic and quantum scattering in one dimension.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Explicit Solution of the 1D Schrödinger Equation\",\"authors\":\"Peter C. Gibson\",\"doi\":\"10.1137/22m1514441\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4466-4493, August 2024. <br/> Abstract. Evaluation of a product integral with values in the Lie group SU(1,1) yields the explicit solution to the impedance form of the Schrödinger equation. Explicit formulas for the transmission coefficient and [math]-matrix of the classical one-dimensional Schrödinger operator with arbitrary compactly supported potential are obtained as a consequence. The formulas involve operator theoretic analogues of the standard hyperbolic functions and provide new tools with which to analyze acoustic and quantum scattering in one dimension.\",\"PeriodicalId\":51150,\"journal\":{\"name\":\"SIAM Journal on Mathematical Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM Journal on Mathematical Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1137/22m1514441\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Mathematical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/22m1514441","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4466-4493, August 2024. Abstract. Evaluation of a product integral with values in the Lie group SU(1,1) yields the explicit solution to the impedance form of the Schrödinger equation. Explicit formulas for the transmission coefficient and [math]-matrix of the classical one-dimensional Schrödinger operator with arbitrary compactly supported potential are obtained as a consequence. The formulas involve operator theoretic analogues of the standard hyperbolic functions and provide new tools with which to analyze acoustic and quantum scattering in one dimension.
期刊介绍：
SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena.
Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere.
Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.
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