{"title":"Energy spectrum of a constrained quantum particle and the Willmore energy of the constraining surface","authors":"Vicent Gimeno i Garcia, Steen Markvorsen","doi":"10.4310/atmp.2023.v27.n8.a3","DOIUrl":null,"url":null,"abstract":"In this paper, we establish geometric and topological upper bounds on the first energy level gap of a particle confined to move on a compact surface in $3$-space. Our main contribution is proving that the first gap in the energy spectrum of a confined particle (a physical property) is bounded above by the Willmore energy of the confining surface (a geometric property). Furthermore, we demonstrate that the only surfaces that permit a confined particle with a stationary and uniformly distributed wave function are surfaces with constant skew curvature.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Theoretical and Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.4310/atmp.2023.v27.n8.a3","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we establish geometric and topological upper bounds on the first energy level gap of a particle confined to move on a compact surface in $3$-space. Our main contribution is proving that the first gap in the energy spectrum of a confined particle (a physical property) is bounded above by the Willmore energy of the confining surface (a geometric property). Furthermore, we demonstrate that the only surfaces that permit a confined particle with a stationary and uniformly distributed wave function are surfaces with constant skew curvature.
期刊介绍:
Advances in Theoretical and Mathematical Physics is a bimonthly publication of the International Press, publishing papers on all areas in which theoretical physics and mathematics interact with each other.