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Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>In this article, we study numerical approximation of eigenvalue problems of the Schr\u00f6dinger operator $-\\Delta u + \\frac{c^2}{|x|^2}u$. There are three stages in our investigation: We start from a ball of any dimension, in which case the exact solution in the radial direction can be expressed by Bessel functions of fractional degrees. This knowledge helps us to design two novel spectral methods by modifying the polynomial basis to fit the singularities of the eigenfunctions. At the second stage, we move to circular sectors in the two-dimensional setting. Again the radial direction can be expressed by Bessel functions of fractional degrees. Only in the tangential direction some modifications are needed from stage one. At the final stage, we extend the idea to arbitrary polygonal domains. We propose a mortar spectral element approach: a polygonal domain is decomposed into several subdomains with each singular corner including the origin covered by a circular sector, in which origin and corner singularities are handled similarly as in the former stages, and the remaining domains are either a standard quadrilateral\/triangle or a quadrilateral\/triangle with a circular edge, in which the traditional polynomial based spectral method is applied. All subdomains are linked by mortar elements (note that we may have hanging nodes). In all three stages, exponential convergence rates are achieved. Numerical experiments indicate that our new methods are superior to standard polynomial based spectral (or spectral element) methods and $hp$-adaptive methods. Our study offers a new and effective way to handle eigenvalue problems of the Schr\u00f6dinger operator including the Laplacian operator on polygonal domains with reentrant corners.<\/jats:p>","DOI":"10.1137\/16m1069596","type":"journal-article","created":{"date-parts":[[2017,1,3]],"date-time":"2017-01-03T11:39:54Z","timestamp":1483443594000},"page":"A114-A140","source":"Crossref","is-referenced-by-count":30,"title":["Efficient Spectral and Spectral Element Methods for  Eigenvalue Problems of  Schr\u00f6dinger Equations with an Inverse Square Potential"],"prefix":"10.1137","volume":"39","author":[{"given":"Huiyuan","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhimin","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,1,3]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.12.059"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107325937"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(95)00946-9"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.201490020"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144503437336"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1155\/S1073792894000437"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-005-0588-3"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-30728-0"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2005.07.010"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.80.797"},{"key":"atypb11","volume-title":"Approximation Theory and Harmonic Analysis on Spherical and Balls","author":"Dai F.","year":"2013"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144595285069"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107786134"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2006.10.019"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1080\/03605300500394439"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/0704008"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.43.36"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.82.2213"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/BF01231320"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1992-00289-6"},{"key":"atypb21","volume-title":"Pitman","author":"Grisvard P.","year":"1985"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389733"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389734"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389735"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/05063756X"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-005-9055-7"},{"key":"atypb27","volume-title":"Computing Ultra-precise Eigenvalues of the Laplacian Within Polygons, arXiv:1602.08636 [math.NA]","author":"Jones R.","year":"2016"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1966.11970915"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0067087"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-014-0628-y"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.3934\/dcdsb.2015.20.1377"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-09-02308-4"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1137\/130940591"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1976-0403235-1"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595295301"},{"key":"atypb36","first-page":"297","volume-title":"AMS","author":"Trefethen L. 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