{"title":"The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula","authors":"Chein-Shan Liu, Botong Li","doi":"10.21136/AM.2024.0005-21","DOIUrl":null,"url":null,"abstract":"<div><p>The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an <i>n</i>-dimensional matrix eigenvalue problem is derived with a special matrix <b>A</b>:= [<i>a</i><sub><i>ij</i></sub>], that is, <i>a</i><sub><i>ij</i></sub> = 0 if <i>i</i> + <i>j</i> is odd.</p><p>Based on the product formula, an integration method with a fictitious time, namely the fictitious time integration method (FTIM), is developed to obtain the higher-index eigenvalues. Also, we recover the symmetric potential function <i>q</i>(<i>x</i>) in the Sturm-Liouville operator by specifying a few lower-index eigenvalues, based on the product formula and the Newton iterative method.</p></div>","PeriodicalId":55505,"journal":{"name":"Applications of Mathematics","volume":"69 3","pages":"355 - 372"},"PeriodicalIF":0.6000,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applications of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.21136/AM.2024.0005-21","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an n-dimensional matrix eigenvalue problem is derived with a special matrix A:= [aij], that is, aij = 0 if i + j is odd.
Based on the product formula, an integration method with a fictitious time, namely the fictitious time integration method (FTIM), is developed to obtain the higher-index eigenvalues. Also, we recover the symmetric potential function q(x) in the Sturm-Liouville operator by specifying a few lower-index eigenvalues, based on the product formula and the Newton iterative method.
期刊介绍：
Applications of Mathematics publishes original high quality research papers that are directed towards applications of mathematical methods in various branches of science and engineering.
The main topics covered include:
- Mechanics of Solids;
- Fluid Mechanics;
- Electrical Engineering;
- Solutions of Differential and Integral Equations;
- Mathematical Physics;
- Optimization;
- Probability
Mathematical Statistics.
The journal is of interest to a wide audience of mathematicians, scientists and engineers concerned with the development of scientific computing, mathematical statistics and applicable mathematics in general.
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