Applications of Mathematics, first online, pp. 1-20


On spectral problems of discrete Schrödinger operators

Chi-Hua Chan, Po-Chun Huang

Received August 13, 2019.   Published online January 25, 2021.

Abstract:  A special type of Jacobi matrices, discrete Schrödinger operators, is found to play an important role in quantum physics. In this paper, we show that given the spectrum of a discrete Schrödinger operator and the spectrum of the operator obtained by deleting the first row and the first column of it can determine the discrete Schrödinger operator uniquely, even though one eigenvalue of the latter is missing. Moreover, we find the forms of the discrete Schrödinger operators when their smallest and largest eigenvalues attain the extrema under certain constraints by use of the notion of generalized directional derivative and the method of Lagrange multiplier.
Keywords:  discrete Schrödinger operator
Classification MSC:  34B09
DOI:  10.21136/AM.2021.0203-19

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Affiliations:   Chi-Hua Chan (corresponding author), Department of Mathematics, National Tsing Hua University, Hsinchu, 30013, Taiwan, e-mail: evolution6423@yahoo.com.tw; Po-Chun Huang, Department of Mathematics and Information Education, National Taipei University of Education, Taipei, 10671, Taiwan, e-mail: d917205@oz.nthu.edu.tw


 
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