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Coupling constant limits of Schrödinger operators with critical potentials

Xiaoyao Jia1* and Yan Zhao2

Author Affiliations

1 Mathematics and Statistics School, Henan University of Science and Technology, No. 263, Luo-Long District, Kai-Yuan Road, Luoyang City, Henan Province, 471023, China

2 College of Civil Engineering and Architecture, Zhejiang University, B505 Anzhong Building, 866 Yuhangtang Road, Hangzhou, Zhejiang Province, 310058, China

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Boundary Value Problems 2013, 2013:62  doi:10.1186/1687-2770-2013-62

Published: 27 March 2013

Abstract

A family of Schrödinger operators, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M1">View MathML</a>, is studied in this paper. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M3">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M4">View MathML</a> is large enough and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M5">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M6">View MathML</a>. We show that each discrete eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> tends to 0 when λ tends to some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8">View MathML</a>. We get asymptotic behavior of the smallest discrete eigenvalue when λ tends to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8">View MathML</a>.

Keywords:
Schrödinger operator; critical potential; asymptotic expansion