An estimate for the number of bound states of the Schrödinger operator in two dimensions

Author:
Mihai Stoiciu

Journal:
Proc. Amer. Math. Soc. **132** (2004), 1143-1151

MSC (2000):
Primary 35P15, 35J10; Secondary 81Q10

DOI:
https://doi.org/10.1090/S0002-9939-03-07257-5

Published electronically:
August 28, 2003

MathSciNet review:
2045431

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Abstract | References | Similar Articles | Additional Information

Abstract: For the Schrödinger operator $-\Delta + V$ on $\mathbb R ^2$ let $N(V)$ be the number of bound states. One obtains the following estimate: \[ N(V) \leq \ 1 \ + \int _{\mathbb R ^2} \int _{\mathbb R ^2} |V(x)| \ |V(y)| \ |C_1 \ln |x-y| + C_2|^2 \ dx dy \] where $C_1 = -\frac {1}{2\pi }$ and $C_2 = \frac {\ln 2 - \gamma }{2 \pi }$ ($\gamma$ is the Euler constant). This estimate holds for all potentials for which the previous integral is finite.

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Additional Information

**Mihai Stoiciu**

Affiliation:
Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125

Email:
mihai@its.caltech.edu

Received by editor(s):
December 17, 2002

Published electronically:
August 28, 2003

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2003
American Mathematical Society