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

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- by Mihai Stoiciu
- Proc. Amer. Math. Soc.
**132**(2004), 1143-1151 - DOI: https://doi.org/10.1090/S0002-9939-03-07257-5
- Published electronically: August 28, 2003
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## 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.## References

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## Bibliographic 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2045431