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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
Published electronically: August 28, 2003
MathSciNet review: 2045431
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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:

\begin{displaymath}N(V) \leq 1 + \int_{\mathbb R ^2} \int_{\mathbb R ^2} \vert V... ...ert V(y)\vert \vert C_1 \ln \vert x-y\vert + C_2\vert^2 dx\,dy \end{displaymath}

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

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07257-5
Received by editor(s): December 17, 2002
Published electronically: August 28, 2003
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society