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

Stoiciu, Mihai

doi:10.1090/S0002-9939-03-07257-5

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.

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