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Published August 28, 2004 | Published
Journal Article Open

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

Stoiciu, Mihai

Abstract

For the Schrödinger operator -Δ + V on R^2 be the number of bound states. One obtains the following estimate: N(V) ≤ 1 + ∫_(R^2)∫_(R^2)|V(x)|V(y)|C_(1)ln|x-y|+C_2|^2 dx dy where C_1 = -1/2π and C_2 = (ln2-γ)/2π (γ is the Euler constant). This estimate holds for all potentials for which the previous integral is finite.

Additional Information

© 2003 American Mathematical Society. Received by editor(s): December 17, 2002; Posted: August 28, 2003; Communicated by: Joseph A. Ball. I would like to thank B. Simon for proposing the problem and both R. Killip and B. Simon for useful discussions. Note added in proof: After the submission of this paper I learned of further related results: N. Setô [15], R. Newton [14] and M. Solomyak [17]. Readers interested in the one-dimensional problem should refer to B. Simon [16] and M. Klaus [13]. I would like to thank P. Exner for bringing some of these papers to my attention.

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August 19, 2023
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