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

Stoiciu, Mihai (2004) An estimate for the number of bound states of the Schrödinger operator in two dimensions. Proceedings of the American Mathematical Society, 132 (4). pp. 1143-1151. ISSN 0002-9939.

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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.

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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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MathSciNet review2045431
Classification Code:MSC (2000): Primary 35P15, 35J10; Secondary 81Q10
Record Number:CaltechAUTHORS:20111012-100552340
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:27181
Deposited By: Jason Perez
Deposited On:12 Oct 2011 20:03
Last Modified:26 Dec 2012 14:15

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