Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

An estimate for the number of bound states of the Schrödinger operator in two dimensions
HTML articles powered by AMS MathViewer

by Mihai Stoiciu PDF
Proc. Amer. Math. Soc. 132 (2004), 1143-1151 Request permission

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
  • Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
  • George B. Arfken and Hans J. Weber, Mathematical methods for physicists, 4th ed., Academic Press, Inc., San Diego, CA, 1995. MR 1423357
  • M. Š. Birman, On the spectrum of singular boundary-value problems, Mat. Sb. (N.S.) 55 (97) (1961), 125–174 (Russian). MR 0142896
  • H. J. Brascamp, Elliott H. Lieb, and J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Functional Analysis 17 (1974), 227–237. MR 0346109, DOI 10.1016/0022-1236(74)90013-5
  • N. N. Khuri, A. Martin, and T. T. Wu, Bound states in n dimensions (especially $n=1$ and $n=2$), Few Body Systems 31 (2002), 83–89.
  • Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
  • Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
  • Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
  • Julian Schwinger, On the bound states of a given potential, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 122–129. MR 129798, DOI 10.1073/pnas.47.1.122
  • Barry Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge-New York, 1979. MR 541149, DOI 10.1007/BFb0064579
  • Barry Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Physics 97 (1976), no. 2, 279–288. MR 404846, DOI 10.1016/0003-4916(76)90038-5
  • B. Simon, On the number of bound states of two-body Schrödinger operators: A review, in Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann, Princeton University Press, Princeton, 1976, pp. 305–326.
  • M. Klaus, On the bound state of Schrödinger operators in one dimension, Ann. Physics 108 (1977), no. 2, 288–300. MR 503200, DOI 10.1016/0003-4916(77)90015-X
  • Roger G. Newton, Bounds on the number of bound states for the Schrödinger equation in one and two dimensions, J. Operator Theory 10 (1983), no. 1, 119–125. MR 715561
  • Noriaki Setô, Bargmann’s inequalities in spaces of arbitrary dimension, Publ. Res. Inst. Math. Sci. 9 (1973/74), 429–461. MR 0340846, DOI 10.2977/prims/1195192566
  • B. Simon, An introduction to the self-adjointness and spectral analysis of Schrödinger operators in The Schrödinger Equation (W. Thirring and P. Urban, eds.), Acta Phys. Aus. Suppl. 17, Springer, Vienna, 1977, pp. 19–42.
  • M. Solomyak, Piecewise-polynomial approximation of functions from $H^l((0,1)^d)$, $2l=d$, and applications to the spectral theory of the Schrödinger operator, Israel J. Math. 86 (1994), no. 1-3, 253–275. MR 1276138, DOI 10.1007/BF02773681
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35P15, 35J10, 81Q10
  • Retrieve articles in all journals with MSC (2000): 35P15, 35J10, 81Q10
Additional 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