Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

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

Author(s): Mihai Stoiciu
Journal: Proc. Amer. Math. Soc. 132 (2004), 1143-1151.
MSC (2000): Primary 35P15, 35J10; Secondary 81Q10
Posted: August 28, 2003
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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.

References:

1.
M. Abramowitz and I. Stegun (editors), Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1972. MR 94b:00012

2.
G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Academic Press, San Diego, CA, 1995. MR 98a:00001

3.
M. S. Birman, On the spectrum of singular boundary-value problems, Mat. Sb. (N.S.) 55 (97) (1961) 125-174, English translation, Amer. Math. Soc. Transl. 53 (1966), 23-80. MR 26:463
4.
H. J. Brascamp, Elliott H. Lieb and J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Funct. Anal. 17 (1974), 227-237. MR 49:10835
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.

6.
M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972. MR 58:12429a

7.
M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self Adjointness, Academic Press, New York, 1975. MR 58:12429b

8.
M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978. MR 58:12429c

9.
J. Schwinger, On the bound states of a given potential, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 122-129. MR 23:B2833

10.
B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Note Series, Vol. 35, Cambridge University Press, Cambridge, 1979. MR 80k:47048

11.
B. Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Phys. 97 (1976), 279-288. MR 53:8646

12.
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.

13.
M. Klaus, On the bound state of Schrödinger operators in one dimension, Ann. Physics 108 (1977), no. 2, 288-300. MR 58:20010

14.
R. 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 85e:81029

15.
N. Setô, Bargmann's inequalities in spaces of arbitrary dimension, Publ. Res. Inst. Math. Sci. 9 (1973/74), 429-461. MR 49:5596

16.
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.

17.
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 95e:35151

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

DOI: 10.1090/S0002-9939-03-07257-5
PII: S 0002-9939(03)07257-5
Received by editor(s): December 17, 2002
Posted: August 28, 2003
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2003, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google