Abstract
Estimates for the number of bound states and their energies, ej ≤ 0, are of obvious importance for the investigation of quantum mechanical Hamiltonians. If the latter are of the single particle form H = -Δ + V(x) in Rn , we shall use available methods to derive the bounds
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Work supported by U. S. National Science Foundation Grant MPS 71–03375-A03.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. H. Lieb and W. E. Thirring, Phys. Rev. Lett. 35, 687(1975).
E. H. Lieb and W. E. Thirring, See Phys. Rev. Lett. 35, 1116(1975) for errata.
M. S. Birman, Mat. Sb. 55(97), 125(1961);
M. S. Birman, Amer. Math. Soc. Translations Ser. 2, 53, 23(1966).
J. Schwinger, Proc. Nat. Acad. Sci. 47, 122(1961).
B. Simon, “Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, ” Princeton University Press. 1971.
B. Simon, “On the Number of Bound States of the Two Body Schrödinger Equation — A Review, ’’ in this volume.
A. Martin; Helv. Phys. Acta 45, 140(1972).
H. Tamura, Proc. Japan Acad. 50, 19(1974).
V. Glaser, A. Martin, H. Grosse and W. Thirring, “A Family of Optimal Conditions for the Absence of Bound States in a Potential/’ in this volume.
S. L. Sobolev, Mat. Sb. 46, 471(1938), in Russian.
S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics, Leningrad (1950), Amer. Math. Soc. Transi, of Monographs, 7(1963).
G. Talenti, Best Constant in Sobolev’s Inequality, Istituto Matemático, Universita Degli Studi Di Firenze, preprint (1975).
G. Rosen, SIAM Jour. Appl. Math. 21, 30(1971).
H. J. Brascamp, E. H. Lieb and J. M. Luttinger, Jour. Funct. Anal. 17, 227(1974).
C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Commun. Pure and Appl Math. 27, 97 (1974).
S. A. Moszkowski, Phys. Rev. 89, 474(1953).
A. E. Green and K. Lee, Phys. Rev. 99, 772(1955).
V. E. Zakharov and L. D. Fadeev, Funkts. Anal, i Ego Pril. 5, 18(1971).
V. E. Zakharov and L. D. Fadeev, Funkts. English translation: Funct. Anal, and its Appl. 5, 280(1971).
H. Epstein, Commun. Math. Phys. 31, 317(1973).
E. Seiler and B. Simon, “Bounds in the Yukawa Quantum Field Theory, ” Princeton preprint (1975).
W. Thirring, T7 Quantenmechanik, Lecture Notes, Institut für Theoretische Physik, University of Vienna.
T. Aubin, C. R. Acad. Sc. Paris 280, 279(1975). The results are stated here without proof; there appears to be a misprint in the expression for Cr, n.
B. Simon, “Weak Trace Ideals and the Bound States of Schrödinger Operators, ” Princeton preprint (1975).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lieb, E.H., Thirring, W.E. (2001). Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04360-8_16
Download citation
DOI: https://doi.org/10.1007/978-3-662-04360-8_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-04362-2
Online ISBN: 978-3-662-04360-8
eBook Packages: Springer Book Archive