Abstract
We give a detailed proof of limiting eigenvalue distribution theorems (LEDT) for eigenvalue clusters associated to suitable bounded perturbations of the spherical Laplacian on Sn and the hydrogen atom Hamiltonian. In order to show the structure of the proof of those theorems, we introduce a LEDT for suitable bounded perturbations of the k-dimensional harmonic oscillator.
Mathematics Subject Classification (2010). 81Q20, 35P20, 44A12, 81R30.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Avron, J.; Herbst, I.; Simon, B. Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45 (1978), no. 4, 847-883.
Avron, J.E.; Herbst, I.W.; Simon, B. Schrödinger operators with magnetic fields. III. Atoms in homogeneous magnetic field. Comm. Math. Phys. 79 (1981), no. 4, 529-572.
Bander M. and Itzykson C. Group theory and the hydrogen atom (I). Reviews of Modern Physics. Volume 38, number 3, April (1966), 330-345,
Bargmann, V. On a Hilbert Space of Analytic Functions and an Associated Integral Transform. Part I. Comm. on Pure and Appl. Math., 14 (1961), 187-214.
De Oliveira, C.R. Intermediate Spectral Theory and Quantum Dynamics. Birk- häuser, Progress in Mathematical Physics, 54, (2009).
Fock, V.A. Zur Theorie des Wasserstoffatoms. Z. Phys., 98, (1935), 145-154.
Guillemin, V. and Sternberg, S. On the spectra of commuting pseudodifferential operators: recent work of Kac-Spencer, Weinstein and others. Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977). Lecture Notes in Pure and Appl. Math., 48, Dekker, New York, (1979), 149-165.
Guillemin, Victor. Band asymptotics in two dimensions. Adv. in Math. 42 (1981), no. 3, 248-282.
Guillemin, V.; Uribe, A.; and Wang, Z. Band invariants for perturbations of the harmonic oscillator, preprint. September, 2011.
Hall, B.C. Holomorphic methods in analysis and mathematical physics. First Summer School in Analysis and Mathematical Physics (Cuernavaca Morelos, 1998). Con- temp. Math., 260, Amer. Math. Soc., Providence, RI, (2000), 1-59.
Helffer, B. and Robert, D. Puits de potentiel généralisés et asymptotique semi- classique. Ann. Inst. H. Poincaré Phys. Théor. 41 (1984), no. 3, 291-331.
Herbst, I.W. Dilation analyticity in constant electric field. I. The two body problem. Comm. Math. Phys. 64 (1979), no. 3, 279-298.
Hislop, P.D., Sigal, I.M. Introduction to spectral theory. With applications to Schrodinger operators. Applied Mathematical Sciences, 113. Springer Verlag, New York, (1996).
Hislop P. and Villegas-Blas C. Semiclassical Szego limit of resonance clusters for the hydrogen atom Stark Hamiltonian. Asymptotic Analysis, 2011, to be published.
Hörmander, L. The Analysis of Linear Partial Differential Operators. Vol. I (Distribution Theory and Fourier Analysis). Springer Verlag, second edition, (1990).
Hörmander, L. The Analysis of Linear Partial Differential Operators. Vol. III (Pseudo-Differential Operators). Springer Verlag, second edition, (1990).
Kato, T. Perturbation Theory for Linear Operators. Classics in Mathematics. Springer Verlag, (1995). Reprinted of the 1980 edition.
Klauder J.R. and Skagerstam B.S. Coherent States: Applications in Physics and Mathematical Physics. Singapore: World Scientific, (1985).
Martinez, A. An Introduction to Semiclassical and Microlocal Analysis. SpringerVerlag, (2002).
Moser, J. Regularization of Kepler’s problem and the averaging method on a manifold. Comm. Pure Appl. Math. 23 (1970), 609-636.
Muller, C. Spherical Harmonics. Lecture Notes in Mathematics 17. Springer Verlag, (1966).
Pushnitski, A.; Raikov G. and Villegas-Blas, C. Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian. October, (2011).
Thomas, L.E.; Villegas-Blas, C. Singular continuous limiting eigenvalue distributions for Schrodinger operators on a 2-sphere. J. Funct. Anal. 141 (1996), no. 1, 249-273.
Thomas, L.E.; and Villegas-Blas, C. Asymptotics of Rydberg States for the Hydrogen Atom. Commun. Math. Phys., 187, (1997), 623-645.
Uribe, A.; Villegas-Blas, C. Asymptotics of spectral clusters for a perturbation of the hydrogen atom. Comm. Math. Phys. 280 (2008), no. 1, 123-144.
Verdiere, Y.C. de. Spectre conjoint d’opérateurs pseudo-différentiels qui commutent. II. Le cas intégrable. Math Z. 171, (1980), 51-73.
Vock, E. and Hunziker, W. Stability of Schroodinger eigenvalue problems. Comm. Math. Phys. 83, no. 2, (1982), 281-302.
Weinstein, A. Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44 (1977), no. 4, 883-892.
Widom, H. Eigenvalue distribution theorems for certain homogeneous spaces. J. Funct. Anal. 32 (1979), no. 2, 139-147.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel
About this paper
Cite this paper
Ojeda-Valencia, D., Villegas-Blas, C. (2012). On Limiting Eigenvalue Distribution Theorems in Semiclassical Analysis. In: Benguria, R., Friedman, E., Mantoiu, M. (eds) Spectral Analysis of Quantum Hamiltonians. Operator Theory: Advances and Applications, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0414-1_11
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0414-1_11
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0413-4
Online ISBN: 978-3-0348-0414-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)