We discuss spectral properties of the selfadjoint operator \( \begin{gathered} - \frac{{{d^2}}}{{d{t^2}}} + {\left( {\frac{{{t^{k + 1}}}}{{k + 1}} - \alpha } \right)^2} \hfill \\ \hfill \\ \end{gathered} \) in L 2(ℝ) for odd integers k. We prove that the minimum over α of the ground state energy of this operator is attained at a unique point which tends to zero as k tends to infinity. We also show that the minimum is nondegenerate. These questions arise naturally in the spectral analysis of Schrödinger operators with magnetic field. Bibliography: 13 titles. Illustrations: 2 figures.
Similar content being viewed by others
References
X.-B. Pan, K.-H. Kwek, “Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains,” Trans. Am. Math. Soc. 354, No. 10, 4201–4227 (electronic) (2002).
B. Helffer, A. Morame, “Magnetic bottles in connection with superconductivity,” J. Funct. Anal. 185, No. 2, 604–680 (2001).
J. Aramaki, “Asymptotics of eigenvalue for the Ginzburg-Landau operator in an applied magnetic field vanishing of higher order,” Int. J. Pure Appl. Math. Sci. 2, No. 2, 257–281 (2005).
B. Helffer, Y. A. Kordyukov, “Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells,” In Mathematical Results in Quantum Mechanics, pp. 137–154. World Sci. Publ., Hackensack, NJ (2008).
B. Helffer, Y. A. Kordyukov, “Semi-classical analysis of Schrödinger operators with magnetic wells,” Contemporary Math. (2009). [To appear]
B. Helffer, Y. A. Kordyukov, “Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: Analysis near the bottom,” J. Funct. Anal. 257, 3043–3081 (2009).
B. Helffer, “The Montgomery model revisited,” In: Colloquium Mathematicum, volume in honor of A. Hulanicki (2009). [To appear]
R. Montgomery, “Hearing the zero locus of a magnetic field,” Comm. Math. Phys. 168, No. 3, 651–675 (1995).
B. Helffer, Y. A. Kordyukov, Complements on Montgomery like model : k > 1, (2009). [Unpublished]
B. Simon, “Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions,” Ann. Inst. H. Poincaré Sect. A (N.S.) 38, No. 3, 295–308 (1983).
B. Helffer, J. Sjöstrand, “Multiple wells in the semiclassical limit. I,” Commun. Partial Differ. Equ. 9, No. 4, 337–408 (1984).
B. Simon, “A canonical decomposition for quadratic forms with applications to monotone convergence theorems,” J. Funct. Anal. 28, No. 3, 377–385 (1978).
S. Flügge, Practical Quantum Mechanics. Springer, Berlin (1999); A translation of the 1947 German original.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problems in Mathematical Analysis 44 January, 2010, pp. 99–114.
Rights and permissions
About this article
Cite this article
Helffer, B., Persson, M. Spectral properties of higher order anharmonic oscillators. J Math Sci 165, 110–126 (2010). https://doi.org/10.1007/s10958-010-9784-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-010-9784-5