On the Stable Eigenvalues of Perturbed Anharmonic Oscillators in Dimension Two

We study the asymptotic behavior of the spectrum of a quantum system which is a perturbation of a spherically symmetric anharmonic oscillator in dimension 2. We prove that a large part of its eigenvalues can be obtained by Bohr-Sommerfeld quantization rule applied to the normal form Hamiltonian and also admit an asymptotic expansion at infinity. The proof is based on the generalization to the present context of the normal form approach developed in [BLM20b] (see also [PS10]) for the particular case of Td.

[1]  Riccardo Montalto,et al.  On the spectrum of the Schrödinger operator on 𝕋d: a normal form approach , 2019, Communications in Partial Differential Equations.

[2]  Riccardo Montalto,et al.  Spectral asymptotics of all the eigenvalues of Schrödinger operators on flat tori , 2020, Nonlinear Analysis.

[3]  Dario Bambusi,et al.  Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators , 1996 .

[4]  R. Shterenberg,et al.  Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schrödinger operators , 2010, 1004.2939.

[5]  Riccardo Montalto,et al.  Growth of Sobolev norms for unbounded perturbations of the Laplacian on flat tori. , 2020 .

[6]  Einzelwerken Muster,et al.  Invent , 2021, Encyclopedic Dictionary of Archaeology.

[7]  Yulia Karpeshina.  Perturbation series for the Schrödinger operator with a periodic potential near planes of diffraction , 1996 .

[8]  B. Helffer,et al.  Proprietes asymptotiques du spectre dioperateurs pseuwdifferentiels sur IRn , 1982 .

[9]  A. Charbonnel.  Calcul fonctionnel a plusieurs variables pour des operateurs pseudodifferentiels dansRn , 1983 .

[10]  野村栄一,et al.  2 , 1900, The Hatak Witches.

[11]  Nonlinearity , 2021, Encyclopedia of Mathematical Geosciences.

[12]  Nicolas Roy.  A Semi-Classical K.A.M. Theorem , 2004, math-ph/0403045.

[13]  Johannes J. Duistermaat,et al.  On global action‐angle coordinates , 1980 .

[14]  H. Knörrer,et al.  The perturbatively stable spectrum of a periodic Schrödinger operator , 1990 .

[15]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[16]  L. Bates,et al.  No monodromy in the champagne bottle, or singularities of a superintegrable system , 2016 .

[17]  Leonid Parnovski,et al.  Bethe–Sommerfeld Conjecture , 2008, 0801.3096.

[18]  K. Allegaert,et al.  (Preprint) , 2018 .

[19]  D. Bambusi,et al.  Exponential Stability in the Perturbed Central Force Problem , 2017, Regular and Chaotic Dynamics.

[20]  A. Charbonnel.  Localisation et developpment asymptotique des elements du spectre conjoint d'operateurs pseudodifferentiels qui commutent , 1986 .

[21]  M. Berti,et al.  Degenerate KAM theory for partial differential equations , 2011 .

[22]  Jean,et al.  Henri Poincare，为科学服务的一生 , 2006 .

[23]  B. M. Fulk.  MATH , 1992 .

[24]  D. Bambusi,et al.  Nekhoroshev theorem for perturbations of the central motion , 2016, 1610.02262.

[25]  T. Paul,et al.  Normal Forms and Quantization Formulae , 1999 .

[26]  A. Sobolev,et al.  Bethe-Sommerfeld conjecture for periodic operators with strong perturbations , 2009, 0907.0887.

[27]  Diophantine tori and spectral asymptotics for nonselfadjoint operators , 2005, math/0502032.

[28]  Sur le théorème de Bertrand (d'après Michael Herman) , 2004, Ergodic Theory and Dynamical Systems.

[29]  H. Knörrer,et al.  Perturbatively unstable eigenvalues of a periodic Schrödinger operator , 1991 .

[30]  B. Helffer,et al.  Asymptotique des niveaux d’énergie pour des hamiltoniens a un degre de liberté , 1982 .

[31]  O. Veliev.  Multidimensional Periodic Schrödinger Operator , 2019 .

[32]  D. Robert,et al.  Reducibility of the Quantum Harmonic Oscillator in $d$-dimensions with Polynomial Time Dependent Perturbation , 2017, 1702.05274.