Karlheinz Bay et al 1997 J. Phys. A: Math. Gen. 30 3057 doi:10.1088/0305-4470/30/9/017
Avoided crossings of the quartic oscillator
Karlheinz Bay
, Wolfgang Lay and Alexey Akopyan
The phenomenon of avoided crossings of energy levels in the spectrum of quantum systems is well known. However, being of an exponentially small order it is hard to calculate. In particular, this is the case when the potential is generating a Schrödinger equation of a type which is beyond the hypergeometric one. Recently, there have been attempts to understand this phenomenon in connection with Heun-type differential equations. The most famous example of this class is the quantum quartic oscillator which is governed by the triconfluent case of Heun's differential equation. In the following we consider situations where the fourth-order potential has two minima and we calculate the avoided crossings of its eigenvalue curves in dependence on the asymmetry and the barrier height between the two wells. The results are compared with those obtained from an asymptotic approach of the problem for large values of the control parameter that governs the barrier height.
- PACS
-
03.65.Ge Solutions of wave equations: bound states
02.60.Lj Ordinary and partial differential equations; boundary value problems
- MSC
-
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
81Q15 Perturbation theories for operators and differential equations
- Subjects
- Dates
-
Issue 9 (7 May 1997)
Received 21 January 1997
-
Avoided crossings of the quartic oscillator
Karlheinz Bay et al 1997 J. Phys. A: Math. Gen. 30 3057
-
Changes in relative light fluence measured during laser heating: implications for optical monitoring and modelling of interstitial laser photocoagulation
L C L Chin et al 2001 Phys. Med. Biol. 46 2407
-
Interview: Earth science gets to the core
2008 Phys. Educ. 43 212
-
The black hole final state
Gary T. Horowitz and Juan Maldacena JHEP02(2004)008
-
Temperature-Controllable Preparation of ZnS Nanosaws on Si Substrate
Wu Xiang et al 2008 Chinese Phys. Lett. 25 737
-
Two-frame 3D particle tracking
Francisco Pereira et al 2006 Meas. Sci. Technol. 17 1680
-
Initial yield process around a spherical inclusion in single-crystalline aluminium
T Tsuru and Y Shibutani 2007 J. Phys. D: Appl. Phys. 40 2183
-
Functional imaging in bulk tissue specimens using optical emission tomography: fluorescence preservation during optical clearing
H S Sakhalkar et al 2007 Phys. Med. Biol. 52 2035
-
Atomistic simulations of strain distributions in quantum dot nanostructures
Christopher Kohler 2003 J. Phys.: Condens. Matter 15 133
-
Scaling and formulary of cross-sections for ion–atom impact ionization
Igor D Kaganovich et al 2006 New J. Phys. 8 278
Related review articles
What's this?- Integrable structure of box–ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry
- Theory of fields of extremals for multiple integrals
- Spectral problems in open quantum chaos