Nodal properties of the scaled quartic anharmonic oscillator

Shanley, Paul E.

doi:10.1016/0003-4916(88)90004-8

Cited by 14 publications

(11 citation statements)

References 32 publications

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“…In some quantum models presented here there are resonances and associated metastable states. The notion of resonance in quantum mechanics is similar to the same notion in acoustics as discussed by Pietro Mengoli [1670] (see also [Martinez and Nédélec 2012]).…”

Section: Introductionmentioning

confidence: 77%

“…The spectral theorem [Kato 1966;Reed and Simon 1975; allows one to associate the self-adjoint Hamiltonian with the real energy spectrum. A state is a class of equivalence of vectors ψ ∈ Ᏼ of unitary norm, with ψ ∼ λψ, |λ| = 1.…”

Section: Essentials Of Quantum Mechanicsmentioning

confidence: 99%

“…Actually, in usual quantum mechanics with a family of self-adjoint Hamiltonians for all real parameters, the delocalization of the states is stable and the level crossings are forbidden [Kato 1966;Reed and Simon 1975;. Also in the case of single-well PT Hamiltonians [Shin 2002;Caliceti 2000;Grecchi and Martinez 2013;Grecchi et al 2009;2010] the states are always localized about the well and the crossings are absent.…”

Section: The Cubic P T -Symmetric Double Wellmentioning

confidence: 99%

See 2 more Smart Citations

Quantum mechanics: some basic techniques for some basic models, I: The models

Grecchi

2016

Math. Mech. Compl. Sys.

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This is a short review of the main results on the oscillators considered basic models in quantum mechanics. Since the potential is polynomial, it is possible to extend the stationary states to entire functions on the complex plane where the semiclassical theory works better. The control on the energy levels is based on the isolation of the nodes, the zeros stable at the unperturbed limit. A new and simple model for the process of racemization of chiral molecules is added.

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Section: Introductionmentioning

confidence: 77%

Section: Essentials Of Quantum Mechanicsmentioning

confidence: 99%

Section: The Cubic P T -Symmetric Double Wellmentioning

confidence: 99%

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Quantum mechanics: some basic techniques for some basic models, I: The models

Grecchi

2016

Math. Mech. Compl. Sys.

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“…In the case of anharmonic oscillator Ly = −y + x 4 y, By = x 2 y, x ∈ R a question about structure of SRS and its branching points has been raised and solved (!) by Bender and Wu [1]; see also [25][26][27][28]34,35]. The case of Mathieu-Hill operators could be deduced to Example (1.3) + (1.4); it has a longer history (see [2,3,10,19,20,37,39,40]).…”

Section: Examplementioning

confidence: 97%

Trace formula and Spectral Riemann Surfaces for a class of tri-diagonal matrices

Djakov

Mityagin

2006

Journal of Approximation Theory

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“…Such results are useful and complementary to the rigorous results we are showing here. Indeed we believe that only the nodal analysis, begun in the papers [10][11][12][13], can give a clear and exhaustive analysis of the level crossings, for which a generalization of the method of control of the zeros by Loeffel-Martin [1] as well as a generalized semiclassical theory are useful tools. Unfortunately, it is not easy to prove the existence of these crossings, and it is even harder to give the selection rule on the two pair of levels (at least) involved in a crossing.…”

Section: Introductionmentioning

confidence: 99%

Level crossings in aPT-symmetric double well

Giachetti¹,

Grecchi²

2016

J. Phys. A: Math. Theor.

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We consider a PT -symmetric cubic oscillator with an imaginary double well. We prove the existence of an infinite number of level crossings with a definite selection rule. Decreasing the positive parameter from large values, at a parameter n we find the crossing of the pair of levels (E 2n+1 ( ), E 2n ( )) becoming the pair of levels (E + n ( ), E − n ( )). For large parameters, a level is a holomorphic function E m ( ) with different semiclassical behaviors, E ± j ( ), along different paths. The corresponding m-nodes delocalized state ψ m ( ) behaves along the same paths as the semiclassical j-nodes states ψ ± j ( ), localized at one of the wells x ± respectively. In particular, if the crossing parameter n is by-passed from above, the levels E 2n+(1/2)±(1/2) ( ) have respectively the semiclassical behaviors of the levels E ∓ n ( ) along the real axis. These results are obtained by the control of the nodes. There is evidence that the parameters n accumulate at zero and the accumulation point of the corresponding energies is an instability point of a subset of the Stokes complex called the monochord, consisting of the vibrating string and the sound board.

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Nodal properties of the scaled quartic anharmonic oscillator

Cited by 14 publications

References 32 publications

Quantum mechanics: some basic techniques for some basic models, I: The models

Quantum mechanics: some basic techniques for some basic models, I: The models

Trace formula and Spectral Riemann Surfaces for a class of tri-diagonal matrices

Level crossings in aPT-symmetric double well

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