Equivalence of unstable anharmonic oscillators and double wells

Buslaev, V. S.; Grecchi, V.

doi:10.1088/0305-4470/26/20/035

Cited by 216 publications

(368 citation statements)

References 7 publications

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“…coupling on the one hand to the double-well potential at positive coupling with a symmetry-breaking term on the other hand [18,19]. (This result, initially conjectured on the basis of numerical evidence, was first proven later by path integral manipulations [20] or recursion formulae [21], and later generalized to arbitrary j [19]. )…”

Section: O(ν)-anharmonic Oscillator and Fokker-planck Equationmentioning

confidence: 88%

Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions

2004

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We consider specific quantum mechanical model problems for which perturbation theory fails to explain physical properties like the eigenvalue spectrum even qualitatively, even if the asymptotic perturbation series is augmented by resummation prescriptions to "cure" the divergence in large orders of perturbation theory. Generalizations of perturbation theory are necessary which include instanton configurations, characterized by nonanalytic factors exp(−a/g) where a is a constant and g is the coupling. In the case of one-dimensional quantum mechanical potentials with two or more degenerate minima, the energy levels may be represented as an infinite sum of terms each of which involves a certain power of a nonanalytic factor and represents itself an infinite divergent series. We attempt to provide a unified representation of related derivations previously found scattered in the literature. For the considered quantum mechanical problems, we discuss the derivation of the instanton contributions from a semi-classical calculation of the corresponding partition function in the path integral formalism. We also explain the relation with the corresponding WKB expansion of the solutions of the Schrödinger equation, or alternatively of the Fredholm determinant det(H − E) (and some explicit calculations that verify this correspondence). We finally recall how these conjectures naturally emerge from a leading-order summation of multi-instanton contributions to the path integral representation of the partition function. The same strategy could result in new conjectures for problems where our present understanding is more limited.

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Section: O(ν)-anharmonic Oscillator and Fokker-planck Equationmentioning

confidence: 88%

Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions

2004

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“…Only recently, the first attempts in this direction have been made [9,10]. A significant improvement of our understanding of the underlying complex dynamics has been offered by Bender and Boettcher [11].…”

Section: Pt Symmetric Non-hermitian Modelsmentioning

confidence: 99%

Supersymmetry Without Hermiticity

Lévai

2004

Czechoslovak Journal of Physics

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“…[6,7] for the γ = 0 weaker-singularity cases. One should add that at γ = 0 the regularization proved comparatively easy as long as it could rely upon the centrifugal-force nature of the γ = 0 repulsion.…”

Section: Introductionmentioning

confidence: 99%

“…One should add that at γ = 0 the regularization proved comparatively easy as long as it could rely upon the centrifugal-force nature of the γ = 0 repulsion. At the same time, one may expect that at γ > 0 and g 2 > 0 the regularization of the singular spike may still be achieved by the same (in fact, by the Buslaev's and Grecchi's [6]) complex shift of the coordinate x = s −iε in which the real shift parameter ε > 0 is a constant while the value of s is kept variable, s ∈ (−∞, ∞).…”

Section: Introductionmentioning

confidence: 99%

The Large −g Observability of the Low-Lying Energies in the Strongly Singular Potentials V (x) = x 2 + g 2/x 6 after their PT −symmetric Regularization

Znojil

2014

Int J Theor Phys

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The large−g observability of the low-lying energies in the strongly singular potentials V (x) = x 2 + g 2 /x 6 after their PT −symmetric regularization Miloslav ZnojilNuclear Physics Institute ASCR, 250 68Řež, Czech Republic e-mail: znojil@ujf.cas.cz AbstractThe elementary quadratic plus inverse sextic interaction V (x) = x 2 +g 2 /x 6 containing a strongly singular repulsive core in the origin is made regular by a complex shift of coordinate x = s − iε. The shift ε > 0 is fixed while the value of s is kept real and potentially observable, s ∈ (−∞, ∞). The low-lying energies of bound states are found in closed form for the large couplings g ≫ 1. Within the asymptotically vanishing O(g −1/4 ) error bars these energies are real so that the time-evolution of the system may be expected unitary in an ad hoc physical Hilbert space.

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Equivalence of unstable anharmonic oscillators and double wells

Cited by 216 publications

References 7 publications

Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions

Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions

Supersymmetry Without Hermiticity

The Large −g Observability of the Low-Lying Energies in the Strongly Singular Potentials V (x) = x 2 + g 2/x 6 after their PT −symmetric Regularization

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