Perturbation theory of odd anharmonic oscillators

Caliceti, Emanuela; Graffi, Sandro; Maioli, Marco

doi:10.1007/bf01962591

Cited by 220 publications

(304 citation statements)

References 11 publications

Supporting

Mentioning

298

Contrasting

Unclassified

Order By: Relevance

“…also a few general relevant remarks on this topic in [17]). Our observation that the imaginary odd powers of s in (9) cannot contribute to the Rayleigh-Schrödinger series only supports the reality of the spectrum in the limit R → ∞.…”

Section: Discussionmentioning

confidence: 96%

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

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 96%

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

View full text Add to dashboard Cite

show abstract

“…Certainly, the latter family is not small. Pars pro toto it contains Hamiltonians of relativistic quantum mechanics [41,42], the well-known P -symmetric imaginary cubic oscillator [43][44][45][46] (which appears, after a more detailed scrutiny, strongly nonlocal [31,47]), its power-law generalizations [10][11][12]48] as well as exactly solvable models [49][50][51][52], models with methodical relevance in the context of supersymmetry [53,54], realistic and computation-friendly interacting-boson models of heavy nuclei [2], benchmark candidates for classification of quantum catastrophes [55][56][57], and so forth.…”

Section: A2 Physical Inner Productsmentioning

confidence: 99%

Hermitian–Non-Hermitian Interfaces in Quantum Theory

Znojil

2018

Advances in High Energy Physics

View full text Add to dashboard Cite

In the global framework of quantum theory, the individual quantum systems seem clearly separated into two families with the respective manifestly Hermitian and hiddenly Hermitian operators of their Hamiltonian. In the light of certain preliminary studies, these two families seem to have an empty overlap. In this paper, we will show that whenever the interaction potentials are chosen to be weakly nonlocal, the separation of the two families may disappear. The overlaps alias interfaces between the Hermitian and non-Hermitian descriptions of a unitarily evolving quantum system in question may become nonempty. This assertion will be illustrated via a few analytically solvable elementary models.

show abstract

“…It was shown in [DDT01] that the spectrum of H c is real and positive. Moreover, H c is closed and has compact resolvent [CGM80,Mez01] so the spectrum is also discrete. On the other hand, Novak recently obtained the following result Theorem 1.1 ( [Nov14]).…”

Section: Non-selfadjoint Operators and Pseudospectramentioning

confidence: 99%

A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

Dondl

Dorey

Rösler

2016

Appl Math Res Express

View full text Add to dashboard Cite

We are concerned with the non-normal Schrödinger operatorThe spectrum of this operator is discrete and contained in the positive half plane. In general, the ε-pseudospectrum of H will have an unbounded component for any ε > 0 and thus will not approximate the spectrum in a global sense.By exploiting the fact that the semigroup e −tH is immediately compact, we show a complementary result, namely that for every δ > 0, R > 0 there exists an ε > 0 such that the ε-pseudospectrum{z : |z − λ| < δ}.In particular, the unbounded part of the pseudospectrum escapes towards +∞ as ε decreases.Additionally, we give two examples of non-selfadjoint Schrödinger operators outside of our class and study their pseudospectra in more detail.Mathematics Subject Classification (2010): 35P99, 47B44.

show abstract

Perturbation theory of odd anharmonic oscillators

Cited by 220 publications

References 11 publications

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

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

Hermitian–Non-Hermitian Interfaces in Quantum Theory

A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

Contact Info

Product

Resources

About