Dual oscillators and quantum pendulums: Spectrum and correlators

Andrianov, Alexander A.

doi:10.1103/physrevd.76.025003

Cited by 17 publications

(21 citation statements)

References 26 publications

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“…This observation has previously been made in [58]. As discussed in [11], the approach of [10] also leads to the same conclusion.…”

supporting

confidence: 71%

“…The notion of the transpose used in [37,38,42] corresponds to choosing the time-reversal operator T as τ . 11 This will be explained in detail in Subsection 2.3.…”

mentioning

confidence: 99%

“…11 This means that using different inner products leads to physically identical theories, or more correctly to different representations of a single physical theory. In conventional QM, one eliminates the chance of employing these alternative representations by adopting the usual (L 2 -) inner product as the only viable choice.…”

mentioning

confidence: 99%

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Pseudo-Hermitian Representation of Quantum Mechanics

Mostafazadeh

2010

Int. J. Geom. Methods Mod. Phys.

865

1,210

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A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as PT , the true meaning and significance of the so-called charge operators C and the CPT -inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between localnon-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos, and biophysics.

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“…This observation has previously been made in [58]. As discussed in [11], the approach of [10] also leads to the same conclusion.…”

supporting

confidence: 71%

“…The notion of the transpose used in [37,38,42] corresponds to choosing the time-reversal operator T as τ . 11 This will be explained in detail in Subsection 2.3.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Pseudo-Hermitian Representation of Quantum Mechanics

Mostafazadeh

2010

Int. J. Geom. Methods Mod. Phys.

865

1,210

View full text Add to dashboard Cite

show abstract

“…(10). The equivalence between a non-hermitean Hamiltonian which preserves the P T symmetry and an Hermitean one was already discussed by a number of authors [35][36][37] . The Schrödinger equation (9) allow us to obtain the asymptotic free states.…”

Section: A Geometry Effectsmentioning

confidence: 99%

Electronic properties of bilayer graphene catenoid bridge

et al. 2020

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We study the properties of an electron on a catenoid surface. The catenoid is understood as a realization of a bridge connecting two graphene layer by a smooth surface. The curvature induces a symmetrical reflectionless potential well around the bridge with one bound-state for m = 0. For m = 0, a centrifugal potential barrier arises controlling the tunnelling between the layers. An external electric field breaks the parity symmetry and provides a barrier that controls the conductance from one layer to another. By applying a constant magnetic field the effective potential exhibits a confining double-well potential nearby the bridge. We obtain the corresponding bound states and study the effects of the curvature on the Landau levels.

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“…The spectral equivalence of these two Hamiltonians has long been known and has been examined in many papers [8,9,10,11,12,13,14]. Spectral equivalences between pairs of non-Hermitian and Hermitian Hamiltonians have been discussed in the past [15,16] and a number of approximately equivalent pairs of Hamiltonians have been constructed (see, for example, Refs.…”

mentioning

confidence: 99%

Exact isospectral pairs of {\cal P}{\cal T} symmetric Hamiltonians

Bender¹,

Hook²

2008

J. Phys. A: Math. Theor.

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Abstract.A technique for constructing an infinite tower of pairs of PT -symmetric Hamiltonians,Ĥ n andK n (n = 2, 3, 4, . . .), that have exactly the same eigenvalues is described and illustrated by means of three examples (n = 2, 3, 4). The eigenvalue problem for the first HamiltonianĤ n of the pair must be posed in the complex domain, so its eigenfunctions satisfy a complex differential equation and fulfill homogeneous boundary conditions in Stokes' wedges in the complex plane. The eigenfunctions of the second HamiltonianK n of the pair obey a real differential equation and satisfy boundary conditions on the real axis. This equivalence constitutes a proof that the eigenvalues of both Hamiltonians are real. Although the eigenvalue differential equation associated withK n is real, the HamiltonianK n exhibits quantum anomalies (terms proportional to powers of ). These anomalies are remnants of the complex nature of the equivalent HamiltonianĤ n . For the cases n = 2, 3, 4 in the classical limit in which the anomaly terms inK n are discarded, the pair of Hamiltonians H n, classical and K n, classical have closed classical orbits whose periods are identical.

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Dual oscillators and quantum pendulums: Spectrum and correlators

Cited by 17 publications

References 26 publications

Pseudo-Hermitian Representation of Quantum Mechanics

Pseudo-Hermitian Representation of Quantum Mechanics

Electronic properties of bilayer graphene catenoid bridge

Exact isospectral pairs of {\cal P}{\cal T} symmetric Hamiltonians

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