Exploring branched Hamiltonians for a class of nonlinear systems

Bagchi, B.; Modak, Subhrajit; Panigrahi, Prasanta K.; Ruzicka, Frantisek; Znojil, Miloslav

doi:10.1142/s0217732315502132

Cited by 6 publications

(4 citation statements)

References 43 publications

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“…It is important that the construction circumvents the necessity of the analysis of the influence of the essential singularity upon the very definition of the (nonlinear) Hamiltonian operator [45].…”

Section: Nonlinear Equations On the Complex Contoursmentioning

confidence: 99%

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

2017

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Schrödinger equations with non-Hermitian, but P T -symmetric quantum potentials V(x) found, recently, a new field of applicability in classical optics. The potential acquired there a new physical role of an "anomalous" refraction index. This turned attention to the nonlinear Schrödinger equations in which the interaction term becomes state-dependent, V(x) → W(ψ(x), x). Here, the state-dependence in W(ψ(x), x) is assumed logarithmic, and some of the necessary mathematical assumptions, as well as some of the potential phenomenological consequences of this choice are described. Firstly, an elementary single-channel version of the nonlinear logarithmic model is outlined in which the complex self-interaction W(ψ(x), x) is regularized via a deformation of the real line of x into a self-consistently constructed complex contour C. The new role played by P T -symmetry is revealed. Secondly, the regularization is sought for a multiplet of equations, coupled via the same nonlinear self-interaction coupling of channels. The resulting mathematical structures are shown to extend the existing range of physics covered by the logarithmic Schrödinger equations.

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Section: Nonlinear Equations On the Complex Contoursmentioning

confidence: 99%

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

2017

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“…For these reasons we skip the problems connected with the ambiguity of transition from Lagrangians to Hamiltonians [53] and we restrict our attention just to one of the specific, PU-related quantum Hamiltonian(s), viz., to the operator picked up for analysis, e.g., in Ref. [48], H = −∂ …”

Section: The Context Of Unbounded Spectramentioning

confidence: 99%

Markov constant and quantum instabilities

Pelantová

Starosta

Znojil

2016

J. Phys. A: Math. Theor.

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For a qualitative analysis of spectra of certain two-dimensional rectangular-well quantum systems several rigorous methods of number theory are shown productive and useful. These methods (and, in particular, a generalization of the concept of Markov constant known in Diophantine approximation theory) are shown to provide a new mathematical insight in the phenomenologically relevant occurrence of anomalies in the spectra. Our results may inspire methodical innovations ranging from the description of the stability properties of metamaterials and of certain hiddenly unitary quantum evolution models up to the clarification of the mechanisms of occurrence of ghosts in quantum cosmology.

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“…Models of classical systems with branched structures [1], in either coordinate (x) space or in its momentum (p) counterpart, have of late been a subject of active theoretical inquiry [2][3][4][5][6][7][8][9]. The key idea is that classical Lagrangians possessing time derivatives in excess of quadratic powers inevitably lead to p becoming a multi-valued function of velocity (v), thereby yielding a multi-valued class of Hamiltonian systems.…”

Section: Introductionmentioning

confidence: 99%

“…Following it, Curtright and Zachos [3] analyzed certain representative models for a classical Lagrangian described by a pair of convex, smoothly tied functions of v. Proceeding to the quantum domain shows that the double-valued Hamiltonian obtained has an inherent feature of being expressible in a supersymmetric form in the p space. Subsequently, a class of nonlinear systems whose Hamiltonians exhibit branching was explored by Bagchi et al [4] who also considered the possibility of quantization for some specific cases of the underlying coupling parameter.…”

Section: Introductionmentioning

confidence: 99%

Branched Hamiltonians for a Class of Velocity Dependent Potentials

Bagchi

Kamil

Tummuru

et al. 2017

J. Phys.: Conf. Ser.

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Hamiltonians that are multivalued functions of momenta are of topical interest since they correspond to the Lagrangians containing higher-degree time derivatives. Incidentally, such classes of branched Hamiltonians lead to certain not too well understood ambiguities in the procedure of quantization. Within this framework, we pick up a model that samples the latter ambiguities and simultaneously turns out to be amenable to a transparent analytic and perturbative treatment.

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Exploring branched Hamiltonians for a class of nonlinear systems

Cited by 6 publications

References 43 publications

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

Markov constant and quantum instabilities

Branched Hamiltonians for a Class of Velocity Dependent Potentials

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