Intertwined Hamiltonians in two-dimensional curved spaces

Samani, Keivan Aghababaei; Zarei, Mina

doi:10.1016/j.aop.2004.11.002

Cited by 47 publications

(10 citation statements)

References 14 publications

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“…Remark 13. The relevance of intertwining relations and of the intertwining operators is discussed in detail, for instance, in [6,25,26,30]. They turn out to be very useful in the deduction of the eigenvalues and eigenvectors for certain pairs of Hamiltonians, obeying suitable intertwining relations.…”

Section: Working With Riesz Basesmentioning

confidence: 99%

Hamiltonians defined by biorthogonal sets

Багарелло¹,

Bellomonte²

2017

J. Phys. A: Math. Theor.

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In some recent papers, the studies on biorthogonal Riesz bases has found a renewed motivation because of their connection with pseudo-hermitian Quantum Mechanics, which deals with physical systems described by Hamiltonians which are not self-adjoint but still may have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed is some previous papers. However, in many physical models, one has to deal not with o.n. bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of Gquasi basis and we show a series of conditions under which a definition of non self-adjoint Hamiltonian with purely point real spectra is still possible.

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Section: Working With Riesz Basesmentioning

confidence: 99%

Hamiltonians defined by biorthogonal sets

Багарелло¹,

Bellomonte²

2017

J. Phys. A: Math. Theor.

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“…• Formulation of multidimensional SUSY QM in arbitrary curvilinear coordinates was developed in [202] (see also [203][204][205]). This formalism could be useful for the investigation of SUSY design of different cosmological and brain-world models.…”

Section: Perspectives Some Applications and Missing Pointsmentioning

confidence: 99%

Nonlinear supersymmetric quantum mechanics: concepts and realizations

Andrianov¹,

Ioffe²

2012

J. Phys. A: Math. Theor.

128

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Abstract. Nonlinear SUSY approach to spectral problems in Quantum Mechanics is reviewed. Its building from the chains (ladders) of linear SUSY systems is outlined and different one-dimensional and two-dimensional realizations are described. For various two-dimensional models it is elaborated how Nonlinear SUSY provides two new methods of supersymmetric separation of variables. In the framework of these methods partial and/or complete solution of some two-dimensional models becomes possible. The full classification of ladder-reducible and irreducible chains of SUSY algebras in one-dimensional QM is given. Emergence of hidden symmetries and spectrum generating algebras is elucidated in the context of Nonlinear SUSY in one-dimensional stationary and non-stationary as well as in two-dimensional QM.

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“…Remark. It might be interesting to recall that the intertwining operators, such as S ϕ and S Ψ , are quite useful in quantum mechanics, PT -symmetric or not [38][39][40][41][42], in order to deduce eigenvectors of certain Hamiltonians connected by intertwining relations. For instance, let us assume that ϕ n is an eigenstate of a certain operator H 1 with eigenvalue E n : H 1 ϕ n = E n ϕ n , and let us also assume that two other operators H 2 and X exist such that ϕ n / ∈ ker(X) and that the intertwining relation XH 1 = H 2 X is satisfied.…”

Section: Appendix a Some General Facts For Non-hermitian Hamiltoniansmentioning

confidence: 99%

PT-symmetric graphene under a magnetic field

Багарелло

Hatano

2016

Proc. R. Soc. A.

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We propose a PT -symmetrically deformed version of the graphene tight-binding model under a magnetic field. We analyse the structure of the spectra and the eigenvectors of the Hamiltonians around the K and K points, both in the PT -symmetric and PT -broken regions. In particular, we show that the presence of the deformation parameter V produces several interesting consequences, including the asymmetry of the zeroenergy states of the Hamiltonians and the breakdown of the completeness of the eigenvector sets. We also discuss the biorthogonality of the eigenvectors, which turns out to be different in the PT -symmetric and PT -broken regions.

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Intertwined Hamiltonians in two-dimensional curved spaces

Cited by 47 publications

References 14 publications

Hamiltonians defined by biorthogonal sets

Hamiltonians defined by biorthogonal sets

Nonlinear supersymmetric quantum mechanics: concepts and realizations

PT-symmetric graphene under a magnetic field

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