Symmetry, Integrability and Geometry: Methods and Applications, 2011, том 7
http://dspace.nbuv.gov.ua:80/handle/123456789/146008
2023-01-28T16:02:01ZProjective Metrizability and Formal Integrability
http://dspace.nbuv.gov.ua:80/handle/123456789/148091
Projective Metrizability and Formal Integrability
Bucataru, I.; Muzsnay, Z.
The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential operator P₁ and a set of algebraic conditions on semi-basic 1-forms. We discuss the formal integrability of P₁ using two sufficient conditions provided by Cartan-Kähler theorem. We prove in Theorem 4.2 that the symbol of P₁ is involutive and hence one of the two conditions is always satisfied. While discussing the second condition, in Theorem 4.3 we prove that there is only one obstruction to the formal integrability of P₁, and this obstruction is due to the curvature tensor of the induced nonlinear connection. When the curvature obstruction is satisfied, the projective metrizability problem reduces to the discussion of the algebraic conditions, which as we show are always satisfied in the analytic case. Based on these results, we recover all classes of sprays that are known to be projectively metrizable: flat sprays, isotropic sprays, and arbitrary sprays on 1- and 2-dimensional manifolds. We provide examples of sprays that are projectively metrizable without being Finsler metrizable.
2011-01-01T00:00:00ZBreaking Pseudo-Rotational Symmetry through H₊² Metric Deformation in the Eckart Potential Problem
http://dspace.nbuv.gov.ua:80/handle/123456789/148090
Breaking Pseudo-Rotational Symmetry through H₊² Metric Deformation in the Eckart Potential Problem
Leija-Martinez, N.; Alvarez-Castillo, D.E.; Kirchbach, M.
The peculiarity of the Eckart potential problem on H₊² (the upper sheet of the two-sheeted two-dimensional hyperboloid), to preserve the (2l+1)-fold degeneracy of the states typical for the geodesic motion there, is usually explained in casting the respective Hamiltonian in terms of the Casimir invariant of an so(2,1) algebra, referred to as potential algebra. In general, there are many possible similarity transformations of the symmetry algebras of the free motions on curved surfaces towards potential algebras, which are not all necessarily unitary. In the literature, a transformation of the symmetry algebra of the geodesic motion on H₊² towards the potential algebra of Eckart's Hamiltonian has been constructed for the prime purpose to prove that the Eckart interaction belongs to the class of Natanzon potentials. We here take a different path and search for a transformation which connects the (2l+1) dimensional representation space of the pseudo-rotational so(2,1) algebra, spanned by the rank-l pseudo-spherical harmonics, to the representation space of equal dimension of the potential algebra and find a transformation of the scaling type. Our case is that in so doing one is producing a deformed isometry copy to H₊² such that the free motion on the copy is equivalent to a motion on H₊², perturbed by a coth interaction. In this way, we link the so(2,1) potential algebra concept of the Eckart Hamiltonian to a subtle type of pseudo-rotational symmetry breaking through H₊²metric deformation. From a technical point of view, the results reported here are obtained by virtue of certain nonlinear finite expansions of Jacobi polynomials into pseudo-spherical harmonics. In due places, the pseudo-rotational case is paralleled by its so(3) compact analogue, the cotangent perturbed motion on S2. We expect awareness of different so(2,1)/so(3) isometry copies to benefit simulation studies on curved manifolds of many-body systems.
2011-01-01T00:00:00ZResolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs
http://dspace.nbuv.gov.ua:80/handle/123456789/148089
Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs
Sokolov, A.V.
This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed of biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration are taken with continuous spectrum and the following cases are examined: an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum and an exceptional point situated inside of continuous spectrum. In the present work the rigorous proofs are given for the resolutions of identity in both cases.
2011-01-01T00:00:00ZResolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
http://dspace.nbuv.gov.ua:80/handle/123456789/148088
Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
Andrianov, A.A.; Sokolov, A.V.
Resolutions of identity for certain non-Hermitian Hamiltonians constructed from biorthogonal sets of their eigen- and associated functions are given for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration possess the continuous spectrum and the following peculiarities are investigated: (1) the case when there is an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum; (2) the case when there is an exceptional point situated inside of continuous spectrum. The reductions of the derived resolutions of identity under narrowing of the classes of employed test functions are revealed. It is shown that in the case (1) some of associated functions included into the resolution of identity are normalizable and some of them may be not and in the case (2) the bounded associated function corresponding to the exceptional point does not belong to the physical state space. Spectral properties of a SUSY partner Hamiltonian for the Hamiltonian with an exceptional point are examined.
2011-01-01T00:00:00Z