Factorization of nonlinear supersymmetry in one-dimensional quantum mechanics. II: Proofs of theorems on reducibility

In this paper, we continue to study factorization of supersymmetric (SUSY) transformations in one-dimensional Quantum Mechanics into chains of elementary Darboux transformations with nonsingular coefficients. We define a class of potentials that are invariant under the Darboux-Crum transformations and prove a number of lemmas and theorems substantiating the formulated conjectures on reducibility of differential operators for spectral equivalence transformations. Analysis of the general case is performed with all the necessary proofs. Bibliography: 27 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 214–237.     

Factorization of nonlinear supersymmetry in one-dimensional quantum mechanics. III: precise classification of irreducible intertwining operators

In this part of the work, we present a detailed classification of first order intertwining operators and of really irreducible intertwining second order operators of I, II, and III types. This classification is constructed in dependence of kernel structures of these operators and relations between spectra of intertwined Hamiltonians. It was shown earlier that one can construct from such operators any intertwining operator of arbitrary order with the help of chain (ladder) construction. Bibliography: 25 titles.     

Extended supersymmetry and hidden symmetries in one-dimensional matrix quantum mechanics

We study properties of nonlinear supersymmetry algebras realized in the one-dimensional quantum mechanics of matrix systems. Supercharges of these algebras are differential operators of a finite order in derivatives. In special cases, there exist independent supercharges realizing an (extended) supersymmetry of the same super-Hamiltonian. The extended supersymmetry generates hidden symmetries of the super-Hamiltonian. Such symmetries have been found in models with (2×2)-matrix potentials.     

Darboux transformation,factorization, and supersymmetry in one-dimensional quantum mechanics

We introduce an N-order Darboux transformation operator as a particular case of general transformation operators. It is shown that this operator can always be represented as a product of N first-order Darboux transformation operators. The relationship between this transformation and the factorization method is investigated. Supercharge operators are introduced. They are differential operators of order N. It is shown that these operators and super-Hamiltonian form a superalgebra of order N. For N=2, we have a quadratic superalgebra analogous to the Sklyanin quadratic algebras. The relationship between the transformation introduced and the inverse scattering problem in quantum mechanics is established. An elementary N-parametric potential that has exactly N predetermined discrete spectrum levels is constructed. The paper concludes with some examples of new exactly soluble potentials.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 2, pp. 356–367, August, 1995.     

Deformed supersymmetry,q-oscillator algebra, and related scattering problems in quantum mechanics

We describe extensions of the supersymmetric quantum mechanics (SSQM) (in one dimension) which are characterized by deformed algebras. The supercharges involving higher-order derivatives are introduced, leading to a deformed algebra which incorporates a higher-order polynomial of the Hamiltonian. When supplementing them with dilatations, one finds the class of q-deformed SUSY systems. For a special choice of q-self-similar potentials, the energy spectrum is (partially) generated by the q-oscillator algebra. In contrast to the standard harmonic oscillators, these systems exhibit a continuous spectrum. We investigate the scattering problem in the q-deformed SSQM and introduce the notion of self-similarity in the momentum space for scattering data. An explicit model for the scattering amplitude of a q-oscillator is constructed in terms of a hypergeometric function. This model corresponds to a reflectionless potential with infinitely, many bound states. A general method of realization of the q-oscillator algebra on the space of wave functions for a one-dimensional Schrödinger Hamiltonian is developed. It shows the existence of non-Fock irreducible representations associated with the continuous part of the spectrum and directly related to the deformation. Bibliography: 24 titles.     

Polynomial supersymmetry and dynamical symmetries in quantum mechanics

A polynomial generalization of supersymmetry in quantum mechanics in one and two dimensions is proposed. Polynomial superalgebras in one dimension are classified. In two dimensions, a detailed analysis is made for supercharges of second order with respect to derivatives and it is shown that in all cases the binomial superalgebra leads to hidden dynamical symmetry generated by the central charge.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 463–478, September, 1995.     

Higher-order supersymmetry in quantum mechanics and integrability of two-dimensional Hamiltonians

A new method of finding the dynamic symmetry operators in two-dimensional quantum systems is proposed. This method is based on the SUSY algebra with superchanges of higher order in derivatives. The symmetry operators arise when closing the SUSY algebra for a wide class of potentials, and, in some cases, they are of second order in derivatives. The solutions for potentials admitting symmetry operators of fourth order are also obtained. The investigation of the quasiclassical limit of the superalgebra results in new classical integrals of motion for a certain type of systems. Bibliography: 9 titles. In memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 68–80. Translated by V. A. Andrianov.     

An algebra of pseudodifferential operators and the asymptotics of quantum mechanics

We discuss in detail the regularity properties of a class of pseudodifferential operators on $\text{R}$^l introduced by Grossmann, Loupias, and Stein, which are more regular and more symmetrical than usual pseudodifferential operators. Those operators that are self-adjoint form a suitable class of smooth observables for a nonrelativistic quantum theory. If their symbols are allowed to depend smoothly upon Planck's constant $\text{h}\text{?}$, those operators provide the framework for regular asymptotics expansions as $\text{h}\text{?}\text{→ 0}$ of quantum mechanics around classical mechanics.     

Bogolyubov averaging and normalization procedures in nonlinear mechanics. I

We suggest a new method for asymptotic analysis of nonlinear dynamical systems based on group-the-oretic methods. On the basis of the Bogolyubov averaging method, we develop a new normalization procedure — asymptotic decomposition. We clarify the contribution of this procedure to the interpretation and development of the averaging method for systems in the standard form and systems with several fast variables. According to this method, the centralized system is regarded as a direct analog of the system averaged in Bogolyubov's sense. The operation of averaging is interpreted as the Bogolyubov projector, i.e., the operation of projection of an operator onto the algebra of centralizer.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1171–1188, September, 1994.     

Group analysis of the one-dimensional Boltzmann equation: I. symmetry groups

We consider the one-dimensional Boltzmann equation f _t + cf _x + Ff _c = 0, where the functions f and F are assumed to depend on three variables t, x, and c. We obtain relations defining the symmetry algebra in the general case and also under the additional conditions of conservation of the relations dx = c dt and dc = F dt, which arise from physical considerations. We show that the widest symmetry algebra is obtained in the case of conservation of both relations. This algebra is infinite-dimensional, and its structure is independent of the form of the function F.     

Preparation and third-order nonlinear optical properties of self-organized complex oxide Ce: BaTiO3 quantum dots

Pulsed laser ablation epitaxial Strancky-Krastanow growth of self-organized complex oxide Ce:BaTiO₃ quantum dots on MgO(lOO) substrates is demonstrated. Atomic force microscopy and X-ray diffraction are used to observe the structure and the growth process of the self-organized complex oxide quantum dots. The average dimension, height, and the density of self-organized complex oxide Ce:BaTiO₃ quantum dots are given. The nonlinear refractive indexes of the wetting layer and the selforganized ordered quantum dots are determined by the single beam Z-scan method. The mechanisms of the nonlinear effect enhancement for these low-dimensional structure are discussed.     

Preparation and third-order nonlinear optical properties of self-organized complex oxide Ce:BaTiO3 quantum dots

Pulsed laser ablation epitaxial Strancky-Krastanow growth of self-organized complex oxide Ce:BaTiO3 quantum dots on MgO(100) substrates is demonstrated.Atomic force microscopy and X-ray diffraction are used to observe the structure and the growth process of the self-organized complex oxide quantum dots.The average dimension,height,and the density of self-organized complex oxide Ce:BaTiO3 quantum dots are given.The nonlinear refractive indexes of the wetting layer and the self-organized ordered quantum dots are determined by the single beam Z-scan method.The mechanisms of the nonlinear effect enhancement for these low-dimensional structure are discussed.     

Mathematical problems of stochastic quantum mechanics: Harmonic analysis on phase space and quantum geometry

In this paper we review the mathematical methods and problems that are specific to the programme of stochastic quantum mechanics and quantum spacetime. The physical origin of these problems is explained, and then the mathematical models are developed. Three notions emerge as central to the programme: positive operator-valued (POV) measures on a Hilbert space, reproducing kernel Hilbert spaces, and fibre bundle formulations of quantum geometries. A close connection between the first two notions is shown to exist, which provides a natural setting for introducing a fibration on the associated overcomplete family of vectors. The introduction of group covariance leads to an extended version of harmonic analysis on phase space. It also yields a theory of induced group representations, which extends the results of Mackey on imprimitivity systems for locally compact groups to the more general case of systems of covariance. Quantum geometries emerge as fibre bundles whose base spaces are manifolds of mean stochastic locations for quantum test particles (i.e., spacetime excitons) that display a phase space structure, and whose fibres and structure groups contain, respectively, the aforementioned overcomplete families of vectors and unitary group representations of phase space systems of covariance.Work supported in part by the Natural Science and Engineering Research Council of Canada (NSERC) grants.     

Factorization of the $$ \mathcal{R} $$-matrix for the quantum algebra Uq( sℓ3)

The Yang-Baxter operator is obtained as a product of operators that permute representation parameters in the Lax operators. The construction relies on a factorization of the Lax operator into triangular matrices. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 88–106.     

Limit-point / limit-circle classification of second-order differential operators arising in PT quantum mechanics

We consider a second-order differential equation −y^″ + q(x)y(x) = λy(x) with complex-valued potential q and eigenvalue parameter λ ∈ ℂ. In PT quantum mechanics the potential q is given by q(x) = −(ix)^N+2 on a contour Γ ⊂ ℂ. Via a parametrization we obtain two differential equations on [0, ∞) and (−∞, 0]. We give a limit-point/limit-circle classification of this problem via WKB-analysis. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Preparation and third-order nonlinear optical properties of self-organized complex oxide Ce: BaTiO3 quantum dots

Pulsed laser ablation epitaxial Strancky-Krastanow growth of self-organized complex oxide Ce:BaTiO₃ quantum dots on MgO(lOO) substrates is demonstrated. Atomic force microscopy and X-ray diffraction are used to observe the structure and the growth process of the self-organized complex oxide quantum dots. The average dimension, height, and the density of self-organized complex oxide Ce:BaTiO₃ quantum dots are given. The nonlinear refractive indexes of the wetting layer and the selforganized ordered quantum dots are determined by the single beam Z-scan method. The mechanisms of the nonlinear effect enhancement for these low-dimensional structure are discussed.     

The thermodynamic limit of quantum Coulomb systems Part I. General theory

This article is the first in a series dealing with the thermodynamic properties of quantum Coulomb systems.In this first part, we consider a general real-valued function E defined on all bounded open sets of R³. Our aim is to give sufficient conditions such that E has a thermodynamic limit. This means that the limit E(Ωn)|Ωn|⁻¹ exists for all ‘regular enough’ sequence Ωn with growing volume, |Ωn|→∞, and is independent of the considered sequence.The sufficient conditions presented in our work all have a clear physical interpretation. In the next paper, we show that the free energies of many different quantum Coulomb systems satisfy these assumptions, hence have a thermodynamic limit.