Publications (7)0 Total impact
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ABSTRACT: In the present review we deal with the recently introduced method of spectral
parameter power series (SPPS) and show how its application leads to an explicit
form of the characteristic equation for different eigenvalue problems involving
Sturm-Liouville equations with variable coefficients. We consider
Sturm-Liouville problems on finite intervals; problems with periodic potentials
involving the construction of Hill's discriminant and Floquet-Bloch solutions;
quantum-mechanical spectral and transmission problems as well as the eigenvalue
problems for the Zakharov-Shabat system. In all these cases we obtain a
characteristic equation of the problem which in fact reduces to finding zeros
of an analytic function given by its Taylor series. We illustrate the
application of the method with several numerical examples which show that at
present the SPPS method is the easiest in the implementation, the most accurate
and efficient. We emphasize that the SPPS method is not a purely numerical
technique. It gives an analytical representation both for the solution and for
the characteristic equation of the problem. This representation can be
approximated by different numerical techniques and for practical purposes
constitutes a powerful numerical method but most important it offers additional
insight into the spectral and transmission problems.
12/2011;
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ABSTRACT: We determine the kind of parametric oscillators that are generated in the
usual factorization procedure of second-order linear differential equations
when one introduces a constant shift of the Riccati solution of the classical
harmonic oscillator. The mathematical results show that some of these
oscillators could be of physical nature. We give the solutions of the obtained
second-order differential equations and the values of the shift parameter
providing strictly periodic and antiperiodic solutions. We also notice that
this simple problem presents parity-time (PT) symmetry. Possible applications
are mentioned
08/2010;
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ABSTRACT: We provide the representation of quasi-periodic solutions of periodic Dirac
equations in terms of the spectral parameter power series (SPPS) recently
introduced by V.V. Kravchenko (2008,2009). We also give the SPPS form of the
Dirac Hill discriminant under the Darboux nodeless transformation using the
SPPS form of the discriminant and apply the results to one of Razavy's
quasi-exactly solvable periodic potentials
06/2010;
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ABSTRACT: We establish a series representation of the Hill discriminant based on the spectral parameter power series (SPPS) recently introduced by V. Kravchenko. We also show the invariance of the Hill discriminant under a Darboux transformation and employing the Mathieu case the feasibility of this type of series for numerical calculations of the eigenspectrum Comment: 13 pages, 2 figures, 19 references with titles, a few minor changes in the text
02/2010;
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ABSTRACT: We consider two closely related Riccati equations of constant parameters whose particular solutions are used to construct the corresponding class of supersymmetrically-coupled second-order differential equations. We solve analytically these parametric periodic problems along the positive real axis. Next, the analytically solved model is used as a case study for a powerful numerical approach that is employed here for the first time in the investigation of the energy band structure of periodic not necessarily regular potentials. The approach is based on the well-known self-matching procedure of James (1949) and implements the spectral parameter power series solutions introduced by Kravchenko (2008). We obtain additionally an efficient series representation of the Hill discriminant based on Kravchenko's series Comment: 15 pages, 4 eps figures, a few minor changes, version accepted to Ann. Phys
10/2009;
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ABSTRACT: In the context of bilayer graphene we use the simple gauge model of Jackiw and Pi to construct its numerical solutions in powers of the bias potential V according to a general scheme due to Kravchenko. Next, using this numerical solutions, we develop the Ermakov-Lewis approach for the same model. This leads us to numerical calculations of the Lewis-Riesenfeld phases that could be of forthcoming experimental interest for bilayer graphene. We also present a generalization of the Ioffe-Korsch nonlinear Darboux transformation
01/2009;
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ABSTRACT: We establish a series representation of the Hill discriminant based on the spectral parameter power series (SPPS) recently introduced by Kravchenko. We also show the invariance of the Hill discriminant under a Darboux transformation and employing the Mathieu case the feasibility of this type of series for numerical calculations of the eigenspectrum.
Annals of Physics.
