23 research outputs found
Spectral parameter power series method for discontinuous coefficients
Let (a,b) be a finite interval and 1/p, q, r be functions from L1(a,b). We
show that a general solution (in the weak sense) of the equation (pu')'+qu =
zru on (a,b) can be constructed in terms of power series of the spectral
parameter z. The series converge uniformly on [a,b] and the corresponding
coefficients are constructed by means of a simple recursive procedure. We use
this representation to solve different types of eigenvalue problems. Several
numerical tests are discussed
Shifted Riccati Procedure: Application to Conformal Barotropic FRW Cosmologies
In the case of barotropic FRW cosmologies, the Hubble parameter in conformal
time is the solution of a simple Riccati equation of constant coefficients. We
consider these cosmologies in the framework of nonrelativistic supersymmetry
that has been effective in the area of supersymmetric quantum mechanics.
Recalling that Faraoni [Amer. J. Phys. 67 (1999), 732-734] showed how to reduce
the barotropic FRW system of differential equations to simple harmonic
oscillator differential equations, we set the latter equations in the
supersymmetric approach and divide their solutions into two classes of
'bosonic' (nonsingular) and 'fermionic' (singular) cosmological zero-mode
solutions. The fermionic equations can be considered as representing
cosmologies of Stephani type, i.e., inhomogeneous and curvature-changing in the
conformal time. We next apply the so-called shifted Riccati procedure by
introducing a constant additive parameter, denoted by S, in the common Riccati
solution of these supersymmetric partner cosmologies. This leads to barotropic
Stephani cosmologies with periodic singularities in their spatial curvature
indices that we call U and V cosmologies, the first being of bosonic type and
the latter of fermionic type. We solve completely these cyclic singular
cosmologies at the level of their zero modes showing that an acceptable shift
parameter should be purely imaginary, which in turn introduces a parity-time
(PT) property of the partner curvature indices
Periodic Sturm-Liouville problems related to two Riccati equations of constant coefficients
"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 an-alytically 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 thefirst time in the investigation of the en-ergy band structure of periodic not necessarily regular potentials. The approach is based on the well-known self-matching procedure of James (1949) and imple-ments the spectral parameter power series solutions introduced by Kravchenko (2008). We obtain additionally an efficient series representation of the Hill dis-criminant based on on Kravchenko´s series.
Parametric oscillators from factorizations employing a constant-shifted Riccati solution of the classical harmonic oscillator
"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.
Recovery of a potential on a quantum star graph from Weyl's matrix
The problem of recovery of a potential on a quantum star graph from Weyl's
matrix given at a finite number of points is considered. A method for its
approximate solution is proposed. It consists in reducing the problem to a
two-spectra inverse Sturm-Liouville problem on each edge with its posterior
solution. The overall approach is based on Neumann series of Bessel functions
(NSBF) representations for solutions of Sturm-Liouville equations, and, in
fact, the solution of the inverse problem on the quantum graph reduces to
dealing with the NSBF coefficients. The NSBF representations admit estimates
for the series remainders which are independent of the real part of the square
root of the spectral parameter. This feature makes them especially useful for
solving direct and inverse problems requiring calculation of solutions on large
intervals in the spectral parameter. Moreover, the first coefficient of the
NSBF representation alone is sufficient for the recovery of the potential. The
knowledge of the Weyl matrix at a set of points allows one to calculate a
number of the NSBF coefficients at the end point of each edge, which leads to
approximation of characteristic functions of two Sturm-Liouville problems and
allows one to compute the Dirichlet-Dirichlet and Neumann-Dirichlet spectra on
each edge. In turn, for solving this two-spectra inverse Sturm-Liouville
problem a system of linear algebraic equations is derived for computing the
first NSBF coefficient and hence for recovering the potential. The proposed
method leads to an efficient numerical algorithm that is illustrated by a
number of numerical tests.Comment: arXiv admin note: substantial text overlap with arXiv:2210.1250