a b s t r a c tWe prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials, solutions of the Klein-Gordon equation with a variable coefficient. Using the transmutation (transformation) operators and their recently discovered mapping properties we prove the completeness of the generalized wave polynomials and use them for an explicit construction of the solution of the Cauchy problem for the Klein-Gordon equation. Based on this result we develop a numerical method for solving the Cauchy problem and test its performance.
For the one‐dimensional Schrödinger equation with short‐range potential on a half‐line x>0, the knowledge of the Jost solution e(ρ,x)∼eiρx, Imρ ≥ 0, x→∞ allows one to solve corresponding spectral problems. In the present work, a new series representation for e(ρ,x) is derived with the aid of the Levin formula for the Jost solution and a recently proposed Fourier‐Laguerre series expansion of the integral kernel from the Levin formula. The representation for e(ρ,x) has the form
efalse(ρ,xfalse)=eiρxfalse(1+∑n=0∞znbnfalse(xfalse)false), where, for the coefficients bn(x), a simple recurrent integration procedure is obtained and the parameter
z:=()12+iρfalse/()12−iρ belongs to the unit disk. An analogous representation is derived for the derivative of the Jost solution as well.
With the aid of the series representations, numerical solution of the classical spectral problem on the half‐line becomes an easy task. Indeed, computation of the eigenvalues reduces to finding zeros of a polynomial for z∈(−1,1). For computing corresponding normalizing constants, a simple formula is derived. Moreover, computation of the (usually) most difficult spectral characteristics, the spectral density function (or its derivative) for all positive values of the spectral parameter reduces to calculation of an easily computable function along the unitary upper half‐circle. Numerical illustrations of the efficiency of the computational method based on the obtained results are presented.
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.
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.
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