Bloch Solutions of Periodic Dirac Equations in SPPS Form

Khmelnytskaya, Kira V.; Rosu, H. C.

doi:10.1007/978-3-0348-0346-5_9

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2012

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“…For a proof not requiring pseudoanalytic functions we refer to and . The representation turned to be an appropriate tool for solving different Sturm–Liouville and related problems and . As is well known (see, e.g., ) under certain regularity conditions, a solution to the initial value problem for the Sturm–Liouville equation is an analytic function of the spectral parameter and hence, admits a Taylor series expansion in powers of the spectral parameter.…”

Section: Introductionmentioning

confidence: 99%

Complete systems of recursive integrals and Taylor series for solutions of Sturm–Liouville equations

Kravchenko

Morelos

Tremblay

2012

Math Methods in App Sciences

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Given a regular nonvanishing complex valued solution y0 of the equation yMathClass-rel′MathClass-rel′MathClass-bin+ q(x)y MathClass-rel= 0, x ∈ (a,b), assume that it is n times differentiable at a point x0 ∈ [a,b]. We present explicit formulas for calculating the first n derivatives at x0 for any solution of the equation uMathClass-rel′MathClass-rel′MathClass-bin+ q(x)u MathClass-rel= λu. That is, a map transforming the Taylor expansion of y0 into the Taylor expansion of u is constructed. The result is obtained with the aid of the representation for solutions of the Sturm‐Liouville equation in terms of spectral parameter power series. Copyright © 2012 John Wiley & Sons, Ltd.

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