Construction of transmutation operators and hyperbolic pseudoanalytic functions Vladislav V. Kravchenko and Sergii M. Torba Abstract A representation for integral kernels of Delsarte transmutation operators is obtained in the form of a functional series with the exact formulae for the terms of the series. It is based on the application of hyperbolic pseudoanalytic function theory and recent results on mapping properties of the transmutation operators. The kernel K1 of the transmutation operator relating A = - d 2 dx 2 + q1(x) and B = - d 2 dx 2 results to be one of the complex components of a bicomplex-valued hyperbolic pseudoanalytic function satisfying a Vekua-type hyperbolic equation of a special form. The other component of the pseudoanalytic function is the kernel of the transmutation operator relating C = - d 2 dx 2 + q2(x) and B where q2 is obtained from q1 by a Darboux transformation. We prove the expansion theorem and a Runge-type theorem for this special hyperbolic Vekua equation and using several known results from hyperbolic pseudoanalytic function theory together with the recently discovered mapping properties of the transmutation operators obtain the new representation for their kernels. Several examples are given. Moreover, based on the presented results approaches for numerical computation of the transmutation kernels and for numerical solution of spectral problems are proposed. 1. Introduction The notion of a transmutation operator relating two linear differential operators was introduced in 1938 by J. Delsarte [12] and nowadays represents a widely used tool in the theory of linear differential equations (see, e.g., [1], [7], [33], [35], [44], [46]). Very often in literature the transmutation operators are called the transformation operators. Here we keep the original term introduced by Delsarte and Lions [13]. It is well known that under certain regularity conditions the transmutation operator transmuting the operator A = − d 2 dx 2 + q(x) into B = − d 2 dx 2 is a Volterra integral operator with good properties. Its integral kernel can be obtained as a solution of a certain Goursat problem for the Klein-Gordon equation with a variable coefficient. There exist very few examples of the transmutation kernels available in a closed form (see [26]). In the present work we obtain a representation for the kernels of the transmutation operators for regular Sturm-Liouville operators (with complex valued coefficients) in the form of a functional series with the exact formulae for the terms of the series. The result is based on several new observations. We use our recent result on the construction of the kernel of the transmutation operator corresponding to a Darboux associated Schr¨ odinger operator [26] to find out that a bicomplex-valued function whose one complex component is the transmutation kernel K 1 (x,t) (for a Schr¨ odinger operator d 2 dx 2 − q 1 (x), q 1 ∈ C [−b,b]) and the other complex component is K 2 (x,t) (for a Schr¨ odinger operator d 2 dx 2 − q 2 (x), with q 2 obtained from q 1 by a 2000 Mathematics Subject Classification 30B60, 30G20, 34A25, 35A35, 35C10 (primary), 30B10, 34L16, 34A45, 35L10, 41A50, 47N20, 65N99 (secondary). Research was supported by CONACYT, Mexico. Research of second named author was supported by DFFD, Ukraine (GP/F32/030) and by SNSF, Switzerland (JRP IZ73Z0 of SCOPES 2009–2012). arXiv:1208.6166v2 [math.AP] 19 Jul 2013