Construction of Transmutation Operators and Hyperbolic Pseudoanalytic Functions

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• 作者：Vladislav V. Kravchenko (1)
  Sergii M. Torba (1)
  
  1. Department of Mathematics ; CINVESTAV del IPN ; Unidad Queretaro ; Libramiento Norponiente No. 2000 ; Fracc. Real de Juriquilla ; 76230聽 ; Quer茅taro ; QRO ; Mexico
  
• 关键词：Transmutation operator ; Hyperbolic variable ; Bicomplex function ; Pseudoanalytic function ; Goursat problem ; Vekua equation ; Sturm鈥揕iouville problem ; Numerical solution of spectral problems ; Primary 30B60 ; 30G20 ; 34A25 ; 35A35 ; 35C10 ; Secondary 30B10 ; 34L16 ; 34A45 ; 35L10 ; 41A50 ; 47N20 ; 65N99
• 刊名：Complex Analysis and Operator Theory
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：9
• 期：2
• 页码：379-429
• 全文大小：1,258 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Operator Theory
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1661-8262

文摘

A representation for integral kernels of Delsarte transmutation operators is obtained in the form of a functional series with 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 \(K_{1}\) of the transmutation operator relating \(A=-\frac{d^{2} }{dx^{2}}+q_{1}(x)\) and \(B=-\frac{d^{2}}{dx^{2}}\) turns out 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 \(C=-\frac{d^{2}}{dx^{2}}+q_{2}(x)\) and \(B\) where \(q_{2}\) is obtained from \(q_{1}\) by a Darboux transformation. We prove an 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 we obtain a new representation for their kernels. Several examples are given. Moreover, approaches for numerical computation of the transmutation kernels and for numerical solution of spectral problems are proposed.