Function theories for the Yukawa and Helmholtz equations
In preceding papers, the authors have developed analogous function theories for the heat, the wave, and the Laplace equations [Rocky Mt. J. Math. 13, 191-214 (1983; Zbl 0554.35012)] and for the heat, the GASPT, and the Euler-Poisson-Darboux equations [ibid. 14, 383-396 (1984; Zbl 0581.35025)]. The present paper is concerned with function theories for the Yukawa and Helmholtz equations. Here, basic solutions and associated functions for the Yukawa and Helmholtz equations are obtained via transmutations acting on heat polynomials and associated heat functions. In detail, in \S 2, transmutations \((T_1,\dots, T_4)\) are introduced which relate solutions of the heat equation to solutions of the Dirichlet and Neumann problems for the Yukawa equation \((T_1, T_2)\) and the damped wave equation \((T_3, T_4)\). For the case of the Yukawa equation, the transmutation operators \(T_1\) and \(T_2\) are applied in \S 3 in order to obtain basic functions, associated functions, generating functions, recurrence relations and generalized Cauchy-Riemann equations. Similar results are proved in \S 4 for the case of the Helmholtz equation by means of \(T_3\) and \(T_4\). In order to indicate that results for the Laplace equation have their counterparts also in the cases of the Yukawa and Helmholtz equations, solutions with analytic data (\S 5) and with entire data (\S 6) are treated, and series expansions as well as integral type formulas for the solutions are generated. Finally, in \S 7, Fourier transform criteria are obtained for defining expansions of solutions in terms of associated functions.
scientific article; zbMATH DE number 878061
scientific article