# On the Anisotropic Wiener-Hopf Operator, Connected with Helmholtze-Sohrodinger Equation

Volume 4, Issue 3, pp 301--309
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### Authors

S. A. H. Matikolai - Department of Mathematics, Yerevan State University, Yerevan, Armenia H. Jafari - Department of Mathematics, University of Mazandaran, Babolsar, Iran R. Lotfikar - Islamic Azad University Ilam branch

### Abstract

In this article, solvability of one the anisotropic Helmholtz-Shrodinger equation with the boundary conditions of the first and second type is investigated in the upper and lower half –space, (x5>0, x5<0), in 5 dimensions. Solvability of these boundary problems reduces to solvability of Rieman- Hilbert boundary problem, in general necessary and sufficient conditions for the correctness of the problem in the Sobolev space are presented as well as explicit formulas for a factorization of the Fourier symbol matrix of the one-medium problem. The solvability analysis is based on the factorization problem of some matrix-function.

### Keywords

• Helmholtz-Shrodinger equation
• Factorization of matrix-function
• Boundary value problem
• Wiener- Hopf equation.

•  47B35
•  47A53

### References

• [1] B. Noble, Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations, Pergamon Press, 7 (1958), 223--230

• [2] A. D. Rawlins, The Explicit Wiener - Hopf Factorization of a Special Matrix, Z. Angew, Math. Mech., 61 (1981), 527--528

• [3] F. O. Speck, Mixed Boundary Value Problems Of the Type of Sommerfeld’s Half- Plane problem, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 104 (1986), 261--277

• [4] V. G. Daniel, On the Solutionof two Coupled Winer-Hopf Equations, SIAM J. Appl. Math., 44 (1984), 667--680

• [5] S. A. Hosseini Matikolai, Solvability of the Boundary Value Problem Coordinated with the Anisotropic Helmholtz-Shrodinger Equation, Word Applied science Journal, 11 (2010), 1348--1352

• [6] S. A. Hosseini Matikolai, R. Lotfikar, Solvability of the Boundary Value Problem Coordinated with the Anisotropic Helmholtz-Shrodinger Equation in case of $k_+=k_-$, World Alpplied sciences Journal, 11 (2011), 1500--1506

• [7] S. A. Hooseini Matikoni , On the Structure of Wiener Operation Corresponting to the Anisotropic Boundary Value Connected with Helmhoitz Shrodinger equation with boundaru conditions of the first and second type, Proc. of the Yerevan State University, 2 (2011), 22--26

• [8] G. I. Eskin, Boundary Value Problems for Ellipit Pseudodifferntinl Equntican Mathematical Society, Providence, 1981 (1981), 283--300

• [9] F. O. Speck, Sommerfel Diffraction Problems with First and Second kind Boundary Condition Society for industrial and Applied Mathematics, SIAM Journal on Mathematical Analysis, 20 (1989), 396--407

• [10] A. E. Heins, The sommerfeld half‐plane problem revisited, II the factoring of a matrix of analytic functions, Mah. Methods Appl. Sci., 5 (1983), 14--21

• [11] N. I. Muskhelishvili, Singular integral Equations, Nauka, 1968 (1968), 126--140

• [12] F. D. Gakhov, Boundary Problems, Fizmatgiz, 1963 (1963), 221--243

• [13] G. S. Litvinchnk, I. M. Spitkovskii, Factorization of measurable matrix functions, Akademie Verlag, 1987 (1987), 182--196

• [14] K. F. Glancey, I. Gohberg, Factorization of Matrix Functions and Singular integral Operators, Springer, 191 (1981), 211--230