From Sommerfeld Diffraction Problems to Operator Factorisation

Abstract

This article presents a brief survey devoted to the interplay between diffraction and operator theory, particularly between the so-called canonical diffraction problems (exemplified by Sommerfeld half-plane problems) on one hand and operator factorisation theory on the other hand. It is shown how operator factorisation concepts appear in a natural way from applications and how they can help to find solutions rigorously in case of well-posed problems or how to normalize problems by an adequate change of function spaces.

Keywords

• Sommerfeld problem,diffraction,Wiener-Hopf method,Wiener- Hopf operator,factorisation,intermediate space,generalised inverse,op- erator relation

References

1. I. D. Abrahams, The application of Pad'e approximants to Wiener-Hopf factorization. IMA J. Appl. Math., 65, 2000, 257-281.
2. I. D. Abrahams, On the application of the Wiener-Hopf technique to problems in dynamic elasticity. Wave Motion, 36, 2002, 311-333.
3. M. S. Agranovich, M. I. Vishik Elliptic problems with a parameter and parabolic problems of general type. Russian Math. Surveys 19 1964, 53-157. Translation from Usp. Mat. Nauk 19 1964, 53-161, Russian.
4. H. Bart, I. Gohberg rm and M. Kaashoek The coupling method for solving integral equations. In Topics in operator theory systems and networks, Workshop Rehovot/Isr. 1983, Oper. Theory Adv. Appl. 2, Birkhauser, Basel 1984, 39-73. Addendum in Integr. Equ. Op. Theory 8 1985, 890-891.
5. H. Bart, V. E. Tsekanovskii Matrical coupling and equivalence after extension. In Operator Theory and Complex Analysis, Eds T. Ando et al., Oper. Theory Adv. Appl. 59, Birkhauser, Basel 1991, 143-160.
6. H. Bart, V. E. Tsekanovskii Complementary Schur complements. Linear Algebra Appl. 197 1994, 651-658.
7. M. A. Bastos, A.F. dos Santos rm and R. Duduchava Finite interval convolution operators on the Bessel potential spaces H_ps Math. Nachr. 173 1995, 49-63.
8. M. A. Bastos, Yu. I. Karlovich, A. F. dos Santos rm and P. M. Tishin The corona theorem and the canonical factorization of triangular AP matrix functions – Effective criteria and explicit formulas. J. Math. Anal. Appl. 223 1998, 523-550.

9. A. Bottcher, Yu. I. Karlovich rm and I. M. Spitkovsky Convolution Operators and Factorization of Almost Periodic Matrix Functions. Oper. Theory Adv. Appl. 131, Birkhauser, Basel 2002.
10. A. Bottcher, B. Silbermann Analysis of Toeplitz Operators. Springer, Berlin 2006.
11. A. Bottcher, F. -O. Speck On the symmetrization of general Wiener-Hopf operators.J. Operator Theory 76 2016, 335-349.
12. A. Bottcher, I. M. Spitkovsky The factorization problem some known results and open questions., In Operator Theory, Operator Algebras and Applications Eds A. Almeida et al., Oper. Theory Adv. Appl. 229, Birkhauser, Basel 2013, 101-122.
13. L. Boutet de Monvel Boundary problems for pseudo-differential operators. Acta Math. 126 1971, 11-51.
14. M. C. Camara, A. B. Lebre rm and F. -O. Speck Meromorphic factorization, partial index estimates and elastodynamic diffraction problems. Math. Nachr. 157 1992, 291-317.
15. M. C. Camara, M. T. Malheiro Meromorphic factorization revisited and application to some groups of matrix functions. Complex Anal. Oper. Theory 2 2008, 299-326.
16. M. C. Camara, A. F. dos Santos rm and M. A. Bastos generalized factorization for Daniele-Khrapkov matrix functions – explicit formulas. J. Math. Anal. Appl. 190 1995, 295-328.
17. M. C. Camara, A. F. dos Santos rm and P. F. dos Santos Matrix Riemann-Hilbert problems and factorization on Riemann surfaces. J. Funct. Anal. 255 2008, 228-254.
18. L. P. Castro The Characterization of the Intermediate Space in generalized Factorizations. MSc thesis, Universidade T'ecnica de Lisboa 1994, vii, 90 p., Portuguese.
19. L. P. Castro Relations between Singular Operators and Applications. PhD thesis, Universidade T'ecnica de Lisboa 1998, xix, 163 p.
20. L. P. Castro, R. Duduchava rm and F. -O. Speck Localization and minimal normalization of some basic mixed boundary value problems. In Factorization, Singular Operators and Related Problems, Eds S. Samko et al., Kluwer, Dordrecht 2003, 73-100.
21. L. P. Castro, R. Duduchava rm and F. -O. Speck Asymmetric factorizations of matrix functions on the real line. In Modern Operator Theory and Applications. The Igor Borisovich Simonenko Anniversary Volume Eds Y.M. Erusalimskii et al., Oper. Theory Adv. Appl. 170, Birkhauser, Basel 2006, 53-74.
22. L. P. Castro, R. Duduchava rm and F. -O. Speck Diffraction from polygonal-conical screens –- an operator approach. In Operator Theory, Operator Algebras and Applications Eds A. Bastos et al., Oper. Theory Adv. Appl. 242, Birkhauser, Basel 2014, 113-137.
23. L. P. Castro, F. -O. Speck On the characterization of the intermediate space in generalized factorizations. Math. Nachr. 176 1995, 39-54.
24. L. P. Castro, F. -O. Speck Regularity properties and generalized inverses of delta-related operators. Z. Anal. Anwend. 17 1998, 577-598.
25. L. P. Castro, F. -O. Speck Relations between convolution type operators on intervals and on the half-line. Integral Equations Oper. Theory 37 2000, 169-207.
26. L. P. Castro, F. -O. Speck Inversion of matrix convolution type operators with symmetry.Port. Math. N.S. 62 2005, 193-216.
27. L. P. Castro, F. -O. Speck rm and F. S. Teixeira On a class of wedge diffraction problems posted by Erhard Meister. In Operator theoretical methods and applications to mathematical physics. The Erhard Meister memorial volume Eds. I. Gohberg et al., Oper. Theory Adv. Appl. 147 2004, 213-240.
28. L. P. Castro, F. -O. Speck rm and F.S. Teixeira Mixed boundary value problems for the Helmholtz equation in a quadrant. Integr. Equ. Oper. Theory 56 2006, 1-44.
29. G. N. Chebotarev Several remarks on the factorization of operators in a Banach space and the abstract Wiener-Hopf equation. Mat. Issled. 2 1968, 215-218, Russian.
30. K. F. Clancey, I. Gohberg Factorization of Matrix Functions and Singular Integral Operators. Oper. Theory Adv. Appl. 3, Birkhauser, Basel 1981.
31. E. T. Copson On an integral equation arising in the theory of diffraction. Q. J. Math., Oxf. Ser. 17 1946, 19-34.
32. V. G. Daniele On the solution of two coupled Wiener-Hopf equations. SIAM J. Appl. Math. 44 1984, 667-680.
33. A. Devinatz, M. Shinbrot General Wiener-Hopf operators. Trans. AMS 145 1969, 467-494.
34. R. Duduchava Integral Equations with Fixed Singularities. Teubner, Leipzig 1979.
35. R. Duduchava, F. -O. Speck Bessel potential operators for the quarter-plane. Appl. Anal. 45 1992, 49-68.%
36. R. Duduchava, F. -O. Speck Pseudodifferential operators on compact manifolds with Lipschitz boundary. Math. Nachr. 160 1993, 149-191.
37. R. Duduchava, W. L. Wendland The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems. Integr. Equ. Oper. Theory 23 1995, 294-335.
38. T. Ehrhardt, A. P. Nolasco rm and F. -O. Speck Boundary integral methods for wedge diffraction problems the angle 2 pi/n, Dirichlet and Neumann conditions. Operators and Matrices 5 2011, 1-40.
39. T. Ehrhardt, A. P. Nolasco rm and F. -O. Speck A Riemannn surface approach for diffraction from rational wedges. Operators and Matrices 8 2014, 301-355.
40. T. Ehrhardt, F. -O. Speck Transformation techniques towards the factorization of non-rational 2x2 matrix functions. Linear Algebra Appl. 353 2002, 53-90.
41. G. I. Eskin Boundary Value Problems for Elliptic Pseudodifferential Equations. AMS, Providence 1981, Russian edition 1973.
42. C. Gohberg, I. A. Fel’dman Convolution equations and projection methods for their solution.Translations of Mathematical Monographs 41. American Mathematical Society, Providence, R. I., 1974.
43. I. Z. Gohberg, M. G. Krein Systems of integral equations on a half-line with kernel depending on the difference of arguments. AMS Trans. 14 1960, 217-287, Russian edition 1958.
44. I. Gohberg, N. Krupnik One-Dimensional Linear Singular Integral Equations I, II. Birkhauser, Basel 1992, German edition 1979, Russian edition 1973.
45. GolGoh60L. S. Goldenstein, I. C. Gohberg On a multidimensional integral equation on a half-space whose kernel is a function on the difference of the arguments, and on a discrete analogue of this equation. Sov. Math. Dokl. 1 1960, 173-176.
46. P. Grisvard Elliptic Problems in Nonsmooth Domains. Pitman, London, 1985.
47. J. Hadamard Sur les probl`emes aux d'eriv'ees partielles et leur signification physique. In Princeton University Bulletin 13, No. 4, 1902, 49-52.
48. A. Heins Systems of Wiener-Hopf integral equations and their application to some boundary value problems in electromagnetic theory. Proc. Symp. Appl. Math. 2 1950, 76-81.
49. A. Heins The Sommerfeld half-plane problem revisited. I The solution of a pair of coupled Wiener-Hopf integral equations.Math. Methods Appl. Sci. 4 1982, 74-90.
50. S. ter Horst, M. Messerschmidt rm and A. C. M. Ran Equivalence after extension for compact operators on Banach spaces. J. Math. Anal. Appl. 431 2015, 136-149.
51. S. ter Horst, A. C. M. Ran Equivalence after extension and matricial coupling coincide with Schur coupling, on separable Hilbert spaces. Linear Algebra Appl. 439 2013, 793-805.
52. G. C. Hsiao, W. L. Wendland Boundary Integral Equations. Springer, Berlin 2008.
53. R. A. Hurd The Wiener-Hopf-Hilbert method for diffraction problems. Can. J. Phys. 54 1976, 775-780.
54. R. A. Hurd The explicit factorization of 2 x 2 Wiener-Hopf matrices. Preprint 1040, Fachbereich Mathematik, Technische Hochschule Darmstadt 1987, 24 p.
55. R. A. Hurd, E. Meister rm and F. -O. Speck Sommerfeld diffraction problems with third kind boundary conditions. SIAM J. Math. Anal. 20 1989, 589-607.
56. M. Idemen A new method to obtain exact solutions of vector Wiener-Hopf equations. Z. Angew. Math. Mech. 59 1979, 656-658.
57. D. S. Jones A simplifying technique in the solution of a class of diffraction problems. Quart.J. Math. 3 1952, 189-196.
58. D. S. Jones The Theory of Electromagnetism. Pergamon Press, Oxford 1964.
59. D. S. Jones Commutative Wiener–Hopf factorization of a matrix. Proc. Roy. Soc. London, 1984, 185-192.
60. A. A. Khrapkov Certain cases of the elastic equilibrium of an infinite wedge with a nonsymmetric notch at the vertex, subjected to concentrated forces. J. Appl. Math. Mech. 35 1971, 625-637, English, translation from Prikl. Mat. Mekh. 35 1971, 677-689, Russian.
61. A. V. Kisil Stability analysis of matrix Wiener-Hopf factorization of Daniele-Khrapkov class and reliable approximate factorization. Proc. A, R. Soc. Lond. 471 2015, 15 p.
62. M. G. Krein Integral equations on a half-line with kernel depending on the difference of arguments. AMS Trans. 22 1962, 163-288, Russian edition 1958.
63. J. B. Lawrie, I. D. Abrahams A brief historical perspective of the Wiener-Hopf technique. J. Eng. Math. 59 2007, 351-358.
64. A. B. Lebre, A. Moura Santos rm and F. -O. Speck Factorization of o class of matrices generated by Sommerfeld diffraction problems with oblique derivatives. Math. Meth. Appl. Sciences 20 1997, 1185-1198.
65. G. S. Litvinchuk, I. M. Spitkovskii Factorization of Measurable Matrix Functions. Oper. Mathematical Research 37, Akademie-Verlag, Berlin 1987, and Theory Adv. Appl. 25, Birkhauser, Basel 1987.
66. E. Luneburg The Sommerfeld problem Methods, generalization and frustrations. In E. Meister ed., Modern Mathematical Methods in Diffraction Theory and its Applications in Engineering. Proceedings of the Sommerfeld ’96 workshop. Peter Lang, Europ. Verlag der Wissenschaften, Frankfurt am Main 1997. Methoden Verfahren Math. Phys. 42, 145-162.
67. E. Luneburg, R. A. Hurd On the diffraction problem on a half-plane with different face impedances. Can. J. Phys. 62 1984, 853-860.
68. V. Maz’ya, T. Shaposhnikova Jacques Hadamard, a universal mathematician. History of Mathematics 14, American Mathematical Society, Providence, RI 1998.
69. E. Meister Randwertaufgaben der Funktionentheorie. Mit Anwendungen auf singulare Integralgleichungen und Schwingungsprobleme der mathematischen Physik. Leitfaden der Angewandten Mathematik und Mechanik 59. Teubner, Stuttgart 1983, German.
70. E. Meister Integraltransformationen mit Anwendungen auf Probleme der mathematischen Physik. Lang, Frankfurt 1983, German.
71. E. Meister Some solved and unsolved canonical problems of diffraction theory. In Differential equations and mathematical physics, Proc. Int. Conf., Birmingham/Ala. 1986, Lect. Notes Math. 1285 1987, 320-336. Short English, version of citeMei87.
72. E. Meister Einige geloste und ungeloste kanonische Probleme der mathematischen Beugungstheorie. Expo. Math. 5 1987, 193-237, German.
73. E. Meister ed. Modern Mathematical Methods in Diffraction Theory and its Applications in Engineering. Proceedings of the Sommerfeld’96 workshop at Freudenstadt, Germany. Methoden und Verfahren der Mathematischen Physik 42. Peter Lang, Europ. Verlag der Wissenschaften, Frankfurt 1997.
74. E. Meister, F. Penzel On the reduction of the factorization of matrix functions of Daniele- Khrapkov type to a scalar boundary value problem on a Riemann surface. Complex Variables, Theory Appl. 18 1992, 63-71.
75. E. Meister, F. -O. Speck Some multidimensional Wiener-Hopf equations with applications.In Trends in applications of pure mathematics to mechanics, Vol. 2 1979, Proceedings of a Symposium at Kozubnik Poland 1977, 217-262.
76. E. Meister, F. -O. Speck Diffraction problems with impedance conditions.Appl. Anal. 22 1986, 193-211.
77. E. Meister, F. -O. Speck Modern Wiener-Hopf methods in diffraction theory. In Ordinary and Partial Differential Equations, Vol. 2 Eds B.D. Sleeman et al., Longman, London 1989, 130-171.
78. E. Meister, F. -O. Speck The explicit solution of elastodynamical diffraction problems by symbol factorization. Z. Anal. Anw. 8 1989, 307-328.%
79. S. G. Mikhlin, S. Prossdorf Singular Integral Operators. Springer, Berlin 1986, German edition Akademie-Verlag, Berlin 1980.
80. G. Mishuris, S. Rogosin Factorization of a class of matrix-functions with stable partial indices. Math. Methods Appl. Sci. 39 2016, 3791-3807.
81. A. Moura Santos, F. -O. Speck Sommerfeld diffraction problems with oblique derivatives. Math. Meth. Appl. Sciences 20 1997, 635-652.
82. A. Moura Santos, F. -O. Speck rm and F. S. Teixeira Compatibility conditions in some diffraction problems. Pitman Research Notes in Mathematics Series 361, Longman, London 1996, 25-38.
83. A. Moura Santos, F. -O. Speck rm and F. S. Teixeira Minimal normalization of Wiener-Hopf operators in spaces of Bessel potentials. J. Math. Anal. Appl. 225 1998, 501-531.
84. R. J. Nagem, M. Zampolli rm and G. Sandri Mathematical Theory of Diffraction. Birkhauser, Boston 2004.
85. M. Z. Nashed, L. B. Rall Annotated bibliography on generalized inverses and applications. In Generalized Inverses and Applications Ed M.Z. Nashed, Academic Press, New York 1976, 771-1041.
86. B. Noble Methods Based on the Wiener-Hopf Technique. Pergamon Press, London 1958.
87. F. Penzel, F. -O. Speck Asymptotic expansion of singular operators on Sobolev spaces. Asymptotic Anal. 7 1993, 287-300.
88. H. Poincaré Sur la polarisation par diffraction. Acta Math. 16 1892, 297-339, French.
89. L. Primachuk, S. Rogosin Factorization of triangular matrix-functions of an arbitrary order. Lobachevskii Journal of Mathematics 39 2018, 809–817.
90. S. Prossdorf, F. -O. Speck A factorisation procedure for two by two matrix functions on the circle with two rationally independent entries.Proc. R. Soc. Edinb., Sect. A 115 1990, 119-138.
91. A. D. Rawlins The solution of a mixed boundary value problem in the theory of diffraction by a semi-infinite plane. Proc. Roy. Soc. London A 346 1975, 469-484. A. D. Rawlins The explicit Wiener-Hopf factorisation of a special matrix. Z. Angew. Math. Mech. 61 1981, 527-528.
92. A. D. Rawlins The solution of a mixed boundary value problem in the theory of diffraction. J. Eng. Math. 18 1984, 37-62.
93. A. D. Rawlins, W. E. Williams Matrix Wiener-Hopf factorisation. Q. J. Mech. Appl. Math. 34 1981, 1-8.
94. S. Rogosin, G. Mishuris Constructive methods for factorization of matrix-functions.IMA J. Appl. Math. 81 2016, 365-391.
95. A. F. dos Santos, F. S. Teixeira The Sommerfeld problem revisited Solution spaces and the edge conditions. J. Math. Anal. Appl. 143 1989, 341-357.
96. T. B. A. Senior Diffraction by a semi-infinite metallic sheet. Proc. R. Soc. Lond., Ser. A 213 1952, 436-458. Ser96 A. H. Serbest ed., S. R. Cloude ed. Direct and inverse electromagnetic scattering. Proceedings of the workshop, September 24–30, 1995, Gebze, Turkey. Mathematics Series 361. Longman, Harlow 1996.
97. E. Shamir Mixed boundary value problems for elliptic equations in the plane. The Lp theory. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 17 1963, 117-139.
98. E. Shamir Elliptic systems of singular integral operators. I. The half-space case. Trans. AMS 127 1967, 107-124.
99. M. Shinbrot On singular integral operators. J. Math. Mech. 13 1964, 395-406.
100. I. B. Simonenko Some general questions in the theory of the Riemann boundary problem. Izv. Akad. Nauk SSSR, Ser. Mat. 32 1968, 1138-1146, Russian. Math. USSR, Izv. 2 1970, 1091-1099, English.
101. A. Sommerfeld Mathematische Theorie der Diffraction. Math. Ann. 47 1896, 317-374, German. Annotated translation in English see citeNZS04.
102. A. Sommerfeld Partial differential equations in physics. Academic Press, New York 1949.
103. F. -O. Speck On the generalized invertibility of Wiener-Hopf operators in Banach spaces. Integr. Equ. Oper. Theory 6 1983, 458-465.
104. F. -O. Speck General Wiener-Hopf Factorization Methods. Pitman, London 1985.
105. F. -O. Speck Mixed boundary value problems of the type of Sommerfeld half-plane problem. Proc. Royal Soc. Edinburgh 104 A 1986, 261-277.
106. F. -O. Speck Sommerfeld diffraction problems with first and second kind boundary conditions. SIAM J. Math. Anal. 20 1989, 396-407.
107. F. -O. Speck In memory of Erhard Meister. In Operator Theoretical Methods and Applications to Mathematical Physics I, Eds. I. Gohberg et al., Oper. Theory Adv. Appl. 147, Birkhauser, Basel 2004, 27-46.
108. F. -O. Speck On the reduction of linear systems related to boundary value problems. In Operator theory, pseudo-differential equations, and mathematical physics, Eds. Yu.I. Karlovich et al., The Vladimir Rabinovich anniversary volume. Oper. Theory Adv. Appl. 228, Birkhauser, Basel 2013, 391-406.
109. F. -O. Speck Diffraction from a three-quarter-plane using an abstract Babinet principle. Z. Angew. Math. Mech. 93 2013, 485-491.
110. F. -O. Speck Wiener-Hopf factorization through an intermediate space. Integr. Equ. Oper. Theory 82 2015, 395-415.
111. F. -O. Speck A class of interface problems for the Helmholtz equation in mRn. Math. Meth. Appl. Sciences 40 2017, 391-403.
112. F. -O. Speck Paired operators in asymmetric space setting. In Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, Eds. D. Bini et al., The Albrecht Bottcher Anniversary Volume. Oper. Theory Adv. Appl. 259, Birkhauser, Basel 2017, 681-702.
113. F. -O. Speck On the reduction of general Wiener-Hopf operators.In Operator Theory, Analysis, and the State Space Approach, Eds. H. Bart et al., In Honor of Rien Kaashoek. Oper. Theory Adv. Appl. 271, Birkhauser, Basel 2018, 399-419.
114. I. M. Spitkovsky, A. M. Tashbaev On the problem of effective factorization of matrix functions. Izv. Vyssh. Uchebn. Zaved., Mat. 4 1989, 69-76, Russian. Sov. Math. 33 1989, 85-93, English.
115. G. Talenti Sulle equazioni integrali di Wiener-Hopf. Boll. Unione Mat. Ital., IV. Ser. 7, Suppl. al Fasc. 1 1973, 18-118, Italian.
116. D. Timotin Schur coupling and related equivalence relations for operators on a Hilbert space. Linear Algebra Appl. 452 2014, 106-119.
117. H. Triebel Theory of function spaces. Akademische Verlagsgesellschaft, Leipzig 1983.
118. M. I. Vishik, G. I. Eskin Equations in convolutions in a bounded region. Russian Math. Surveys 20 1965, 85-151. Translation from Usp. Mat. Nauk 20 1965, 89-152.
119. L. A. Weinstein The Theory of Diffraction and the Factorization Method. Golem Press, Boulder, Colorado 1969.
120. N. Wiener, E. Hopf Uber eine Klasse singularer Integralgleichungen. Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 30-32 1931, 696-706, German.
121. W. E. Williams Recognition of some readily “Wiener–Hopf” factorizable matrices. IMA J. Appl. Math. 32 1984, 367–378.
122. J. Wloka Partial Differential Equations. Cambridge University Press, 1987.