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Systems, scattering and operators, and their connections with number theory. (English) Zbl 1471.93131

Banaszak, Grzegorz (ed.) et al., Arithmetic methods in mathematical physics and biology II. Proceedings of the 2nd international conference, Będlewo, Poland, August 5–11, 2018. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 124, 123-141 (2021).
Summary: We review the relation between linear system theory and Lax-Phillips scattering theory. We also suggest their connections with number theory, although they are very preliminary.
For the entire collection see [Zbl 1470.11004].

MSC:

93C05 Linear systems in control theory
47A40 Scattering theory of linear operators
47A11 Local spectral properties of linear operators
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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[1] V. M. Adamjan, D. Z. Arov, A class of scattering operators and of characteristic operator-functions of contractions, Dokl. Akad. Nauk SSSR 160 (1965), 9-12; English transl.: Soviet Math. Dokl. 6 (1965), 1-5. · Zbl 0125.35001
[2] V. M. Adamjan, D. Z. Arov, On unitary couplings of semiunitary operators, Mat. Issled. 1 (1966), no. 2, 3-64; English transl.: Amer. Math. Soc. Transl. 95 (1970), 75-129. · Zbl 0258.47012
[3] A. C. Antoulas (ed.), Mathematical System Theory: The Influence of R. E. Kalman, Springer, Berlin, 1991.
[4] J. A. Ball, P. T. Carroll, Y. Uetake, Lax-Phillips scattering theory and well-posed linear systems: a coordinate-free approach, Math. Control Signals Systems 20 (2008), 37-79. · Zbl 1145.93024
[5] G. Banaszak, Y. Uetake, Abstract intersection theory and operators in Hilbert space, Commun. Number Theory Phys. 5 (2011), 699-712. · Zbl 1254.47002
[6] G. Banaszak, Y. Uetake, Standard models of abstract intersection theory for operators in Hilbert space, Bull. Pol. Acad. Sci. Math. 63 (2015), 149-175. · Zbl 1345.47004
[7] G. Banaszak, Y. Uetake, Abstract intersection theory for zeta-functions: geometric as-pects, Funct. Approx. Comment. Math. 64 (2021), 251-265. · Zbl 1473.11166
[8] G. Banaszak, Y. Uetake, Scattering theory for automorphic forms on adele groups and a spectral interpretation of automorphic L-functions, in preparation.
[9] B. Conrey, Riemann’s hypothesis, in: Colloquium De Giorgi 2013 and 2014, Colloquia 5, Pisa, 2015, 109-117. · Zbl 1382.11062
[10] R. F. Curtain, H. Zwart, An Introduction to Infinite-Dimensional Linear Systems The-ory, Texts Appl. Math. 21, Springer, Berlin, 1995. · Zbl 0839.93001
[11] S. Eilenberg, Automata, Languages and Machines, Vols. A, B, Pure Appl. Math. 58, 59, Academic Press, New York, 1974, 1976. · Zbl 0317.94045
[12] P. Fleig, H. P. A. Gustafsson, A. Kleinschmidt, D. Persson, Eisenstein Series and Automorphic Representations, with Applications in String Theory, Cambridge Stud. Adv. Math. 176, Cambridge Univ. Press, Cambridge, 2018. · Zbl 1435.11001
[13] C. Foiaş, Contractive intertwining dilations and waves in layered media, in: Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, 605-613. · Zbl 0454.47006
[14] Y. UETAKE
[15] I. M. Gelfand, Automorphic functions and the theory of representations, in: Proceed-ings of the International Congress of Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, 74-85. · Zbl 0138.07102
[16] J. W. Helton, Discrete time systems, operator models, and scattering theory, J. Func-tional Analysis 16 (1974), 15-38. · Zbl 0282.93033
[17] J. W. Helton, Systems with infinite-dimensional state space: The Hilbert space approach, Proc. IEEE 64 (1976), 145-160.
[18] M. Ikawa, Scattering Theory (in Japanese), Iwanami Shoten, 1999.
[19] S. Ikehara, An extension of Landau’s theorem in the analytical theory of numbers, J. Math. Phys. Mass. Inst. Tech. 10 (1931), 1-12. · JFM 57.0212.01
[20] K. Iwasawa, On Z -extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246-326. · Zbl 0285.12008
[21] R. E. Kalman, Mathematical description of linear dynamical systems, J. SIAM Control Ser. A 1 (1963), 152-192. · Zbl 0145.34301
[22] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss. 132, Springer-Verlag, 1966, 1976, 1980. · Zbl 0148.12601
[23] D. Kazhdan, Introduction to QFT, in: Quantum Fields and Strings: A Course for Mathematicans, Vol. 1, Amer. Math. Soc., Providence, RI, 1999, 377-418. · Zbl 1170.81407
[24] K. Kodaira, The eigenvalue problem for ordinary differential equations of the second order and Heisenberg’s theory of S-matrices, Amer. J. Math. 71 (1949), 921-945. · Zbl 0035.27101
[25] N. Kurokawa, Special values of Selberg zeta functions, in: Algebraic K-Theory and Algebraic Number Theory (Honolulu, HI, 1987), Contemp. Math. 83, Amer. Math. Soc., Providence, RI, 1989, 133-150. · Zbl 0684.10038
[26] R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics 544, Springer, Berlin, 1976. · Zbl 0332.10018
[27] R. P. Langlands, Where stands functoriality today? in: Representation Theory and Automorphic Forms (Edinburgh, 1996), Proc. Sympos. Pure Math. 61, Amer. Math. Soc., Providence, RI, 1997, 457-471. · Zbl 0901.11032
[28] P. D. Lax, Functional Analysis, Pure Appl. Math. (N.Y.), John Wiley & Sons, New York, 2002. · Zbl 1009.47001
[29] P. D. Lax, R. S. Phillips, Scattering Theory for Automorphic Functions, Ann. of Math. Stud. 87, Princeton Univ. Press, Princeton, NJ, 1976. · Zbl 0362.10022
[30] P. D. Lax, R. S. Phillips, Scattering theory for automorphic functions, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 261-295. · Zbl 0442.10018
[31] P. D. Lax, R. S. Phillips, Scattering Theory, second ed., Pure Appl. Math. 26, Academic Press, New York, 1989. · Zbl 0697.35004
[32] Yu. I. Manin, Reflections on arithmetical physics, in: Conformal Invariance and String Theory (Poiana Braşov, 1987), Perspect. Phys., Academic Press, Boston, MA, 1989, 293-303. · Zbl 0674.53068
[33] P. R. Masani, Norbert Wiener 1894-1964, Vita Math. 5, Birkhäuser, Basel, 1990.
[34] B. Mazur, A. Wiles, Class fields of abelian extensions of Q, Invent. Math. 76 (1984), 179-330. · Zbl 0545.12005
[35] R. B. Melrose, Geometric Scattering Theory, Stanford Lectures, Cambridge Univ. Press, Cambridge, 1995. · Zbl 0849.58071
[36] J. S. Milne, The Work of John Tate, in: The Abel Prize 2008-2012, Springer, Heidelberg, 2014, 259-340. · Zbl 1317.01011
[37] W. Mlak, On a theorem of Lebow, Ann. Polon. Math. 35 (1977), 107-109. · Zbl 0371.47007
[38] E. Nordgren, H. Radjavi, P. Rosenthal, Weak resolvents of linear operators, Indiana Univ. Math. J. 36 (1987), 913-934. · Zbl 0644.47002
[39] B. S. Pavlov, L. D. Faddeev, Scattering theory and automorphic functions, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 27 (1972), 161-193; English transl.: J. Soviet Math. 3 (1975), 522-548. · Zbl 0343.35004
[40] F. Shahidi, Langlands-Shahidi method, in: Automorphic Forms and Applications, IAS/Park City Math. Ser., Amer. Math. Soc., Providence, RI, 2007, 297-330. · Zbl 1116.11003
[41] F. Shahidi, Eisenstein Series and Automorphic L-Functions, Amer. Math. Soc. Colloq. Publ. 58, Amer. Math. Soc., Providence, RI, 2010. · Zbl 1215.11054
[42] O. J. Staffans, Well-Posed Linear Systems, Encyclopedia Math. Appl. 103, Cambridge Univ. Press, 2005. · Zbl 1057.93001
[43] O. J. Staffans, Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view), in: Mathematical Systems Theory in Biology, Communication, Computation, and Finance, IMA Vol. Math. Appl. 134, Springer, New York, 2003, 375-413. · Zbl 1156.93326
[44] O. J. Staffans, G. Weiss, Transfer functions of regular linear systems. II. The system operator and the Lax-Phillips semigroup, Trans. Amer. Math. Soc. 354 (2002), 3229-3262. · Zbl 0996.93012
[45] B. Sz.-Nagy, C. Foiaş, H. Bercovici, L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, second ed., Universitext, Springer, New York, 2010. · Zbl 1234.47001
[46] Y. Uetake, Some local properties of spectrum of linear dynamical systems in Hilbert space, Integral Equations Operator Theory 51 (2005), 283-302. · Zbl 1083.93026
[47] Y. Uetake, The Lax-Phillips infinitesimal generator and the scattering matrix for au-tomorphic functions, Ann. Polon. Math. 92 (2007), 99-122. · Zbl 1201.11053
[48] Y. Uetake, Lax-Phillips scattering for automorphic functions based on the Eisenstein transform, Integral Equations Operator Theory 60 (2008), 271-288. · Zbl 1132.11323
[49] Y. Uetake, Spectral scattering theory for automorphic forms, Integral Equations Oper-ator Theory 63 (2009), 439-457. · Zbl 1182.11023
[50] A. Weil, Sur la théorie du corps de classes, J. Math. Soc. Japan 3 (1951), 1-35. · Zbl 0044.02901
[51] A. Weil, Oeuvres Scientifiques -Collected Papers I 1926-1951, Springer, Berlin, 1979. · Zbl 0424.01027
[52] A. Weil, Oeuvres Scientifiques -Collected Papers II 1951-1964, Springer, Berlin, 1979. · Zbl 0424.01028
[53] N. Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), 1-100. · JFM 58.0226.02
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