June  2018, 13(2): 191-215. doi: 10.3934/nhm.2018009

Functional model for extensions of symmetric operators and applications to scattering theory

1. 

Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK

2. 

Institute of Physics and Mathematics, Dragomanov National Pedagogical University, 9 Pyrohova St, Kyiv, 01601, Ukraine

3. 

Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México C.P. 04510, México D.F. México

* Corresponding author: Kirill D. Cherednichenko

To the memory of Professor Boris Pavlov

Received  September 2017 Revised  December 2017 Published  May 2018

On the basis of the explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices, we derive a new representation for the scattering matrix for pairs of such extensions. We use this representation to explicitly recover the coupling constants in the inverse scattering problem for a finite non-compact quantum graph with $δ$-type vertex conditions.

Citation: Kirill D. Cherednichenko, Alexander V. Kiselev, Luis O. Silva. Functional model for extensions of symmetric operators and applications to scattering theory. Networks & Heterogeneous Media, 2018, 13 (2) : 191-215. doi: 10.3934/nhm.2018009
References:
[1]

V. M. Adamjan and D. Z. Arov, Unitary couplings of semi-unitary operators, (Russian) Mat. Issled., 1 (1966), 3–64; English translation in Amer. Math Soc. Transl. Ser., 2 (1970), p95.  Google Scholar

[2]

V. M. Adamyan and B. S. Pavlov, Zero-radius potentials and M. G. Kreĭn's formula for generalized resolvents, (Russian)translated from Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 149 (1986), Issled. Lineĭn. Teor. Funktsiĭ. XV, 7–23, 186; J. Soviet Math. 42 (1988), 1537–1550. doi: 10.1007/BF01665040.  Google Scholar

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J. BehrndtM. M. Malamud and H. Neidhardt, Scattering theory for open quantum systems with finite rank coupling, Math. Phys. Anal. Geom., 10 (2007), 313-358.  doi: 10.1007/s11040-008-9035-x.  Google Scholar

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G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs volume, 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013.  Google Scholar

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M. Š. Birman, On the theory of self-adjoint extensions of positive definite operators, Math. Sb. N. S., 38 (1956), 431-450.   Google Scholar

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M. Š. Birman, Birman, Existence conditions for wave operators, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 27 (1963), 883-906.   Google Scholar

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M. Š. Birman and M. G. Kreĭn, On the theory of wave operators and scattering operators, (Russian), Dokl. Akad. Nauk SSSR, 144 (1962), 475-478.   Google Scholar

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M. Š. Birman and M. Z. Solomjak, Spectral Theory of Selfadjoint Operators in Hilbert Space, Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, 1987. Translated from the 1980 Russian original by S. Khrushchëv and V. Peller.  Google Scholar

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G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, (German), Acta Math., 78 (1946), 1-96.  doi: 10.1007/BF02421600.  Google Scholar

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[18]

Y. Y. ErshovaI. I. Karpenko and A. V. Kiselev, On inverse topology problem for Laplace operators on graphs, Carpathian Math. Publ., 6 (2014), 230-236.  doi: 10.15330/cmp.6.2.230-236.  Google Scholar

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K. O. Friedrichs, On the perturbation of continuous spectra, Communications on Appl. Math., 1 (1948), 361-406.  doi: 10.1002/cpa.3160010404.  Google Scholar

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V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006.  Google Scholar

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M. G. Kreĭn, Theory of self-adjoint extensions of semi-bounded Hermitian operators and applications Ⅱ, (Russian), Mat. Sb. N. S., 21 (1947), 365-404.   Google Scholar

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M. G. Kreĭn, Solution of the inverse Sturm-Liouville problem, (Russian), Doklady Akad. Nauk SSSR (N.S.), 76 (1951), 21-24.   Google Scholar

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show all references

References:
[1]

V. M. Adamjan and D. Z. Arov, Unitary couplings of semi-unitary operators, (Russian) Mat. Issled., 1 (1966), 3–64; English translation in Amer. Math Soc. Transl. Ser., 2 (1970), p95.  Google Scholar

[2]

V. M. Adamyan and B. S. Pavlov, Zero-radius potentials and M. G. Kreĭn's formula for generalized resolvents, (Russian)translated from Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 149 (1986), Issled. Lineĭn. Teor. Funktsiĭ. XV, 7–23, 186; J. Soviet Math. 42 (1988), 1537–1550. doi: 10.1007/BF01665040.  Google Scholar

[3]

J. BehrndtM. M. Malamud and H. Neidhardt, Scattering theory for open quantum systems with finite rank coupling, Math. Phys. Anal. Geom., 10 (2007), 313-358.  doi: 10.1007/s11040-008-9035-x.  Google Scholar

[4]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs volume, 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013.  Google Scholar

[5]

M. Š. Birman, On the theory of self-adjoint extensions of positive definite operators, Math. Sb. N. S., 38 (1956), 431-450.   Google Scholar

[6]

M. Š. Birman, Birman, Existence conditions for wave operators, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 27 (1963), 883-906.   Google Scholar

[7]

M. Š. Birman and M. G. Kreĭn, On the theory of wave operators and scattering operators, (Russian), Dokl. Akad. Nauk SSSR, 144 (1962), 475-478.   Google Scholar

[8]

M. Š. Birman and M. Z. Solomjak, Spectral Theory of Selfadjoint Operators in Hilbert Space, Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, 1987. Translated from the 1980 Russian original by S. Khrushchëv and V. Peller.  Google Scholar

[9]

G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, (German), Acta Math., 78 (1946), 1-96.  doi: 10.1007/BF02421600.  Google Scholar

[10]

G. Borg, Uniqueness theorems in the spectral theory of y"+(λ-q(x))y = 0, In Den 11te Skandinaviske Matematikerkongress, Trondheim, 1949, pages 276–287. Johan Grundt Tanums Forlag, Oslo, 1952.  Google Scholar

[11]

M. BrownM. MarlettaS. Naboko and I. Wood, Boundary triples and $M$ -functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices, J. Lond. Math. Soc. (2), 77 (2008), 700-718.  doi: 10.1112/jlms/jdn006.  Google Scholar

[12]

K. Cherednichenko, A. Kiselev and L. Silva, Scattering theory for non-selfadjoint extensions of symmetric operators, Preprint, arXiv: 1712.09293, 2017. Google Scholar

[13]

P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math., 32 (1979), 121-251.   Google Scholar

[14]

V. Derkach, Boundary triples, Weyl functions, and the Kreĭn formula, Operator Theory: Living Reference Work, Springer Basel, 2014, 1–33. doi: 10.1007/978-3-0348-0692-3_32-1.  Google Scholar

[15]

V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal., 95 (1991), 1-95.  doi: 10.1016/0022-1236(91)90024-Y.  Google Scholar

[16]

Y. ErshovaI. I. Karpenko and A. V. Kiselev, Isospectrality for graph Laplacians under the change of coupling at graph vertices, J. Spectr. Theory, 6 (2016), 43-66.  doi: 10.4171/JST/117.  Google Scholar

[17]

Y. ErshovaI. I. Karpenko and A. V. Kiselev, Isospectrality for graph Laplacians under the change of coupling at graph vertices: necessary and sufficient conditions, Mathematika, 62 (2016), 210-242.  doi: 10.1112/S0025579314000394.  Google Scholar

[18]

Y. Y. ErshovaI. I. Karpenko and A. V. Kiselev, On inverse topology problem for Laplace operators on graphs, Carpathian Math. Publ., 6 (2014), 230-236.  doi: 10.15330/cmp.6.2.230-236.  Google Scholar

[19]

P. Exner, A duality between Schrödinger operators on graphs and certain Jacobi matrices, Ann. Inst. H. Poincaré Phys. Théor., 66 (1997), 359-371.   Google Scholar

[20]

L. D. Faddeev The inverse problem in the quantum theory of scattering. II. (Russian) Current problems in mathematics, Vol. 3 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, (1974), 93–180. English translation in: J. Sov. Math., 5 (1976), 334–396.  Google Scholar

[21]

L. D. Faddeyev, The inverse problem in the quantum theory of scattering, J. Mathematical Phys., 4 (1963), 72-104.  doi: 10.1063/1.1703891.  Google Scholar

[22]

K. O. Friedrichs, On the perturbation of continuous spectra, Communications on Appl. Math., 1 (1948), 361-406.  doi: 10.1002/cpa.3160010404.  Google Scholar

[23]

I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, (Russian), Izvestiya Akad. Nauk SSSR. Ser. Mat., 15 (1951), 309-360.   Google Scholar

[24]

I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonself-Adjoint Operators, Translations of Mathematical Monographs, vol. 18 AMS, Providence, R. I., 1969.  Google Scholar

[25]

M. L. Gorbachuk and V. I. Gorbachuk, The theory of selfadjoint extensions of symmetric operators; entire operators and boundary value problems, Ukraïn. Mat. Zh., 46 (1994), 55-62.  doi: 10.1007/BF01057000.  Google Scholar

[26]

V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, volume 48 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1991. Translated and revised from the 1984 Russian original. doi: 10.1007/978-94-011-3714-0.  Google Scholar

[27]

I. Kac and M. G. Kreĭn, $R$ -functions-analytic functions mapping upper half-plane into itself, Amer. Math. Soc. Transl. Series 2, 103 (1974), 1-18.   Google Scholar

[28]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Second edition, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132.  Google Scholar

[29]

T. Kato and S. T. Kuroda, The abstract theory of scattering, Rocky Mountain J. Math., 1 (1971), 127-171.  doi: 10.1216/RMJ-1971-1-1-127.  Google Scholar

[30]

A. N. Kočubeĭ, Extensions of symmetric operators and of symmetric binary relations, Mat. Zametki, 17 (1975), 41-48.   Google Scholar

[31]

A. N. Kočubeĭ, Characteristic functions of symmetric operators and their extensions (in Russian), Izv. Akad. Nauk Arm. SSR Ser. Mat., 15 (1980), 219-232.   Google Scholar

[32]

V. Kostrykin and R. Schrader, The inverse scattering problem for metric graphs and the traveling salesman problem, Preprint, arXiv: math-ph/0603010, 2006. Google Scholar

[33]

V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006.  Google Scholar

[34]

M. G. Kreĭn, Theory of self-adjoint extensions of semi-bounded Hermitian operators and applications Ⅱ, (Russian), Mat. Sb. N. S., 21 (1947), 365-404.   Google Scholar

[35]

M. G. Kreĭn, Solution of the inverse Sturm-Liouville problem, (Russian), Doklady Akad. Nauk SSSR (N.S.), 76 (1951), 21-24.   Google Scholar

[36]

M. G. Kreĭn, On the transfer function of a one-dimensional boundary problem of the second order, (Russian), Doklady Akad. Nauk SSSR (N.S.), 88 (1953), 405-408.   Google Scholar

[37]

M. G. Kreĭn, On determination of the potential of a particle from its S-function, (Russian), Dokl. Akad. Nauk SSSR (N.S.), 105 (1955), 433-436.   Google Scholar

[38]

P. Kurasov, Inverse problems for Aharonov-Bohm rings, Math. Proc. Cambridge Philos. Soc., 148 (2010), 331-362.  doi: 10.1017/S030500410999034X.  Google Scholar

[39]

P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, 26. Academic Press, New York-London, 1967.  Google Scholar

[40]

N. Levinson, The inverse Sturm-Liouville problem, Mat. Tidsskr. B., 1949 (1949), 25-30.   Google Scholar

[41]

M. S. Livshitz, On a certain class of linear operators in Hilbert space, Rec. Math. [Mat. Sbornik] N.S., 19 (1946), 239-262.   Google Scholar

[42]

V. A. Marčenko, On reconstruction of the potential energy from phases of the scattered waves, (Russian), Dokl. Akad. Nauk SSSR (N.S.), 104 (1955), 695-698.   Google Scholar

[43]

S. N. Naboko, Absolutely continuous spectrum of a nondissipative operator, and a functional model. Ⅰ, Zap. Naučn. Sem. Leningrad. Otdel Mat. Inst. Steklov. (LOMI), 65 (1976), 90-102,204-205.   Google Scholar

[44]

S. N. Naboko, Functional model of perturbation theory and its applications to scattering theory, Trudy Mat. Inst. Steklov., 147 (1980), 86-114,203.   Google Scholar

[45]

S. N. Naboko, Nontangential boundary values of operator $R$ -functions in a half-plane, Algebra i Analiz, 1 (1989), 197-222.   Google Scholar

[46]

S. N. Naboko, On the conditions for existence of wave operators in the nonselfadjoint case, Wave propagation. Scattering theory, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, 157 (1993), 127–149. doi: 10.1090/trans2/157/09.  Google Scholar

[47]

B. S. Pavlov, Conditions for separation of the spectral components of a dissipative operator, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 123–148, 240. English translation in: Math. USSR Izvestija, 9 (1975), 113–137.  Google Scholar

[48]

B. S. Pavlov, Dilation theory and the spectral analysis of non-selfadjoint differential operators, Proc. 7th Winter School, Drogobych, 1974, TsEMI, Moscow, 1976, 3–69. English translation: Transl., II Ser., Am. Math. Soc., 115 (1981), 103–142.  Google Scholar

[49]

B. S. Pavlov, Selfadjoint dilation of a dissipative Schrödinger operator, and expansion in its eigenfunction, (Russian), Mat. Sb. (N.S.), 102 (1977), 511-536,631.   Google Scholar

[50]

D. B. Pearson, Conditions for the existence of the generalized wave operators, J. Mathematical Phys, 13 (1972), 1490-1499.  doi: 10.1063/1.1665869.  Google Scholar

[51]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. III, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1979.  Google Scholar

[52]

M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1985. Oxford Science Publications.  Google Scholar

[53]

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