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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[30]

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

[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.

[32]

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

[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.

[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.

[35]

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

[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.

[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.

[38]

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

[39]

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

[40]

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

[41]

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

[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.

[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.

[44]

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

[45]

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

[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.

[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.

[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.

[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.

[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.

[51]

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

[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.

[53]

M. Rosenblum, Perturbation of the continuous spectrum and unitary equivalence, Pacific J. Math., 7 (1957), 997-1010. doi: 10.2140/pjm.1957.7.997.

[54]

V. Ryzhov. Functional model of a class of non-selfadjoint extensions of symmetric operators, In Operator Theory, Analysis and Mathematical Physics, 174 of Oper. Theory Adv. Appl., Birkhäuser, Basel, (2007), 117–158. doi: 10.1007/978-3-7643-8135-6_9.

[55]

K. Schmüdgen, Unbounded Self-Adjoint Operators on Hilbert Space, Volume 265 of Graduate Texts in Mathematics. Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-4753-1.

[56]

B. Sz.-Nagy, C. Foias, H. Bercovici and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, Universitext. Springer, New York, Second enlarged edition, 2010. doi: 10.1007/978-1-4419-6094-8.

[57]

W. T. Tutte, Graph theory. With a foreword by C. St. J. A. Nash-Williams, Reprint of the 1984 original. Encyclopedia of Mathematics and its Applications, 21. Cambridge University Press, Cambridge, 2001.

[58]

M. I. Višik, On general boundary problems for elliptic differential equations (Russian), Trudy Moskov. Mat. Obšč., 86 (1952), 645-648.

[59]

J. von Neumann, Über adjungierte Funktionaloperatoren, Ann. Math., 33 (1932), 294-310. doi: 10.2307/1968331.

[60]

J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955. Translated by Robert T. Beyer.

[61]

R. Weder, Scattering theory for the matrix Schrödinger operator on the half line with general boundary conditions J. Math. Phys., 56 (2015), 092103, 24 pp. doi: 10.1063/1.4930293.

[62]

R. Weder, Trace formulas for the matrix Schrödinger operator on the half-line with general boundary conditions J. Math. Phys., 57 (2016), 112101, 11 pp. doi: 10.1063/1.4964447.

[63]

J. Weidmann, Linear Operators in Hilbert Spaces, volume 68 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1980. Translated from the German by Joseph Szücs.

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D. R. Yafaev, Mathematical Scattering Theory, volume 105 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992. General theory, Translated from the Russian by J. R. Schulenberger.

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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[30]

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

[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.

[32]

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

[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.

[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.

[35]

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

[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.

[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.

[38]

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

[39]

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

[40]

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

[41]

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

[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.

[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.

[44]

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

[45]

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

[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.

[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.

[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.

[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.

[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.

[51]

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

[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.

[53]

M. Rosenblum, Perturbation of the continuous spectrum and unitary equivalence, Pacific J. Math., 7 (1957), 997-1010. doi: 10.2140/pjm.1957.7.997.

[54]

V. Ryzhov. Functional model of a class of non-selfadjoint extensions of symmetric operators, In Operator Theory, Analysis and Mathematical Physics, 174 of Oper. Theory Adv. Appl., Birkhäuser, Basel, (2007), 117–158. doi: 10.1007/978-3-7643-8135-6_9.

[55]

K. Schmüdgen, Unbounded Self-Adjoint Operators on Hilbert Space, Volume 265 of Graduate Texts in Mathematics. Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-4753-1.

[56]

B. Sz.-Nagy, C. Foias, H. Bercovici and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, Universitext. Springer, New York, Second enlarged edition, 2010. doi: 10.1007/978-1-4419-6094-8.

[57]

W. T. Tutte, Graph theory. With a foreword by C. St. J. A. Nash-Williams, Reprint of the 1984 original. Encyclopedia of Mathematics and its Applications, 21. Cambridge University Press, Cambridge, 2001.

[58]

M. I. Višik, On general boundary problems for elliptic differential equations (Russian), Trudy Moskov. Mat. Obšč., 86 (1952), 645-648.

[59]

J. von Neumann, Über adjungierte Funktionaloperatoren, Ann. Math., 33 (1932), 294-310. doi: 10.2307/1968331.

[60]

J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955. Translated by Robert T. Beyer.

[61]

R. Weder, Scattering theory for the matrix Schrödinger operator on the half line with general boundary conditions J. Math. Phys., 56 (2015), 092103, 24 pp. doi: 10.1063/1.4930293.

[62]

R. Weder, Trace formulas for the matrix Schrödinger operator on the half-line with general boundary conditions J. Math. Phys., 57 (2016), 112101, 11 pp. doi: 10.1063/1.4964447.

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