Abstract
When the singular finite rank perturbations of an unbounded selfadjoint operator Ao in a Hilbert space c, formally defined by A(α) = A 0 + GαG*, are lifted to an exit Pontryagin space ℌ by means of an operator model, they become ordinary range perturbations of a self-adjoint operator H ∞ in ℌ ⊃ ℌ0:H T = H ∞-ΩT -1Ω*. Here G is a mapping from ℂd into some scale space ℌ (A 0), ℕ, of generalized elements associated with A 0, while Ω is a mapping from into ℂd the extended space ℌ, where H T , is defined. The connection between these two perturbation formulas is studied.
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References
S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden, Solvable models in quan-tum mechanics,Springer-Verlag, New York, 1988.
S. Albeverio and P. Kurasov, “Finite rank perturbations and distribution theory”, Proc. Amer. Math. Soc., 127 (1999), 1151–1161.
S. Albeverio and P. Kurasov, Singular perturbations of differential operators, London Math. Soc., Lecture Notes Series 271, Cambridge University Press, 1999.
Yu. M. Arlinskiĭ, S. Hassi, Z. Sebestyen, and H. S. V. de Snoo, “On the class of extremal extensions of a nonnegative operator”, Bela Szokefalvi-Nagy memorial volume, Oper. Theory Adv. Appl., 127 (2001), 41–81.
T.Ya. Azizov and I.S. Iokhvidov, Foundations of the theory of linear operators in spaces with an indefinite metric, Nauka, Moscow, 1986 (Russian) (English transla-tion: Wiley, New York, 1989).
Yu. M. Berezanskiĭ, Expansions in eigenfunctions of self-adjoint operators, Naukova Dumka, Kiev, 1965 (Russian) (English translation: Translations of Mathematical Monographs, Volume 17, American Mathematical Society, 1968).
V.A. Derkach, “On generalized resolvents of Hermitian relations”, J. Math. Sciences, 97 (1999), 4420–4460.
V.A. Derkach and S. Hassi, “A reproducing kernel space model for N„-functions”, Proc. Amer. Math. Soc., 131 (2003), 3795–3806.
V.A. Derkach, S. Hassi, and H.S.V. de Snoo, “Operator models associated with singular perturbations”, Methods of Functional Analysis and Topology, 7 (2001), 1–21.
V.A. Derkach, S. Hassi, and H.S.V. de Snoo, “Rank one perturbations in a Pontryagin space with one negative square”, J. Funct. Anal., 188 (2002), 317–349.
V.A. Derkach, S. Hassi, and H.S.V. de Snoo, “Singular perturbations of self-adjoint operators”, Mathematical Physics, Analysis and Geometry, 6 (2003), 349–384.
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.
V.A. Derkach and M.M. Malamud, “The extension theory of hermitian operators and the moment problem”, J. Math. Sciences, 73 (1995), 141–242.
J.F. van Diejen and A. Tip, “Scattering from generalized point interaction using selfadjoint extensions in Pontryagin spaces”, J. Math. Phys., 32 (3), (1991), 631–641.
A. Dijksma, H. Langer, Yu.G. Shondin, and C. Zeinstra, “Self-adjoint operators with inner singularities and Pontryagin spaces”, Oper. Theory Adv. Appl., 118 (2000), 105–175.
V.I. Gorbachuk and M.L. Gorbachuk, Boundary value problems for operator differential equations,Naukova Dumka, Kiev, 1984 (Russian) (English translation: Kluwer Academic Publishers, Dordrecht, Boston, London, 1990).
S. Hassi, M. Kaltenback, and H.S.V. de Snoo, “Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass N1 of Nevanlinna functions”, J. Operator Theory, 37 (1997), 155–181.
S. Hassi, H. Langer, and H.S.V. de Snoo, “Self-adjoint extensions for a class of symmetric operators with defect numbers (1, 1)”, 15th OT Conference Proceedings, (1995), 115–145.
S. Hassi and H.S.V. de Snoo, “One-dimensional graph perturbations of self-adjoint relations”, Ann. Acad. Sci. Fenn. A.I. Math., 22 (1997), 123–164.
S. Hassi and H.S.V. de Snoo, “Nevanlinna functions, perturbation formulas and triplets of Hilbert spaces”, Math. Nachr., 195 (1998), 115–138.
M.G. KreIn and H. Langer, “Uber die Q-Funktion eines ir-hermiteschen Operators im Raume IL”, Acta Sci. Math. (Szeged), 34 (1973), 191–230.
P. Kurasov,“ℌ-n-perturbations of self-adjoint operators and Krein’s resolvent formula”, Integr. Equ. Oper. Theory, 45 (2003), 437–460.
Yu.G. Shondin, “Quantum-mechanical models in associated with extensions of the energy operator in Pontryagin space”, Teor. Mat. Fiz., 74 (1988), 331–344) (Russian) (English translation: Theor. Math. Phys., 74 (1988), 220–230).
A.V. Straus, “Extensions and generalized resolvents of a symmetric operator which is not densely defined”, Izv. Akad. Nauk SSSR, Ser. Mat., 34 (1970), 175–202 (Russian) (English translation: Math. USSR-Izvestija, 4 (1970), 179–208).
E.R. Tselcanovskii and Yu. L. Shmulyan, “The theory of bi-extensions of operators on rigged Hilbert spaces. Unbounded operator colligations and characteristic functions”, Uspekhi Mat. Nauk, 32:5 (1977), 69–124 (Russian) (English translation: Russian Math. Surveys, 32:5 (1977), 73–131).
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Derkach, V., Hassi, S., de Snoo, H. (2004). Singular Perturbations as Range Perturbations in a Pontryagin Space. In: Janas, J., Kurasov, P., Naboko, S. (eds) Spectral Methods for Operators of Mathematical Physics. Operator Theory: Advances and Applications, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7947-7_4
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DOI: https://doi.org/10.1007/978-3-0348-7947-7_4
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