Towards theory of C-symmetries

• Autorzy: Kuzhel S. , Sudilovskaya V.

Identyfikatory

DOI

10.7494/OpMath.2017.37.1.65

• Języki publikacji: EN

Abstrakty

EN

The concept of C-symmetry originally appeared in PT-symmetric quantum mechanics is studied within the Krein spaces framework.

Słowa kluczowe

EN

Krein space; J-self-adjoint operator; J-symmetric operator; Friedrichs extension; C-symmetry

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2017
• Tom: Vol. 37, no. 1
• Strony: 65--80
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Kuzhel S.

• AGH University of Science and Technology Faculty of Applied Mathematics al. A. Mickiewicza 30, 30-059 Krakow, Poland

autor

Sudilovskaya V.

• Kyiv Vocational College Kyiv, Ukraina

Bibliografia

• [1] S. Albeverio, S. Kuzhel, PT-Symmetric Operators in Quantum Mechanics: Krein Spaces Methods, [in:] Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects, Fabio Bagarello, Jean-Pierre Gazeau, Franciszek H. Szafraniec, Miloslav Znojil (eds), John Wiley & Sons, Inc. 2015.
• [2] S. Albeverio, A. Motovilov, A. Shkalikov, Bounds on variation of spectraal subspaces under J-self-adjoint perturbations, Integr. Equ. Oper. Theory 64 (2009), 455-486.
• [3] Yu. Arlinskii, E. Tsekanovskii, M. Krein’s research on semi-bounded operators, its contemporary developments, and applications, Operator Theory: Advances and Applications 190 (2009), 65-112.
• [4] Yu. M. Arlinskii, S. Hassi, Z. Sebestyen, H.S.V. de Snoo, On the class of extremal extensions of a nonnegative operator, Recent Advances in Operator Theory and Related Topics The Bela Szokefalvi-Nagy Memorial Volume, Operator Theory: Advances and Applications 127 (2001), 41-81.
• [5] T.Ya. Azizov, I.S. Iokhvidov, Linear Operators in Spaces with Indefinite Metric, Wiley, Chichester, 1989.
• [6] C.M. Bender, Introduction to PT-symmetric quantum theory, Contemp. Phys. 46 (2005) 277-292.
• [7] C.M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70 (2007), 947-1018.
• [8] C.M. Bender, M. Gianfreda, Nonuniqueness of the C operator in PT-symmetric quantum mechanics, J. Phys. A. 46 (2013), 275306-275324.
• [9] C.M. Bender, S.P. Klevansky, Nonunique C operator in PT quantum mechanics, Phys. Lett. A 373 (2009), 2670-2674.
• [10] C.M. Bender, S. Kuzhel, Unbounded C-symmetries and their nonuniqueness, J. Phys. A. 45 (2012), 444005-444019.
• [11] C.M. Bender, B. Tan, Calculation of the hidden symmetry operator for a PT-symmetric square well, J. Phys. A. 39 (2006), 1945-1953.
• [12] B. Curgus, B. Najman, The operator (sgnx) -y is similar to a selfadjoint operator in L2(R), Proc. Amer. Math. Soc. 123 (1995), 1125-1128.
• [13] A. Gheondea, Canonical forms of unbounded unitary operators in Krein space, Publ. RIMS Kyoto Univ. 24 (1988), 205-224.
• [14] A. Grod, S. Kuzhel, V. Sudilovskaja, On operators of transition in Krein spaces, Opuscula Math. 31 (2011), 49-59.
• [15] V. Kostrykin, K.A. Makarov, A.K. Motovilov, Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach, Contemp. Math. 327 (2003), 181-198.
• [16] M.G. Krein, Theory of self-adjoint extensions of semibounded operators and its applications I. Math. Trans. 20 (1947), 431-495.
• [17] S. Kuzhel, On pseudo-Hermitian operators with generalized C-symmetries, Operator Theory: Advances and Applications 190 (2009), 128-135.
• [18] H. Langer, Maximal dual pairs of invariant subspaces of J-self-adjoint operators, Mat. Za- metki 7 (1970), 443-447 [in Russian].
• [19] H. Langer, Spectral functions of definitizable operators in Krein spaces, Functional Analysis Proceedings of a conference held at Dubrovnik, Yugoslavia, November 2-14, 1981, Lecture Notes in Mathematics, 948, Springer-Verlag, Berlin, Heidelberg, New York, 1982, 1-46.
• [20] S.N. Naboko, Conditions for similarity to unitary and self-adjoint operators, Func. Anal. and Appl. 18 (1984) 1, 13-22.
• [21] M. Tomita, Operators and operator algebras in Krein spaces, I Spectral Analysis in Pontrjagin spaces, RIMS-Kokyuuroku 398 (1980), 131-158.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).