Banach Journal of Mathematical Analysis

Phillips symmetric operators and their extensions

Sergii Kuzhel and Leonid Nizhnik

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This article is devoted to the investigation of self-adjoint (and, more generally, proper) extensions of Phillips symmetric operators (PSO). A closed densely defined symmetric operator with equal defect numbers is considered a Phillips symmetric operator if its characteristic function is a constant on C+. We present equivalent definitions of PSO and prove that proper extensions with real spectra of a given PSO are similar to each other. Our results imply that one-point interaction of the momentum operator iddx+αδ(xy) leads to unitarily equivalent self-adjoint operators with Lebesgue spectra. Self-adjoint operators with nontrivial spectral properties can be obtained as a result of more complicated perturbations of the momentum operator. In this way, we study special classes of perturbations which can be characterized as one-point interactions defined by the nonlocal potential γL2(R).

Article information

Banach J. Math. Anal., Volume 12, Number 4 (2018), 995-1016.

Received: 25 November 2017
Accepted: 18 March 2018
First available in Project Euclid: 4 September 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B25: Symmetric and selfadjoint operators (unbounded)
Secondary: 47A10: Spectrum, resolvent

symmetric operator characteristic function wandering subspace bilateral shift Lebesgue spectrum


Kuzhel, Sergii; Nizhnik, Leonid. Phillips symmetric operators and their extensions. Banach J. Math. Anal. 12 (2018), no. 4, 995--1016. doi:10.1215/17358787-2018-0009.

Export citation


  • [1] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, reprint of the 1961 and 1963 translations, Dover, New York, 1993.
  • [2] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2nd ed., with an appendix by P. Exner, Amer. Math. Soc. Chelsea Publ., Providence, 2005.
  • [3] S. Albeverio and S. Kuzhel, “$\mathcal{PT}$-symmetric operators in quantum mechanics: Krein spaces methods” in Non-Selfadjoint Operators in Quantum Physics, Wiley, Hoboken, N.J., 2015, 293–343.
  • [4] S. Albeverio and L. P. Nizhnik, Schrödinger operators with nonlocal potentials, Methods Funct. Anal. Topology 19 (2013), no. 3, 199–210.
  • [5] Y. M. Arlinskii, V. A. Derkach, and E. R. Tsekanovskii, Unitarily equivalent quasi-Hermitian extensions of Hermitian operators (in Russian), Mat. Fiz. 29 (1981), 72–77.
  • [6] T. Y. Azizov and I. S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, Pure Appl. Math. (New York), Wiley, Chichester, 1989.
  • [7] J. Behrndt and M. Langer, On the adjoint of a symmetric operator, J. Lond. Math. Soc. (2) 82 (2010), no. 3, 563–580.
  • [8] O. Christensen, Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 2010.
  • [9] C. R. de Oliveira, Intermediate Spectral Theory and Quantum Dynamics, Progr. Math. Phys. 54, Birkhäuser, Basel, 2009.
  • [10] P. Exner, “Momentum operators on graphs” in Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday, Proc. Sympos. Pure Math. 87, Amer. Math. Soc., Providence, 2013, 105–118.
  • [11] M. L. Gorbachuk, V. I. Gorbachuk, M. G. Krein’s Lectures on Entire Operators, Oper. Theory Adv. Appl. 97, Birkhäuser, Basel, 1997.
  • [12] V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Operator-Differential Equations, Math. Appl. (Soviet Ser.) 48, Kluwer, Dordrecht, 1991.
  • [13] S. Hassi and S. Kuzhel, On $J$-self-adjoint operators with stable $C$-symmetries, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 1, 141–167.
  • [14] H. Helson, The Spectral Theorem, Lecture Notes in Math. 1227, Springer, Berlin, 1986.
  • [15] P. E. T. Jorgensen, S. Pedersen, and F. Tian, Momentum operators in two intervals: Spectra and phase transition, Complex Anal. Oper. Theory 7 (2013), no. 6, 1735–1773.
  • [16] A. N. Kochubei, Symmetric operators commuting with a family of unitary operators, Funktsional. Anal. i Prilozhen. 13 (1979), no. 4, 77–78; English translation in Funct. Anal. Appl. 13 (1980), no. 4, 300-301.
  • [17] A. N. Kochubei, Characteristic functions of symmetric operators and their extensions (in Russian), Izv. Nats. Akad. Nauk Armenii Mat. 15 (1980), no. 3, 219–232; English translation in Soviet J. Contemp. Math. Anal. 15 (1980).
  • [18] A. V. Kuzhel and S. A. Kuzhel, Regular Extensions of Hermitian Operators, VSP, Utrecht, 1998.
  • [19] S. A. Kuzhel, O. Shapovalova, and L. Vavrykovych, On $J$-self-adjoint extensions of the Phillips symmetric operator, Methods Funct. Anal. Topology, 16 (2010), no. 4, 333–348.
  • [20] P. D. Lax and R. F. Phillips, Scattering Theory, 2nd ed., with appendices by C. S. Morawetz and G. Schmidt, Pure Appl. Math. 26, Academic Press, Boston, 1989.
  • [21] L. P. Nizhnik, Inverse spectral nonlocal problem for the first order ordinary differential equation, Tamkang J. Math. 42 (2011), no. 3, 385–394.
  • [22] S. Pedersen, J. D. Phillips, F. Tian, and C. E. Watson, On the spectra of momentum operators, Complex Anal. Oper. Theory 9 (2015), no. 7, 1557–1587.
  • [23] R. S. Phillips, “The extension of dual subspaces invariant under an algebra” in Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem, 1961, 366–398.
  • [24] K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert space, Grad. Texts in Math. 265, Springer, Dordrecht, 2012.
  • [25] A. V. Shtraus, Extensions and characteristic function of a symmetric operator (in Russian), Izv. Akad. Nauk SSSR. Ser. Mat. 32 (1968), 186–207; English translation in Izv. Math. 2 (1968), 181–203.
  • [26] J. G. Sinai, Dynamical systems with countable Lebesgue spectrum, I (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961) 899–924.
  • [27] B. Sz.-Nagy, C. Foias, H. Bercovici, and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, 2nd ed., Universitext, Springer, New York, 2010.