## Banach Journal of Mathematical Analysis

### Non-self-adjoint Schrödinger operators with nonlocal one-point interactions

#### Abstract

We generalize and study, within the framework of quantum mechanics and working with $1$-dimensional, manifestly non-Hermitian Hamiltonians $H=-{d^{2}}/{dx^{2}}+V$, the traditional class of exactly solvable models with local point interactions $V=V(x)$. We discuss the consequences of the use of nonlocal point interactions such that $(Vf)(x)=\int K(x,s)f(s)\,ds$ by means of the suitably adapted formalism of boundary triplets.

#### Article information

Source
Banach J. Math. Anal., Volume 11, Number 4 (2017), 923-944.

Dates
Accepted: 12 January 2017
First available in Project Euclid: 11 September 2017

https://projecteuclid.org/euclid.bjma/1505116817

Digital Object Identifier
doi:10.1215/17358787-2017-0032

Mathematical Reviews number (MathSciNet)
MR3708536

Zentralblatt MATH identifier
06841261

Subjects
Primary: 47B25: Symmetric and selfadjoint operators (unbounded)
Secondary: 35P05: General topics in linear spectral theory

#### Citation

Kuzhel, Sergii; Znojil, Miloslav. Non-self-adjoint Schrödinger operators with nonlocal one-point interactions. Banach J. Math. Anal. 11 (2017), no. 4, 923--944. doi:10.1215/17358787-2017-0032. https://projecteuclid.org/euclid.bjma/1505116817

#### References

• [1] 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., Providence, 2005.
• [2] S. Albeverio, R. O. Hryniv, and L. Nizhnik, Inverse spectral problems for non-local Sturm-Liouville operators, Inverse Problems 23 (2007), no. 2, 523–535.
• [3] S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators: Solvable Schrödinger Type Operators, London Math. Soc. Lecture Note Ser. 271, Cambridge Univ. Press, Cambridge, 2000.
• [4] S. Albeverio and S. Kuzhel, One-dimensional Schrödinger operators with $\mathcal{P}$-symmetric zero-range potentials, J. Phys. A 38 (2005), no. 22, 4975–4988.
• [5] S. Albeverio, S. Kuzhel, and L. Nizhnik, Singular perturbed self-adjoint operators in scales of Hilbert spaces (in Russian), Ukraïn. Mat. Zh. 59 (2007), no. 6, 723–743; English translation in Ukrainian Math. J. 59 (2007), 787–810.
• [6] S. Albeverio and L. Nizhnik, Schrödinger operators with nonlocal point interactions, J. Math. Anal. Appl. 332 (2007), no. 2, 884–895.
• [7] S. Albeverio and L. Nizhnik, Schrödinger operators with nonlocal potentials, Methods Funct. Anal. Topology. 19 (2013), no. 3, 199–210.
• [8] F. Bagarello, J.-P. Gazeau, F. H. Szafraniec, and M. Znojil, eds., Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects, Wiley, Hoboken, 2015.
• [9] H. Baumgärtel, Analytic Perturbation Theory for Matrices and Operators, Oper. Theory Adv. Appl. 15, Birkhäuser, Basel, 1985.
• [10] J. Behrndt, F. Gesztesy, H. Holden, and R. Nichols, Dirichlet-to-Neumann maps, abstract Weyl–Titchmarsh $M$-functions, and a generalized index of unbounded meromorphic operator-valued functions, J. Differential Equations 261 (2016), no. 6, 3551–3587.
• [11] J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal. 243 (2007), no. 2, 536–565.
• [12] J. Behrndt and M. Langer, On the adjoint of a symmetric operator, J. Lond. Math. Soc. (2) 82 (2010), no. 3, 563–580.
• [13] J. Behrndt, L. Leben, F. Martínez-Pería, R. Möws, and C. Trunk, Sharp eigenvalue estimates for rank one perturbations of nonnegative operators in Krein spaces, J. Math. Anal. Appl. 439 (2016), no. 2, 864–895.
• [14] C. M. Bender, “Ghost busting: Making sense of non-Hermitian Hamiltonians” in Algebraic Analysis of Differential Equations (Kyoto, 2005), Springer, Tokyo, 2007, 55–66.
• [15] J. Brasche and L. Nizhnik, One-dimensional Schrödinger operators with general point interactions, Methods Funct. Anal. Topology 19 (2013), no. 1, 4–15.
• [16] V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), no. 1, 1–95.
• [17] V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. (N. Y.) 73 (1995), no. 2, 141–242.
• [18] F. Gesztesy and E. Tsekanovskii, On matrix-valued Herglotz functions, Math. Nachr. 218 (2000), 61–138.
• [19] A. Grod and S. Kuzhel, Schrödinger operators with non-symmetric zero-range potentials, Methods Funct. Anal. Topology 20 (2014), no. 1, 34–49.
• [20] G. Sh. Guseinov, On the concept of spectral singularities, Pramana J. Phys. 73 (2009), no. 3, 587–603.
• [21] A. Kuzhel and S. Kuzhel, Regular Extensions of Hermitian Operators, VSP, Utrecht, 1998.
• [22] A. Mostafazadeh, Delta-function potential with a complex coupling, J. Phys. A 39 (2006), no. 43, 13495–13506.
• [23] A. Mostafazadeh, Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys. 7 (2010), no. 7, 1191–1306.
• [24] A. Mostafazadeh, Spectral singularities of a general point interaction, J. Phys. A 44 (2011), no. 37, Art. ID 375302.
• [25] A. Mostafazadeh, “Physics of spectral singularities” in Geometric Methods in Physics, Trends Math., Birkhäuser/Springer, Cham, 2015, 145–165.
• [26] M. A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis, Amer. Math. Soc. Transl. (2) 16 (1960), 103–193.
• [27] K. Schmüdgen, Unbounded Self-Adjoint Operators on Hilbert Space, Grad. Texts in Math. 265, Springer, Dordrecht, 2012.
• [28] M. Znojil and V. Jakubský, Solvability and $PT$-symmetry in a double-well model with point interactions, J. Phys. A 38 (2005), no. 22, 5041–5056.