Abstract
General point interactions for the second derivative operator in one dimension are studied. In particular, \(\mathcal{P}\mathcal{T}\)-self-adjoint point interactions with the support at the origin and at points ±l are considered. The spectrum of such non-Hermitian operators is investigated and conditions when the spectrum is pure real are presented. The results are compared with those for standard self-adjoint point interactions.
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Albeverio, S., D?browski, L. and Kurasov, P.: Symmetries of Schrödinger operators with point interactions, Lett. Math. Phys. 45 (1998), 33–47.
Albeverio, S., Fei, S. M. and Kurasov, P.: Gauge fields, point interactions and fewbody problems in one dimension, SFB256-Preprint No. 614, Rheinische Friedrich-Wilhelms-Universität-Bonn, September 1999.
Albeverio, S., Fei, S. M. and Kurasov, P.: N-Body Problems with 'Spin'-Related Contact Interactions in One Dimensional, Rep. Math. Phys. 47 (2001), 157–165.
Albeverio, S., Gesztesy, F., Høegh-Krohn, R. and Holden, H.: Solvable Models in Quantum Mechanics, Springer, New York, 1988.
Albeverio, S. and Kurasov, P.: Singular perturbations of differential operators. In: Solvable Schrödinger type operators, London Math. Soc. Lecture Note Ser. 271, Cambridge Univ. Press, Cambridge, 2000.
Bender, C. M. and Boettcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998), 5243–5246.
Bender, C. M., Boettcher, S., Jones, H. F. and Savage, V. M.: Complex square well-a new exactly solvable quantum mechanical model, J. Phys. A 32 (1999), 6771–6781.
Bender, C. M., Boettcher, S. and Meisinger, P. N.: PT-symmetric quantum mechanics, J. Math. Phys. 40 (1999), 2201–2229.
Bender, C. M., Cooper, F., Meisinger, P. N. and Savage, V. M.: Variational ansatz for PT-symmetric quantum mechanics, Phys. Lett. A 259 (1999), 224–231.
Bender, C. M. and Dunne, G. V.: Large-order perturbation theory for a non-Hermitian PT-symmetric Hamiltonian, J. Math. Phys. 40 (1999), 4616–4621.
Bender, C. M., Dunne, G. V. and Meisinger, P. N.: Complex periodic potentials with real band spectra, Phys. Lett. A 252 (1999), 272–276.
Boman, J. and Kurasov, P.: Finite rank singular perturbations and distributions with discontinuous test functions, Proc. Amer. Math. Soc. 126 (1998), 1673–1683.
Coutinho, F. A. B., Nogami, Y. and Lauro Tomio: Time-reversal aspect of the point interactions in one-dimensional quantum mechanics, J. Phys. A 32 (1999), L133–L136.
Demkov, Yu. N. and Ostrovsky, V. N.: Zero-range potentials and their applications in atomic physics, Plenum, New York, 1988.
Dorey, P., Dunning, C. and Tateo, R.: Supersymmetry and the spontaneous breakdown of PT-symmetry, J. Phys. A 34 (2001), L391–L400.
Dorey, P., Dunning, C. and Tateo, R.: Spectral equivalence, bethe ansatz, and reality properties in PT-symmetric quantum mechanics, J. Phys. A 34 (2001), 5679–5704.
Handy, C. R.: Generating converging eigenenergy bounds for the discrete states of the-ix 3 non-Hermitian potential, J. Phys. A 34 (2001), L271–L277.
Handy, C. R., Khan, D., Xiao-Qian Wang, and Tymczak, C. J.: Mutiscale reference function analysis of the PT-symmetry breaking solutions for the P 2 + iX 2 + i?X Hamiltonian, J. Phys. A 34 (2001), 5593–5602.
Kurasov, P.: Distribution theory for discontinuous test functions and differential operators with generalized coefficients, J. Math. Anal. Appl. 201 (1996), 297–323.
Kurasov, P.: Energy dependent boundary conditions and the few-body scattering problem, Rev. Math. Phys. 9 (1997), 853–906.
Lévai, G. and Znojil, M.: Systematic search for PT-symmetric potentials with real energy spectra, J. Phys. A 33 (2000), 7165–7180.
Naboko, S. N.: Functional model of perturbation theory and its applications to scattering theory (Russian) In: BoundaryValue Problems of Math. Phys. 10. Trudy Mat. Inst. Steklov. 147 (1980), 86–114.
Naboko, S. N.: Conditions for the existence of wave operators in the nonselfadjoint case (Russian) In: Wave Propagation. Scattering Theory (Russian), Probl. Mat. Fiz. 12, Leningrad. Univ., Leningrad, 1987, pp. 132–155.
Pavlov, B.: On a non-selfadjoint Schrödinger operator (in Russian), Prob. Math. Phys. I (1966), 102–132.
Pavlov, B. S.: Dilation theory and spectral analysis of nonselfadjoint differential operators (Russian), In: Mathematical Programming and Related Questions, (Proc. Seventh Winter School, Drogobych, 1974), Theory of Operators in Linear Spaces (Russian), Central. Ekonom. Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 3–69.
Šeba, P.: The generalized point interaction in one dimension, Czech. J. Phys. B 36 (1986), 667–673.
Znojil, M.: Non-Hermitian matrix description of the PT-symmetric anharmonic oscillators, J. Phys. A 32 (1999), 7419–7428.
Znojil, M.: Exact solution for Morse oscillator in PT-symmetric quantum mechanics, Phys. Lett. A 264 (1999), 108–111.
Znojil, M.: PT-symmetric harmonic oscillators, Phys. Lett. A 259 (1999), 220–223.
Znojil, M.: Spiked and PT-symmetrized decadic potentials supporting elementary N-plets of bound states, J. Phys. A 33 (2000), 6825–6833.
Znojil, M.: PT-symmetrically regularized Eckart, Pöschl-Teller and Hulthén potentials, J. Phys. A 33 (2000), 4561–4572.
Znojil, M., Cannata, F., Bagchi, B. and Roychoudhury, R. Supersymmetry without Hermiticity within PT symmetric quantum mechanics, Phys. Lett. B 483 (2000), 284–289.
Znojil, M. and Lávai, G.: The Coulomb-harmonic oscillator correspondence in PT symmetric quantum mechanics, Phys. Lett. A 271 (2000), 327–333.
Znojil, M. and Tater, M.: Complex Calogero model with real energies, J. Phys. A 34 (2001), 1793–1803.
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Albeverio, S., Fei, SM. & Kurasov, P. Point Interactions: \(\mathcal{P}\mathcal{T}\)-Hermiticity and Reality of the Spectrum. Letters in Mathematical Physics 59, 227–242 (2002). https://doi.org/10.1023/A:1015559117837
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DOI: https://doi.org/10.1023/A:1015559117837