Abstract
We study spectral properties of the one-dimensional Schrödinger operators \( {\mathrm{H}}_{\mathrm{X},\alpha, \mathrm{q}}:=-\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+\mathrm{q}(x)+{\varSigma_x}_{{}_n}\in X{\alpha}_n\delta \left(x-{x}_n\right) \) with local interactions, d* = 0, and an unbounded potential q being a piecewise constant function, by using the technique of boundary triplets and the corresponding Weyl functions. Under various sufficient conditions for the self-adjointness and discreteness of Jacobi matrices, we obtain the condition of self-adjointness and discreteness for the operator H X,α,q.
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References
N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York, 1965.
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, New York, 1993.
S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, AMS, Providence, RI, 2005.
S. Albeverio, A. Kostenko, and M. Malamud, “Spectral theory of semibounded Sturm–Liouville operators with local interactions on a discrete set,” J. Math. Phys., 51, 102102 (2010).
A. Ananieva, “One-dimensional Schr¨odinger operator with unbounded potential and point interactions,” Math. Notes, 99, 769–773 (2016).
Yu. M. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators, AMS, Providence, RI, 1968.
J. F. Brasche, “Perturbation of Schrödinger Hamiltonians by measures — self-adjointness and semiboundedness,” J. Math. Phys., 26, 621–626 (1985).
J. Bruening, V. Geyler, and K. Pankrashkin, “Spectra of self-adjoint extensions and applications to solvable Schrödinger operators,” Rev. Math. Phys., 20, 1–70 (2008).
R. Carlone, M. Malamud, and A. Posilicano, “On the spectral theory of Gesztesy– Šeba realizations of 1-D Dirac operators with point interactions on a discrete set,” J. Differ. Equa., 254, No. 9, 3835–3902 (2013).
T. Chihara, “Chain sequences and orthogonal polynomials,” Trans. AMS, 104, 1–16 (1962).
C. Shubin Christ and G. Stolz, “Spectral theory of one-dimentional Schrödinger operators with point interactions,” J. Math. Anal. Appl., 184, 491–516 (1994).
V. A. Derkach and M. M. Malamud, “Generalized resolvents and the boundary value problems for Hermitian operators with gaps,” J. Funct. Anal., 95, 1–95 (1991).
V. A. Derkach and M. M. Malamud, “Generalized resolvents and the boundary-value problems for Hermitian operators with gaps,” J. Math. Sci., 73, No. 2, 141–242 (1995).
A. Dijksma and H. S. V. de Snoo, “Symmetric and self-adjoint relations in Kre˘ın spaces,” Oper. Theory: Adv. Appl., 24, 145–166 (1987).
F. Gesztesy and P. Šeba, “New analytically solvable models of relativistic point interactions,” Lett. Math. Phys., 13, 345–358 (1987).
F. Gesztesy andW. Kirsch, “One–dimensional Schrödinger operators with interactions singular on a discrete set,” J. reine Angew. Math., 362, 27–50 (1985).
V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Kluwer, Dordrecht, 1991.
R. S. Ismagilov and A. G. Kostyuchenko, “Spectral asymptotics for the Sturm–Liouville operator with point interaction,” Funct. Anal. Appl., 44, 253–258 (2010).
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966.
A. N. Kochubei, “One-dimensional point interactions,” Ukr. Math. J., 41, 1391–1395 (1989).
A. Kostenko and M. Malamud, “1-D Schrödinger operators with local point interactions on a discrete set,” J. Differ. Equa., 249, 253–304 (2010).
A. S. Kostenko and M. M. Malamud, “1-D Schrödinger operators with local point interactions: a review,” Proc. Symp. Pure Math., 87, 232–262 (2013).
A. Kostenko and M. Malamud, “One-dimensional Schrödinger operator with δ-interactions,” Funct. Anal. Appl., 44, 87–91 (2010).
A. Kostenko, M. Malamud, and N. Natyagaylo, “Schrödinger operators with matrix-valued potentials and point interactions,” Mathem. Notes, 100, 59—77 (2016).
A. G. Kostyuchenko and K. A. Mirzoev, “Complete indefiniteness tests for Jacobi matrices with matrix entries,” Funct. Anal. Appl., 35, 265–269 (2001).
M. G. Krein and H. Langer, “On defect subspaces and generalized resolvents of a Hermitian operator in a space Πϰ,” Funct. Anal. Appl., 5, 136–146 (1971).
V. Lotoreichik and S. Siminov, “Spectral analysis of the half-line Kronig––Penney model with Wigner–von Neumann perturbations,” Rep. Math. Phys., 74, 45–72 (2014).
M. M. Malamud and H. Neidhardt, “On the unitary equivalence of absolutely continuous parts of self-adjoint extensions,” J. Funct. Anal., 260, 613–638 (2011).
M. M. Malamud and H. Neidhardt, “Sturm––Liouville boundary–value problems with operator potentials and unitary equivalence,” J. Diff. Equa., 252, 5875–5922 (2012).
K. A. Mirzoev, “Sturm–Liouville operators,” Trans. Moscow Math. Soc., textbf75, 281–299 (2014).
V. A. Mikhailets, “Schrödinger operator with point δ′-interactions,” Doklady Math., 348, No. 6, 727–730(1996).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, III: Scattering Theory, Academic Press, New York, 1975.
A. M. Savchuk and A. A. Shkalikov, “Inverse problems for the Sturm-Liouville operator with potentials in Sobolev spaces: uniform stability,” Funct. Anal. Appl., 44, No. 4, 270–285 (2010).
C. Shubin Christ and G. Stolz, “Spectral theory of one-dimentional Schr¨odinger operators with point interactions,” J. Math. Anal. Appl., 184, 491–516 (1994).
G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, AMS, Providence, RI, 2000.
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 13, No. 1, pp. 28–67, January–March, 2016.
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Ananieva, A.Y. 1-D Schrödinger Operators with Local Interactions on a Discrete Set with Unbounded Potential. J Math Sci 220, 554–583 (2017). https://doi.org/10.1007/s10958-016-3200-8
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DOI: https://doi.org/10.1007/s10958-016-3200-8