## Zeroes of the spectral density of the Schrödinger operator with the slowly decaying Wigner-von Neumann potential.(English)Zbl 1372.47059

In this paper, the author considers the operator $L_{\alpha}:L_2(\mathbb R_+)\cap H_{\mathrm{loc}}^2(\mathbb R_+)\to L_2(\mathbb R_+)$ as $$L_{\alpha}y=Ly$$, where $Ly:=-y''+q(x)y+q_{WN}(x,\gamma)y+q_1(x)y,\quad x\in \mathbb R_+:=[0,\infty),$ and $y(0)\cos\alpha-y'(0)\sin\alpha=0.$ Here, $$q\in L_1(0,a)$$ is periodic with period $$a$$, \begin{aligned} q_{WN}(x,\gamma)=\begin{cases} c\sin(2\omega x+\delta)/x^{\gamma},\quad &\text{if }\gamma\in(\frac12,1),\\c\sin(2\omega x+\delta)/x+1, &\text{if }\gamma=1,\end{cases}\end{aligned} $$c,\omega,\delta\in\mathbb R$$, $$2a\omega/\pi\not\in\mathbb Z$$, $$q_1\in L_1(\mathbb R_+)$$, $$\alpha\in[0,\pi)$$. The author studies the properties of the spectral density of the operator $$L_{\alpha}$$. The main results are given in Theorem 1.1.

### MSC:

 47E05 General theory of ordinary differential operators 34B20 Weyl theory and its generalizations for ordinary differential equations 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
Full Text:

### References:

 [1] Abramowitz, K., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964) · Zbl 0171.38503 [2] Behncke, H, Absolute continuity of Hamiltonians with von Neumann-Wigner potentials I, Proc. Am. Math. Soc., 111, 373-384, (1991) · Zbl 0726.34072 [3] Behncke, H, Absolute continuity of Hamiltonians with von Neumann-Wigner potentials II, Manuscr. Math., 71, 163-181, (1991) · Zbl 0726.34073 [4] Behncke, H, The m-function for Hamiltonians with Wigner-von Neumann potentials, J. Math. Phys., 35, 1445-1462, (1994) · Zbl 0799.34080 [5] Buslaev, VS; Matveev, VB, Wave operators for the Schrödinger equation with a slowly decreasing potential, Theoret. Math. Phys., 2, 266-274, (1970) [6] Buslaev, VS; Skriganov, MM, Coordinate asymptotic behavior of the solution of the scattering problem for the Schrödinger equation, Theoret. Math. Phys., 19, 465-476, (1974) · Zbl 0296.35023 [7] Capasso, F; Sirtori, C; Faist, J; Sivco, DL; Chu, SNG; Cho, AY, Observation of an electronic bound state above a potential well, Nature, 358, 565-567, (1992) [8] Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955) · Zbl 0064.33002 [9] Cruz-Sampedro, J; Herbst, I; Martínez-Avendaño, R, Perturbations of the Wigner-von Neumann potential leaving the embedded eigenvalue fixed, Ann. Henri Poincaré, 3, 331-345, (2002) · Zbl 1021.81016 [10] Damanik, D; Simon, B, Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics, Invent. Math., 165, 1-50, (2006) · Zbl 1122.47029 [11] Eastham, M.S.P.: The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, Edinburgh (1973) · Zbl 0287.34016 [12] Gilbert, DJ; Pearson, DB, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl., 128, 30-56, (1987) · Zbl 0666.34023 [13] Harris, WA; Lutz, DA, Asymptotic integration of adiabatic oscillators, J. Math. Anal. Appl., 51, 76-93, (1975) · Zbl 0315.34070 [14] Hinton, DB; Klaus, M; Shaw, JK, Embedded half-bound states for potentials of Wigner-von Neumann type, Proc. Lond. Math. Soc., 3, 607-646, (1991) · Zbl 0689.34018 [15] Janas, J; Simonov, S, Weyl-titchmarsh type formula for discrete Schrödinger operator with Wigner-von Neumann potential, Stud. Math., 201, 167-189, (2010) · Zbl 1205.47029 [16] Jitomirskaya, SY; Last, Y, Dimensional Hausdorff properties of singular continuous spectra, Phys. Rev. Lett., 76, 1765-1769, (1996) · Zbl 0935.81018 [17] Klaus, M, Asymptotic behavior of Jost functions near resonance points for Wigner-von Neumann type potentials, J. Math. Phys., 32, 163-174, (1991) · Zbl 0772.35057 [18] Kodaira, K, The eigenvalue problem for ordinary differential equations of the second order and heisenberg’s theory of $$S$$-matrices, Am. J. Math., 71, 921-945, (1949) · Zbl 0035.27101 [19] Kreimer, Y; Last, Y; Simon, B, Monotone Jacobi parameters and non-Szegő weights, J. Approx. Theory, 157, 144-171, (2009) · Zbl 1185.42027 [20] Kurasov, P, Zero-range potentials with internal structures and the inverse scattering problem, Lett. Math. Phys., 25, 287-297, (1992) · Zbl 0761.35075 [21] Kurasov, P, Scattering matrices with finite phase shift and the inverse scattering problem, Inverse Probl., 12, 295-307, (1996) · Zbl 0848.34073 [22] Kurasov, P; Naboko, S, Wigner-von Neumann perturbations of a periodic potential: spectral singularities in bands, Math. Proc. Camb. Philos. Soc., 142, 161-183, (2007) · Zbl 1170.34355 [23] Kurasov, P; Simonov, S, Weyl-titchmarsh type formula for periodic Schrödinger operator with Wigner-von Neumann potential, Proc. R. Soc. Edinb. Sect. A, 143A, 401-425, (2013) · Zbl 1305.34149 [24] Levitan, B., Sargsyan, I.: Introduction to Spectral Theory. AMS, New York (1975) · Zbl 0225.47019 [25] Lukic, M, Orthogonal polynomials with recursion coefficients of generalized bounded variation, Commun. Math. Phys., 306, 485-509, (2011) · Zbl 1237.42017 [26] Lukic, M.: On higher-order Szegő theorems with a single critical point of arbitrary order. Constr. Approx. doi:10.1007/s00365-015-9320-4 · Zbl 1353.47061 [27] Lukic, M; Ong, D, Wigner-von Neumann type perturbations of periodic Schrödinger operators, Trans. Am. Math. Soc., 367, 707-724, (2015) · Zbl 1308.35072 [28] Matveev, VB, Wave operators and positive eigenvalues for a Schrödinger equation with oscillating potential, Theoret. Math. Phys., 15, 574-583, (1973) · Zbl 0278.47004 [29] Naboko, SN; Simonov, S, Zeroes of the spectral density of the periodic Schrödinger operator with Wigner-von Neumann potential, Math. Proc. Camb. Philos. Soc., 153, 33-58, (2012) · Zbl 1257.34072 [30] Nesterov, PN, Averaging method in the asymptotic integration problem for systems with oscillatory-decreasing coefficients, Differ. Equ., 43, 745-756, (2007) · Zbl 1156.34039 [31] Remling, C, Relationships between the $$m$$-function and subordinate solutions of second order differential operators, J. Math. Anal. Appl., 206, 352-363, (1997) · Zbl 0878.34070 [32] Simonov, S, Zeroes of the spectral density of the discrete Schrödinger operator with Wigner-von Neumann potential, Integral Equ. Oper. Theory, 73, 351-364, (2012) · Zbl 1283.47034 [33] Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-Order Differential Equations. Part II. Clarendon Press, Oxford (1946) · Zbl 0061.13505 [34] Neumann, J; Wigner, EP, Über merkwürdige diskrete eigenwerte, Z. Phys., 30, 465-467, (1929) · JFM 55.0520.04 [35] Wasow, W.: Asymptotic Expansions for Ordinary Differential Equations. Dover, New York (1965) · Zbl 0133.35301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.