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Operators with singular continuous spectrum. II: Rank one operators. (English) Zbl 1055.47500
From the text: The main result is the following theorem: Let \(A\) be selfadjoint with cyclic vector \(\phi\) and corresponding rank-one projection \(P_\phi\); then the set of \(\lambda\) for which \(A_\lambda=A+\lambda P_\phi\) has no eigenvalue in the spectrum of \(A\) is a dense \(G_\delta\) in \(\mathbb R\). The authors also obtain a similar result for rank-one domain perturbations.
For Parts I and III–VII see Zbl 0851.47003, Zbl 0830.34074, Zbl 0908.47002, Zbl 0979.34063, Zbl 0846.47023, and Zbl 0848.34069.

47A10 Spectrum, resolvent
34L05 General spectral theory of ordinary differential operators
47A55 Perturbation theory of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Full Text: DOI
[1] Aronszajn, N.: On a problem of Weyl in the theory of singular Sturm-Liouville equations. Am. J. Math.79, 597–610 (1957) · Zbl 0079.10802 · doi:10.2307/2372564
[2] Carmona, R.: Exponential localization in one-dimensional disordered systems. Duke Math. J.49, 191–213 (1982) · Zbl 0491.60058 · doi:10.1215/S0012-7094-82-04913-4
[3] Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. Boston: Birkhäuser, 1990 · Zbl 0717.60074
[4] del Rio, R.: A forbidden set for embedded eigenvalues. To appear Proc. AMS · Zbl 0805.34075
[5] Donoghue, W.: On the perturbation of spectra. Comm. Pure Appl. Math.18, 559–579 (1965) · Zbl 0143.16403 · doi:10.1002/cpa.3160180402
[6] Goldsheid, I.: Asymptotics of the product of random matrices depending on a parameter. Soviet Math. Dokl.16, 1375–1379 (1975) · Zbl 0366.60037
[7] Goldsheid, I., Molchanov, S., Pastur, L.: A pure point spectrum of the stochastic onedimensional Schrödinger equation. Func. Anal. Appl.11, 1–10 (1977) · Zbl 0368.34015 · doi:10.1007/BF01135526
[8] Gordon, A.: On exceptional value of the boundary phase for the Schrödinger equation of a half-line. Russ. Math. Surv.47, 260–261 (1992) · doi:10.1070/RM1992v047n01ABEH000868
[9] Gordon, A.: Pure point spectrum under 1-parameter perturbations and instability of Anderson localization. Commun. Math. Phys., to appear · Zbl 0839.47002
[10] Kirsch, W., Molchanov, S., Pastur, L.: One dimensional Schrödinger operators with high potential barriers. Operator Theory: Advances and Applications, Vol. 57, Basel: Birkhäuser, 1992, pp. 163–170 · Zbl 0883.34078
[11] Kotani, S.: Lyapunov exponents and spectra for one-dimensional random Schrödinger operators. Contemp. Math.50, 277–286 (1986) · Zbl 0587.60054
[12] Simon, B.: Operators with singular continuous spectrum, I. General operators. Ann Math., to appear · Zbl 0851.47003
[13] Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Comm. Pure Appl. Math.39, 75–90 (1986) · Zbl 0609.47001 · doi:10.1002/cpa.3160390105
[14] Stolz, G.: Bounded solutions and absolute continuity of Sturm-Liouville operators, J. Math. Anal. Appl.169, 210–228 (1992) · Zbl 0785.34052 · doi:10.1016/0022-247X(92)90112-Q
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