Boundary conditions of Sturm-Liouville operators with mixed spectra. (English) Zbl 1056.34099
The authors consider Sturm-Liouville operators on the half-line generated by the differential expression \(l(y)\equiv -y''(x)+v(x)y(x)\), \(0\leq x<\infty\), and by the boundary condition \(y(0)\cos\alpha-y'(0)\sin\alpha=0\), \(\alpha\in[0,\pi)\). Spectral properties of these operators are studied. In particular, the authors study bounds on averages of spectral functions corresponding to these operators. As a consequence, constraints are obtained which imply the existence of a singular spectrum embedded in the a.c. spectrum for sets of boundary conditions with positive measure and potentials vanishing on an interval \([0,N]\). These constraints are related to estimates on the measure of sets where the spectral density is positive.
Reviewer: Victor S. Rykhlov (Saratov)
MSC:
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 34L05 | General spectral theory of ordinary differential operators |
References:
| [1] | del Rio, R.; Makarov, N.; Simon, B., Operators with singular continuous spectrum: II. Rank one operators, Commun. Math. Phys., 165, 59-67 (1994) · Zbl 1055.47500 |
| [2] | del Rio, R.; Simon, B., Point spectrum and mixed spectral types for rank one perturbations, Proc. Amer. Math. Soc., 129, 3593-3599 (1997) · Zbl 0888.47008 |
| [3] | del Rio, R.; Fuentes, S.; Poltoratski, A., Coexistence of spectra in rank-one perturbation problems, Bol. Soc. Mat. Mexicana (3), 8, 49-61 (2002) · Zbl 1059.47020 |
| [4] | del Rio, R.; Fuentes, S.; Poltoratski, A., Families of spectral measures with mixed types, Oper. Theory Adv. Appl., 132, 131-140 (2002) · Zbl 1043.34090 |
| [5] | Gesztesy, F.; Simon, B., A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure, Ann. of Math., 152, 593-643 (2000) · Zbl 0983.34013 |
| [6] | Gilbert, D. J.; Pearson, B., On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl., 128, 30-56 (1987) · Zbl 0666.34023 |
| [7] | Gordon, A., Pure point spectrum under 1-parameter perturbations and instability of Anderson localization, Commun. Math. Phys., 164, 489-505 (1994) · Zbl 0839.47002 |
| [8] | Remling, C., Embedded singular continuous spectrum for one-dimensional Schrödinger operators, Trans. Amer. Math. Soc., 351, 2479-2497 (1999) · Zbl 0918.34074 |
| [9] | C. Remling, Universal bounds on spectral measures of one-dimensional Schrödinger operators, available electronically in http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-394; C. Remling, Universal bounds on spectral measures of one-dimensional Schrödinger operators, available electronically in http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-394 |
| [10] | Simon, B., Spectral analysis of rank one perturbations and applications, CMR Proc. Lecture Notes, 8, 109-149 (1995) · Zbl 0824.47019 |
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