Effective Prüfer angles and relative oscillation criteria. (English) Zbl 1167.34009

Krüger, Helge; Teschl, Gerald

J. Differ. Equations 245, No. 12, 3823-3848 (2008).

Authors’ abstract: Summary: We present a streamlined approach to relative oscillation criteria based on effective Prüfer angles adapted to the use at the edges of the essential spectrum.
Based on this we provide a new scale of oscillation criteria for general Sturm-Liouville operators which answer the question whether a perturbation inserts a finite or an infinite number of eigenvalues into an essential spectral gap. As a special case we recover and generalize the Gesztesy-Ünal criterion (which works below the spectrum and contains classical criteria by Kneser, Hartman, Hille, and Weber) and the well-known results by Rofe-Beketov including the extensions by Schmidt.

Reviewer: Qingkai Kong (DeKalb)

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34L05	General spectral theory of ordinary differential operators

Keywords:

Sturm-Liouville operators; oscillation theory

References:

[1]	Babikov, V.V., The method of phase functions in quantum mechanics, (1988), Nauka Moscow
[2]	Calogero, F., Variable phase approach to potential scattering, (1967), Academic Press New York · Zbl 0193.57501
[3]	Eastham, M.S.P., The spectral theory of periodic differential equations, (1973), Scottish Academic Press Edinburgh · Zbl 0287.34016
[4]	Gesztesy, F.; Ünal, M., Perturbative oscillation criteria and Hardy-type inequalities, Math. nachr., 189, 121-144, (1998) · Zbl 0903.34030
[5]	Hartman, P., On the linear logarithmic-exponential differential equation of the second-order, Amer. J. math., 70, 764-779, (1948) · Zbl 0035.18302
[6]	Hartman, P., Ordinary differential equations, (2002), SIAM Philadelphia · Zbl 0125.32102
[7]	Hille, E., Nonoscillation theorems, Trans. amer. math. soc., 64, 234-252, (1948) · Zbl 0031.35402
[8]	Khrabustovskii, V.I., The perturbation of the spectrum of selfadjoint differential operators with periodic matrix-valued coefficients, (), 117-138, (in Russian)
[9]	Khrabustovskii, V.I., The perturbation of the spectrum of selfadjoint differential operators of arbitrary order with periodic matrix coefficients, (), 123-140, (in Russian)
[10]	Khrabustovskii, V.I., The discrete spectrum of perturbed differential operators of arbitrary order with periodic matrix coefficients, Math. notes, 21, 5-6, 467-472, (1977) · Zbl 0399.47042
[11]	Kneser, A., Untersuchungen über die reellen nullstellen der integrale linearer differentialgleichungen, Math. ann., 42, 409-435, (1893) · JFM 25.0522.01
[12]	H. Krüger, G. Teschl, Relative oscillation theory, zeros of the Wronskian, and the spectral shift function, Comm. Math. Phys., in press
[13]	Krüger, H.; Teschl, G., Relative oscillation theory for sturm – liouville operators extended, J. funct. anal., 254, 6, 1702-1720, (2008) · Zbl 1144.34014
[14]	Rofe-Beketov, F.S., A test for the finiteness of the number of discrete levels introduced into gaps of a continuous spectrum by perturbations of a periodic potential, Soviet math. dokl., 5, 689-692, (1964) · Zbl 0117.06004
[15]	Rofe-Beketov, F.S., Spectral analysis of the Hill operator and its perturbations, Funkcional. anal., 9, 144-155, (1977), (in Russian) · Zbl 0449.34013
[16]	F.S. Rofe-Beketov, A generalisation of the Prüfer transformation and the discrete spectrum in gaps of the continuous one, in: Spectral Theory of Operators, Baku, Elm, 1979, pp. 146-153 (in Russian)
[17]	F.S. Rofe-Beketov, Spectrum perturbations, the Kneser-type constants and the effective masses of zones-type potentials, in: Constructive Theory of Functions ’84, Sofia, 1984, pp. 757-766
[18]	Rofe-Beketov, F.S., Kneser constants and effective masses for band potentials, Soviet phys. dokl., 29, 391-393, (1984) · Zbl 0597.34016
[19]	Rofe-Beketov, F.S.; Kholkin, A.M., Spectral analysis of differential operators. interplay between spectral and oscillatory properties, (2005), World Scientific Hackensack · Zbl 1090.47030
[20]	Schmidt, K.M., Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrödinger operators in the plane, Proc. amer. math. soc., 127, 2367-2374, (1999) · Zbl 0918.34039
[21]	Schmidt, K.M., Critical coupling constants and eigenvalue asymptotics of perturbed periodic sturm – liouville operators, Comm. math. phys., 211, 465-485, (2000) · Zbl 0953.34069
[22]	Swanson, C.A., Comparison and oscillation theory of linear differential equations, (1968), Academic Press New York · Zbl 0191.09904
[23]	Weber, H., Die partiellen differential-gleichungen der mathematischen physik, vol. 2, (1912), Vieweg Braunschweig