# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Effective Prüfer angles and relative oscillation criteria. (English) Zbl 1167.34009
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.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34L05 General spectral theory for OD operators
##### Keywords:
Sturm-Liouville operators; oscillation theory
Full Text:
##### References:
 [1] Babikov, V. V.: The method of phase functions in quantum mechanics, (1988) [2] Calogero, F.: Variable phase approach to potential scattering, (1967) · Zbl 0193.57501 [3] Eastham, M. S. P.: The spectral theory of periodic differential equations, (1973) · Zbl 0287.34016 [4] Gesztesy, F.; Ünal, M.: Perturbative oscillation criteria and Hardy-type inequalities, Math. nachr. 189, 121-144 (1998) · Zbl 0903.34030 · doi:10.1002/mana.19981890108 [5] Hartman, P.: On the linear logarithmic-exponential differential equation of the second-order, Amer. J. Math. 70, 764-779 (1948) · Zbl 0035.18302 · doi:10.2307/2372211 [6] Hartman, P.: Ordinary differential equations, (2002) · Zbl 1009.34001 [7] Hille, E.: Nonoscillation theorems, Trans. amer. Math. soc. 64, 234-252 (1948) · Zbl 0031.35402 · doi:10.2307/1990500 [8] Khrabustovskii, V. I.: The perturbation of the spectrum of selfadjoint differential operators with periodic matrix-valued coefficients, , 117-138 (1973) [9] Khrabustovskii, V. I.: The perturbation of the spectrum of selfadjoint differential operators of arbitrary order with periodic matrix coefficients, , 123-140 (1974) [10] Khrabustovskii, V. I.: The discrete spectrum of perturbed differential operators of arbitrary order with periodic matrix coefficients, Math. notes 21, No. 5 -- 6, 467-472 (1977) · Zbl 0399.47042 · doi:10.1007/BF01410176 [11] Kneser, A.: Untersuchungen über die reellen nullstellen der integrale linearer differentialgleichungen, Math. ann. 42, 409-435 (1893) · Zbl 25.0522.01 · http://jfm.sub.uni-goettingen.de/JFM/digit.php?an=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 · Zbl 1186.47009 · doi:10.1007/s00220-008-0600-8 [13] Krüger, H.; Teschl, G.: Relative oscillation theory for Sturm -- Liouville operators extended, J. funct. Anal. 254, No. 6, 1702-1720 (2008) · Zbl 1144.34014 · doi:10.1016/j.jfa.2007.10.007 [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) · 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) · Zbl 0439.34024 [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 · Zbl 0595.47009 [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) · 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 · doi:10.1090/S0002-9939-99-05069-8 [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 · doi:10.1007/s002200050822 [22] Swanson, C. A.: Comparison and oscillation theory of linear differential equations, (1968) · Zbl 0191.09904 [23] Weber, H.: Die partiellen differential-gleichungen der mathematischen physik, vol. 2, (1912) · Zbl 43.0438.12