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| In [[mathematics]], in the field of [[ordinary differential equation]]s, a nontrivial solution to an ordinary differential equation
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| :<math>F(x,y,y',\ \dots,\ y^{(n-1)})=y^{(n)} \quad x \in [0,+\infty)</math>
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| is called '''oscillating''' if it has an infinite number of [[root of a function|root]]s, otherwise it is called '''non-oscillating'''. The differential equation is called '''oscillating''' if it has an oscillating solution. | |
| The number of roots carries also information on the [[Spectrum (functional analysis)|spectrum]] of associated [[boundary value problem]]s.
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| == Examples ==
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| The differential equation
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| :<math>y'' + y = 0\ </math>
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| is oscillating as sin(''x'') is a solution.
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| == Connection with spectral theory ==
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| Oscillation theory was initiated by [[Jacques Charles François Sturm]] in his investigations of [[Sturm–Liouville theory|Sturm–Liouville problems]] from 1836. There he showed that the n'th eigenfunction of a Sturm–Liouville problem has precisely n-1 roots. For the one-dimensional [[Schrödinger equation]] the question about oscillation/non-oscillation answers the question whether the eigenvalues accumulate at the bottom of the continuous spectrum.
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| == Relative oscillation theory ==
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| In 1996 [[Fritz Gesztesy|Gesztesy]]–[[Barry Simon|Simon]]–[[Gerald Teschl|Teschl]] showed that the number of roots of the [[Wronski determinant]] of two eigenfunctions of a Sturm–Liouville problem gives the number of eigenvalues between the corresponding eigenvalues. It was later on generalized by Krüger–Teschl to the case of two eigenfunctions of two different Sturm–Liouville problems. The investigation of the number of roots of the Wronski determinant of two solutions is known as relative oscillation theory.
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| == See also ==
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| Classical results in oscillation theory are:
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| * [[Kneser's theorem (differential equations)]]
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| * [[Sturm–Picone comparison theorem]]
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| * [[Sturm separation theorem]]
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| ==References==
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| *[[Frederick_Valentine_Atkinson|Atkinson, F.V.]] (1964). Discrete and Continuous Boundary Problems, Academic Press.
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| *Gesztesy, F.; Simon, B.; Teschl, G. (1996). Zeros of the Wronskian and renormalized oscillation theory, Am. J. Math. 118, 571–594.
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| *Kreith, K. (1973). Oscillation Theory, Lecture Notes in Mathematics 324, Springer.
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| *Krüger, H; Teschl G. (2009). Relative oscillation theory, weighted zeros of the Wronskian, and the spectral shift function, Commun. Math. Phys. 287, 613–640.
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| *Sturm, J.C.F. (1836). Memoire sur les equations diferentielles lineaires du second ordre, J. Math. Pures Appl. 1, 106–186.
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| *Swanson, C.A. (1968). Comparison and Oscillation Theory of Linear Differential Equations, Academic Press.
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| *{{cite book| last = Teschl| given = G.|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
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| *Weidmann, J. (1987). Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics 1258, Springer.
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| {{DEFAULTSORT:Oscillation (Differential Equation)}}
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| [[Category:Ordinary differential equations]]
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| {{mathanalysis-stub}}
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