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In
mathematics, in the field of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
s, a nontrivial solution to an
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
:
of associated boundary value problems.
Examples
The differential equation
:
is oscillating as sin(''x'') is a solution.
Connection with spectral theory
Oscillation theory was initiated by
Jacques Charles François Sturm Jacques Charles François Sturm (29 September 1803 – 15 December 1855) was a French mathematician.
Life and work
Sturm was born in Geneva (then part of France) in 1803. The family of his father, Jean-Henri Sturm, had emigrated from Strasbourg ...
in his investigations of
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 Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
the question about oscillation/non-oscillation answers the question whether the eigenvalues accumulate at the bottom of the continuous spectrum.
Relative oscillation theory
In 1996
Gesztesy–
Simon–
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.
See also
Classical results in oscillation theory are:
*
Kneser's theorem (differential equations) In mathematics, the Kneser theorem can refer to two distinct theorems in the field of ordinary differential equations:
* the first one, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not;
* ...
*
Sturm–Picone comparison theorem In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-osci ...
*
Sturm separation theorem
References
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Ordinary differential equations
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