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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, in the field of
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s, a nontrivial solution to an
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
:F(x,y,y',\ \dots,\ y^)=y^ \quad x \in
root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
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 A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
of associated boundary value problems.


Examples

The differential equation :y'' + y = 0 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, who made a significant addition to equation theory with his work, Sturm's theorem. Early life Sturm was born in Geneva, France in 1803. The fam ...
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 partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
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) * Sturm–Picone comparison theorem * Sturm separation theorem


References

* * * * * Reprinted in * * * Ordinary differential equations {{mathanalysis-stub