Oscillation Theory
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 contrast w ...
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 contrast w ...
:F(x,y,y',\ \dots,\ y^)=y^ \quad x \in
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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).html" "title="root_of_a_function.html" "title=",+\infty) is called oscillating if it has an infinite number of root of a function">root In vascular plants, the roots are the 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 below the sur ...
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 (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
 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. 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 the ...
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 Simon may refer to: People * Simon (given name), including a list of people and fictional characters with the given name Simon * Simon (surname), including a list of people with the surname Simon * Eugène Simon, French naturalist and the genus ...
– Teschl showed that the number of roots of the
Wronski determinant In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions. Definition The Wronskian o ...
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; * t ...
*
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-oscill ...
*
Sturm separation theorem In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous second order linear differential equations. Basic ...


References

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