High-frequency limit: Difference between revisions
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The '''high-frequency limit''' of an equation concerns the behaviour of the equation for solutions which have very high frequency <math>|\xi| \to \infty</math>. Typically one rescales the equation in order to normalize the solutions to have frequency comparable to one. In that case, the equation acquires a rescaling parameter <math>\hbar</math>, and the limit is usually the same as the [[semi-classical limit]]. The high-frequency limit is also related to the [[propagation of singularities]] for such equations, as singularities have infinite frequency. | The '''high-frequency limit''' of an equation concerns the behaviour of the equation for solutions which have very high frequency <math>|\xi| \to \infty</math>. Typically one rescales the equation in order to normalize the solutions to have frequency comparable to one. In that case, the equation acquires a rescaling parameter <math>\hbar</math>, and the limit is usually the same as the [[semi-classical limit]]. The high-frequency limit is also related to the [[propagation of singularities]] for such equations, as singularities have infinite frequency. | ||
Typically, the effects of lower order terms such as mass, potential terms, and magnetic terms disappear in the high-frequency limit, unless they are amplified in tandem with the frequency parameter. | Typically, the effects of lower order terms such as mass, potential terms, and magnetic terms disappear in the high-frequency limit, unless they are amplified in tandem with the frequency parameter. | ||
The opposite of the high-frequency limit is the [[low-frequency limit]]. | |||
[[Category:limits]] | [[Category:limits]] | ||
Latest revision as of 20:02, 5 August 2006
The high-frequency limit of an equation concerns the behaviour of the equation for solutions which have very high frequency . Typically one rescales the equation in order to normalize the solutions to have frequency comparable to one. In that case, the equation acquires a rescaling parameter , and the limit is usually the same as the semi-classical limit. The high-frequency limit is also related to the propagation of singularities for such equations, as singularities have infinite frequency.
Typically, the effects of lower order terms such as mass, potential terms, and magnetic terms disappear in the high-frequency limit, unless they are amplified in tandem with the frequency parameter.
The opposite of the high-frequency limit is the low-frequency limit.
