Localization of eigenfunctions of a one-dimensional elliptic operator

Source abstract

The localization of vibrations is a widely observed, but little understood physical phenomenon. Roughly speaking, the effect of localization is a confinement of some eigenfunctions of an elliptic operator to a small portion of the original domain in the presence of irregularities of the boundary or of the coefficients of the underlying operator. Until recently, there have been essentially no mathematical results explaining such a behavior. In the present paper the authors establish an asymptotic formula for the localization of eigenfunctions of the elliptic operator L = − d d x A ( x ) d d x L=-\frac {d}{dx} A(x) \frac {d}{dx} associated to a piecewise constant function A A . Quite unexpectedly, this formula expresses the strength of localization purely as a function of f ( x 0 ) f(x_0) , the value of the corresponding eigenfunction at the discontinuity point of the coefficients.