Oscillation estimates of eigenfunctions via the combinatorics of noncrossing partitions

Oscillation estimates of eigenfunctions via the combinatorics of noncrossing partitions, Discrete Analysis 2017:13, 20 pp.

An important phenomenon that often occurs with linear operators is that the complexity of their eigenfunctions is closely related to the size of their eigenvalues. To give a particularly simple example, the differential operator $f\mapsto f''$, defined on the space of smooth functions on the circle, has trigonometric functions as its eigenfunctions. The eigenvalue associated with a function of the form $A\cos (n\theta)+B\sin (n\theta)$ is $-n^2$, while the function itself oscillates $n$ times as it goes round the circle, so as the size of the eigenvalue increases, the function becomes more complex, in the sense that it oscillates more. A nodal domain of an eigenfunction $f$ defined on a manifold is a connected component where $f$ does not change sign. A fundamental theorem of Courant, his nodal domain theorem, states that the $n$th eigenfunction of a Laplacian (under suitable conditions) has at most $n$ nodal domains.

This paper concerns oscillations of the eigenfunctions for a fractional Schrödinger operator on the real line. The authors analyse a related problem concerning functions defined on the upper half plane, where they prove a result similar to the nodal domain theorem. In order to use this to obtain estimates for the number of oscillations of the fractional Schrödinger operator, they use combinatorial arguments in a surprising way.

A noncrossing partition is a partition of a totally ordered set $X$ with the property that if $a < b < c < d$, then it is not possible for $a\sim c$ and $b\sim d$, where $\sim$ is the equivalence relation corresponding to the partition. The name is due to the fact that if the points of $X$ are drawn in order on a circle, then a partition is noncrossing if and only if it is possible to decompose the circle into nonoverlapping connected regions such that each region contains the points from one cell of the partition on its boundary. (Equivalently, but slightly less naturally, the convex hulls of the equivalence classes are disjoint.) Noncrossing partitions were introduced by Kreweras in 1972 and have had applications in a wide variety of areas, to which this paper adds another.

The one-dimensional Schrödinger operator is the operator $-\frac {d^2}{dx^2} + V(x)$, where $V$ is a suitable potential. A fractional Schrödinger operator is an operator of the form $(-\frac{d^2}{dx^2})^{\alpha/2}+V(x)$, where $0<\alpha<2$ and one makes sense of the fractional Laplacian in the usual way: convert it into a multiplier by taking Fourier transforms, raise the multiplier to the power $\alpha/2$, and take the inverse Fourier transform again. There are many reasons to be interested in the behaviour of its eigenfunctions. The motivation for the authors is to understand better the behaviour of travelling waves for equations where the fractional Laplacian models dispersion.

^{_{Article image by Dmitry Belayev}}