Oscillation Estimates of Eigenfunctions via the Combinatorics of Noncrossing Partitions
نویسندگان
چکیده
We study oscillations in the eigenfunctions for a fractional Schrödinger operator on the real line. An argument in the spirit of Courant’s nodal domain theorem applies to an associated local problem in the upper half plane and provides a bound on the number of nodal domains for the extensions of the eigenfunctions. Using the combinatorial properties of noncrossing partitions, we turn the nodal domain bound into an estimate for the number of sign changes in the eigenfunctions. We discuss applications in the periodic setting and the Steklov problem on planar domains.
منابع مشابه
Noncrossing partitions with fixed points
The noncrossing partitions with each of their blocks containing a given element are introduced and studied. The enumeration of these partitions is described through a polynomial of several variables which is proved to satisfy a recursive formula. It is shown that each variable increased by one is a factor of this polynomial.
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