Oscillation estimates of eigenfunctions via the combinatorics of noncrossing partitions

Vera Mikyoung Hur; Mathew A. Johnson; Jeremy L. Martin

doi:10.19086/da.2102

Oscillation estimates of eigenfunctions via the combinatorics of noncrossing partitions

Vera Mikyoung Hur, Mathew A. Johnson, Jeremy L. Martin

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English (US)
Pages (from-to)	1-20
Number of pages	20
Journal	Discrete Analysis
Volume	13
Issue number	2017
DOIs	https://doi.org/10.19086/da.2102
State	Published - 2017

Keywords

Eigenfunction
Fractional Schrödinger
Noncrossing partition
Oscillation

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology
Discrete Mathematics and Combinatorics

Online availability

10.19086/da.2102

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abstract = "We study oscillations in the eigenfunctions for a fractional Schr{\"o}dinger 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.",

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note = "\u2217Supported by the National Science Foundation under the grant CAREER DMS-1352597, an Alfred P. Sloan Research Fellowship, a Simons Fellowship in Mathematics, and the University of Illinois at Urbana-Champaign under the Arnold O. Beckman Research Award RB14100 and a Beckman Fellowship. \u2020Supported by the National Science Foundation under grants DMS-1614785 and DMS-1211183. \u2021Supported by Simons Foundation Collaboration Grant number 315347.",

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N2 - 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.

AB - 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.

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KW - Fractional Schrödinger

KW - Noncrossing partition

KW - Oscillation

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Oscillation estimates of eigenfunctions via the combinatorics of noncrossing partitions

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