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Concavity of solutions to Schrödinger equations

Published on:

5 March 2024

Primary Category:

Differential Geometry

Paper Authors:

Gabriel Khan,

Soumyajit Saha,

Malik Tuerkoen

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Key Details

Establishes log-concavity of principal eigenfunction under conformal connections

Applies this to obtain first fundamental gap bounds for horoconvex domains in H^2

Also provides convexity estimates for torsion problem on spheres

Uses continuity method and barrier functions to prove log-concavity

Studies how curvature affects concavity properties of PDE solutions

AI generated summary

Concavity of solutions to Schrödinger equations

This paper studies the concavity properties of solutions to Schrödinger equations with variable density and potential. It establishes log-concavity estimates for the principal eigenfunction, enabling new bounds on the fundamental spectral gap for various geometries. A key technique is the use of conformal connections to simplify computations. As an application, the paper partially resolves a conjecture on spectral gaps for horoconvex domains in hyperbolic geometry. It also derives a power convexity estimate for the torsion problem on spherical domains.

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