A sharp lower bound for some Neumann eigenvalues of the Hermite operator

This paper deals with the Neumann eigenvalue problem for the Hermite operator defined in a convex, possibly unbounded, planar domain $Ω$, having one axis of symmetry passing through the origin. We prove a sharp lower bound for the first eigenvalue $μ_1^{odd}(Ω)$ with an associated eigenfunction odd with respect to the axis of symmetry. Such an estimate involves the first eigenvalue of the corresponding one-dimensional problem. As an immediate consequence, in the class of domains for which $μ_1(Ω)=μ_1^{odd}(Ω)$, we get an explicit lower bound for the difference between $μ(Ω)$ and the first Neumann eigenvalue of any strip.

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