Timeline for Positivity of Second-Order Elliptic Differential Operators

6 events

when toggle format	what		by	license	comment
Jun 23, 2011 at 23:01	vote	accept	Viktor Bundle
Jun 22, 2011 at 1:49	comment	added	Terry Tao		In the asymptotic limit $c \to \infty$, semiclassical heuristics predict that the number of negative eigenvalues should approximately equal (up to some constant factor involving $2\pi$) the volume of $\{ (x,p) \in T M: \|p\|^2 + ch < 0 \}$ in the tangent bundle (or cotangent bundle, if one prefers). This should be rigorously provable by semiclassical analysis as soon as $h$ and $M$ are smooth enough. In the other direction, for potentials with sufficiently small (in $L^{n/2}$ norm) negative part, there should be positive semi-definiteness from the Sobolev inequality (at least for $n>2$).
Jun 21, 2011 at 4:16	comment	added	Noam D. Elkies		More generally, for each $n$ there exists $c_n$ such that $\Delta + ch$ has at least $n$ negative eigenvalues once $c > c_n$. Proof: let $V$ be an $n$-dimensional space of smooth functions supported on the same neighborhood. Then the quadratic form $⟨f,hf⟩$ is negative-definite on $V$, whence $⟨f,(Δ+ch)f⟩$ is also negative-definite for $c$ large enough, etc.
Jun 21, 2011 at 0:00	history	edited	Noam D. Elkies	CC BY-SA 3.0	Fix mispelling (where --> were)
Jun 20, 2011 at 23:29	vote	accept	Viktor Bundle
Jun 23, 2011 at 22:59
Jun 20, 2011 at 23:05	history	answered	Noam D. Elkies	CC BY-SA 3.0