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Some geometric bounds on eigenvalues of elliptic PDEs. Evans Harrell G eorgia T ech www.math.gatech.edu/~harrell. Spectral Theory Network 25 July, 2004 Cardiff.
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Some geometric bounds on eigenvalues of elliptic PDEs Evans Harrell Georgia Tech www.math.gatech.edu/~harrell Spectral Theory Network 25 July, 2004 Cardiff
• Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet conditions in part, preprint, 2004. • Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators, preprint, 2003.
Derived from one-dimensional “Hardy inequality” • Related inequalities of “Barta” form:
On a (hyper) surface,what object is most likethe Laplacian? ( = the good old flat scalar Laplacian of Laplace)
Answer #1 (Beltrami’s answer): Consider only tangential variations. The Laplace-Beltrami operator is an intrinsic object, and as such is unaware that the surface is immersed.
Answer #2 (The nanoanswer): Perform a singular limit and renormalization to attain the surface as the limit of a thin domain: - + q,
Some other answers • In other physical situations, such as reaction-diffusion, q(x) may be other quadratic expressions in the curvature, usually q(x) ≤ 0. • The conformal answer: q(x) is a multiple of the scalar curvature.
Heisenberg's Answer(if he had thought about it) Note: q(x) ≥ 0 !
Gap bounds for (hyper) surfaces Here h is the sum of the principal curvatures.
Partition function Z(t) := tr(exp(-tH)).
