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The Fundamental Gap

By Mark Ashbaugh


The second main problem to be focused on at the workshop is what we’ll refer to as the Fundamental Gap Problem, which is the problem of finding a sharp lower bound to the gap between the first two eigenvalues of a Schrödinger operator on a bounded convex domain with convex potential in terms of the diameter of the domain. We call the lowest eigenvalue gap, i.e., the gap between the first two eigenvalues, the fundamental gap. We shall describe two versions of this problem, one for the Laplacian alone and the other for Schrödinger operators. The second problem contains the first as a special case, but the first is of course of substantial interest in its own right, and may possibly be more tractable than the general problem. In the first problem one just considers the Laplacian − ∆ on a bounded convex domain Ω with Dirichlet boundary conditions. This problem has a purely discrete spectrum, {λi(Ω)} ∞ i=1, with ∞ as its only point of accumulation. Listed in increasing order, with multiplicities, we have 0 < λ1(Ω) < λ2(Ω) ≤ λ3(Ω) ≤ · · · → ∞. (1) The fundamental gap, Γ(Ω), is given simply a

Year: 2006
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