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Velichkov: Some new problems in spectral optimization
No full textAbstract We present some new problems in spectral optimization. The first one consists in determining the best domain for the Dirichlet energy (or for the first eigenvalue) of the metric Laplacian, and we consider in particular Riemannian or Finsler manifolds, Carnot-Carathéodory spaces, Gaussian spaces. The second one deals with the optimal shape of a graph when the minimization cost is of spectral type. The third one is the optimization problem for a Schrödinger potential in suitable classes
Spectral optimization problems for Schrödinger operators
Get PDFIn this chapter we consider Schrödinger operators of the form −∆+V(x) on the Sobolev space H_0^1(D), where D is an open subset of R^d. We are interested in finding optimal potentials for some suitable criteria; the optimization problems we deal with are then written as
min {F(V) : V∈V}
where F is a suitable cost functional and V is a suitable class of admissible potentials. For simplicity, we consider the case when D is bounded and V ≥ 0; under these conditions the resolvent operator of −∆ + V(x) is compact and the spectrum λ(V) of the Schrödinger operator is discrete and consists of an increasing sequence of positive eigenvalues
λ(V) = (λ_1(V), λ_2(V), ...)
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