Coupling of the continuum and semiclassical limit | Colóquio do DFMA

Abstract: In this talk we analyze the semiclassical d-dimensional Schrödinger operator in the continuum −1/2∆ + λ2N V discretized on a mesh with spacing proportional to 1/N. The semi-classical parameter λN is chosen as λN = N1−γ, with γ ∈ (−1, 1), which ensures that N governs both the semiclassical and continuum limit simultaneously. We prove that all eigenvalues of the discrete operator converge to those of the continuum, as λN → ∞. Beyond this semi-classical domain, in the case of the harmonic oscillator, we further discuss the spectral asymptotics for γ ∈ R \ (−1, 1), thereby fully characterizing the eigenvalue behavior across all possible values of γ ∈ R. Joint work with L. Pettinari and M. Keller.

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