2018/02/08 by Jean‐François Bony, Bony, Jean-Francois, Nicolas Popoff +1 · 2 citations
Computer Science · Mathematics · #35J10 #35P20 #81Q10 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1802.02882
openalex publication_date 2018/02/08 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
In this article, we consider the semiclassical Schr "odinger operator P = -\nh2 \Δ + V in \ℝd with confining non-negative potential V\nwhich vanishes, and study its low-lying eigenvalues \λk ( P ) as h\n\→ 0. First, we give a necessary and sufficient criterion upon V-1 ( 0 )\nfor \λ1 ( P ) h- 2 to be bounded. When d = 1 and V-1 ( 0 ) =\n 0 , we are able to control the eigenvalues \λk ( P ) for\nmonotonous potentials by a quantity linked to an interval Ih, determined\nby an implicit relation involving V and h. Next, we consider the case where\nV has a flat minimum, in the sense that it vanishes to infinite order. We\ngive the asymptotic of the eigenvalues: they behave as the eigenvalues of the\nDirichlet Laplacian on Ih. Our analysis includes an asymptotic of the\nassociated eigenvectors and extends in particular cases to higher dimensions.\n