2025/01/18 by Kristiansen, Kristian Uldall, Szmolyan, Peter
#Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2501.10707
In this paper, we revisit the eigenvalue problem of the one-dimensional Schrödinger equation for smooth single well potentials. In particular, we provide a new interpretation of the Bohr-Sommerfeld quantization formula. A novel aspect of our results, which are based on recent work of the authors on the turning point problem based upon dynamical systems methods, is that we cover all eigenvalues E∈ [0,\mathcal O(1)] and show that the Bohr-Sommerfeld quantitization formula approximates all of these eigenvalues (in a sense that is made precise). At the same time, we provide rigorous smoothness statements of the eigenvalues as functions of ε. We find that whereas the small eigenvalues E=\mathcal O(ε) are smooth functions of ε, the large ones E=\mathcal O(1) are smooth functions of nε∈[c1,c2], 0