On the location of resonances of exponentially decaying Sturm-Liouville potentials (Q5957974)

The authors investigate the location of the resonances in the unphysical sheet for the following problem NEWLINE\[NEWLINE y''(x)+(\lambda^2-q(x))y=0, \quad 0\leq x\leq 0,\qquad y(0)=0, NEWLINE\]NEWLINE where the real-valued potential \(q\) decays exponentially: NEWLINE\[NEWLINE q(x)= O(e^{-\alpha x}),\;\;x\to\infty, \;\;\alpha>0. NEWLINE\]NEWLINE It is well known that in this case the Jost function is analytic in the half-plane \(\text{ Im}\lambda>-{\alpha\over 2}\). The zeros of the Jost function in the strip \(0>\text{Im }\lambda>-{\alpha\over 2}\) are said to be the resonances in the unphysical sheet. The authors set NEWLINE\[NEWLINE q(x)=e^{-\alpha x}p(x), NEWLINE\]NEWLINE where \(p(x)\) is \(2\pi\)-periodic with complex Fourier coefficients \(c_{\nu}\). They prove that if NEWLINE\[NEWLINE\sum_{-\infty}^{\infty}|c_{\nu}|<2\alpha\delta\log 2,\quad \delta>0, NEWLINE\]NEWLINE then the Jost function has no zeros in the complement of \(\sigma(\delta)\), where \(\sigma(\delta)\) stands for the open \(\delta\)-neighbourhood of the set \(\{-i{1\over 2}n\alpha-{1\over 2}\nu\}_{n\in \mathbb{N}}^{\nu=0,\pm 1\pm 2\dots}\). The case of the Bessel equation (\(p(x)\equiv 1\)) is investigated in detail.