Bounds on scattering poles in one dimension (Q1969059)

In this article, the number of resonances (i.e., poles of the transmission coefficient in the open lower \(k\)-half-plane) of the 1D Schrödinger equation with a real potential \(p(x)\) such that \(\int_{-\infty}^\infty e^{a|x|}p(x) dx<\infty\) for any \(a>1\), is studied in detail. In particular, it is proved that their number \(N_p(r)\) with \(|k|\leq r\) has the upper bound \[ N_p(a/2)\leq C(p)\left[1+\iint e^{2a|x-y|}|p(x)p(y)|dx dy\right],\quad a\geq 1, \] where the constant \(C(p)\) depends on \(\|p\|_{L^1({\mathbb{R}})}\), but not on \(a\). Further, for nonnegative potentials of this type sufficient conditions for having resonance free strips in the lower half-plane and a resonance free imaginary line are given.