Heat traces and existence of scattering resonances for bounded potentials (Q332184)

The authors consider the Schrödinger operator NEWLINE\[NEWLINE P_V = -\triangle + V (x),\quad \eqno{(1)} NEWLINE\]NEWLINE where \(V\in L_c^{\infty}(\mathbb{R}^n; \mathbb{R})\) is a bounded, compactly supported, real valued potential and \(n\geq 3\) is an odd number, with the aim to answer the question about the quantity of scattering resonances.NEWLINENEWLINEScattering resonances are defined as poles of the meromorphic continuation of the resolvent for \(n\geq 3\) NEWLINE\[NEWLINER_V (\lambda) := (-\triangle + V - \lambda^2)^{-1}, \quad n \quad \text{odd}, \quad \eqno{(2)}NEWLINE\]NEWLINE from Im\(\lambda\gg 1\) to \(\lambda \in \mathbb{C}\).NEWLINENEWLINEIt is proved for \(n\geq 3\) that, in odd dimensions, any real valued, bounded potential with compact support has at least one scattering resonance that was known earlier only for sufficiently smooth potentials.