Resonances of a \(\lambda\)-rational Sturm-Liouville problem (Q2731109)

Let \(\widetilde A_\vartheta:=\left(\begin{smallmatrix} A & \vartheta v\\ \vartheta v & u\end{smallmatrix}\right),\) and \(Ay:= -y''+py\) with the domain \(D(A)=\{y\mid y,\;y'\in AC[0,1]\), \(y''\in L^2\), \(y(0)=y(1)=0\}\), \(p,u\) and \(v\) are analytic functions on a simply connected neighborhood \(U\) of the interval \([0,1]\), with \(p,u\) and \(v\) real on \([0,1]\) and \(v(x)>0\), \(u'(x)>0\) for \(x\in [0,1]\), and that \(u\) is injective on \(U\).NEWLINENEWLINENEWLINEThe author shows that if \(\widetilde A_0\) has an eigenvalue that is embedded in the essential spectrum, then for \(\vartheta\neq 0\) this eigenvalue in general disappears, but the resolvent of \(\widetilde A_\vartheta\) has a pole on the unphysical sheet of the Riemann surface. The unphysical sheet arises from analytic continuation from the upper half-plane across the essential spectrum.NEWLINENEWLINENEWLINEFurthermore, the asymptotic behavior of this resonance pole for small \(\vartheta\) is investigated. The results are proved by considering the \(\lambda\)-rational Sturm-Liouville problem NEWLINE\[NEWLINEy''+\left( z-py+{v^2 \over u-z}\right)y=0, \quad y(0)=y(1)=0,NEWLINE\]NEWLINE and its Titchmarsh-Weyl coefficient.