On non-self-adjoint Sturm-Liouville operators with matrix potentials (Q2473724)

Consider the vectorial differential operator \(L_{t}\left( Q\right) \), which acts in \(L_{2}^{m}(0,1)\) and generated by \[ l(y)=-y''(x)+Q(x)y(x),\text{ for }0\leq x\leq 1 \] and subject to the boundary conditions \[ y'(1)=e^{it}y'(0)\text{ and }y(1)=e^{it}y(0). \] Here, \(Q\) is a complex valued \ \(m\times m\;\)matrix whose entries are in \(L(0,1).\) The author shows that the spectrum, which is discrete, can be split into \(m\) sequences \(\left\{ \lambda _{n},_{i}\right\} _{n\in \mathbb{Z}}\) for \(i=1,\dots,m\). Also for large \(k,\) the eigenvalues \(\lambda _{k,i\;}\;\) lie within \(O\left( \ln (k)/k\right) \) of the eigenvalues of the operator \(L_{t}(C)\), where \(C=\int_{0}^{1}Q(x)dx\) is the averaged matrix of \(Q\) over \((0,\;1).\)