On the spectra and eigenfunctions of the Schrödinger and Maxwell operators (Q1071179)

The eigenvalues and eigenfunctions to the operator \(-\Delta +B(x)\nabla +q(x),\) and also to the Maxwell operator for an inhomogeneous medium are studied. Essentially it is required that \(B: R^ n\to C^ n\), \(q: R^ n\to C\) are continuous with \(q=q_ 1+s(x)\nabla q_ 2\), \(B=B_ 1+s(x)\nabla B_ 2\), where \(s: R^ n\to R^ n\) is a given function with \(| s| =1\). It is shown that if \(| q_ j| +| B_ j| =o(1)\), \(r\to \infty\), then any eigenfunction belonging to eigenvalues \(\lambda\) with \(\lambda <0\) or Im \(\lambda\) \(\neq 0\) decays exponentially. Under the stricter condition \(| q_ j+| B_ j| =o(r^{- 1}),\) \(r\to \infty\), it is shown that there are no eigenvalues embedded in the continuous spectrum. For the Maxwell operator conditions are given ensuring exponential decay of eigenfunctions in the non-isentropic case. Also, conditions are given which guarantee the absence of eigenvalues in some special cases, which also include the isentropic case.