Bound states for non-symmetric evolution Schrödinger potentials (Q2766241)

Let \((D^2+k^2)\varphi =u\varphi\) be the Schrödinger equation on the real line, where \(k\in\mathbb{C}\), the potential \(u\) and the wave-function \(\varphi\) being square-matrix-valued functions, with entries in the Schwartz class. The author proves that the bound states are localized similarly to the scalar and symmetric cases, but by the zeroes of an analytic matrix-valued function. This is done by simple calculations, by using the well-known theory of complex variables and ordinary differential equations, but without using the fact that the Wronskian of a pair of wavefunctions is constant.