In "Béthuel, F. und J. C. Saut: Travelling waves for the Gross-Pitaevskii equation. I. Ann. Inst. H. Poincaré Phys. Théor., 70(2):147–238, 1999." the authors write on page 150, that one can easily check the operator
$L_0 = \left( \begin{array} -\triangle +2 & c \partial_{x_1} \\ - c \partial_{x_1} & - \triangle \end{array} \right)$
viewed as an unbounded operator in $L^2(\mathbb{R}^2)^2$ has continuous spectrum $[0, \infty)$ if $c < \sqrt{2}$ and $[-\frac{(c^2-2)^2}{4}, \infty)$ if $c > \sqrt{2}$.
How does one easily start with this computation? Is there a standard way to compute the continuous spectrum of an unbounded operator?