On the discrete spectrum of a boundary value problem (Q757718)

Some sufficient conditions for the boundedness of the spectrum and for the absence of the discrete spectrum of the problem \(y''+(\lambda^ 2- 2i\lambda Q(x)-U(x))y=0,\quad y(\lambda,0)=0\) are obtained. Such a problem arises in the wave propagation in inhomogeneous media. The common assumptions are: 1) U(x) is real and continuous on the semi-axis (0,\(\infty)\), \(\int^{\infty}_{0}x| U(x)| dx<\infty\); 2) Q(x) is real and continuously differentiable on the semi-axis \([0,\infty)\), \(\int^{\infty}_{0}| Q(x)| dx<\infty;\quad \int^{\infty}_{0}x| Q'(x)| dx<\infty;\) 3) Q(x)\(\leq 0\). Typical results are the following: Theorem. Let the above conditions be satisfied and suppose that there exist numbers \(\kappa >0\), \(\delta\geq 0\), \(C>0\), such that in the interval \(x\in [\delta,\infty)\) the following inequality holds: \(| U(x)-Q^ 2(x)+Q'(x)| <Ce^{-\kappa x}\). Then the number of eigenvalues for the considered problem is finite. Theorem. If in addition to conditions 1-3 the inequality \[ \int^{\infty}_{0}x| U(x)-Q^ 2(x)+Q'(x)| dx \exp \int^{\infty}_{0}(3| Q(x)| +x| U(x)|)dx<1 \] holds, then the spectrum of the considered problem is continuous.