Distribution of the spectrum of a quadratic pencil
The problem \(Ly=-y''+q(x)y+i\lambda \alpha (x)y'+\lambda^ 2y=0\) \(- \infty <x<\infty,\) where \(\lambda\) is a spectral parameter and \(y=y(x)\), \(| \alpha (x)| \leq 2\) and q(x) is a continuous regular function \((0<q(x)\to \infty\) as \(| x| \to \infty)\) is investigated. The method of investigating is analytical and is based on estimate of the kernel of the resolvent of the operator L(\(\lambda\)) and application of Tauberian theorems. As a result of the asymptotic behavior of the function N(\(\lambda\)), the distribution of eigenvalues is received.
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