Calculation of the regularized trace for the Sturm-Liouville problem with spectral parameter in the boundary conditions (Q1934206)

The authors consider the boundary value problem \(-y''+qy=\lambda y,\) \(P_{11}(\lambda)y(0)+P_{12}(\lambda)y'(0)=0,\) \(P_{21}(\lambda)y(\pi)+P_{22}(\lambda)y'(\pi)=0,\) where \(q\in W_2^1[0,\pi]\) is a real function, \(P_{ij},\) \(i,j=1,2\), are polynomials (in the variable \(\lambda\)) with real coefficients. The first regularized trace of this BVP is calculated. Four cases are distinguished according to mutual relations between the degrees of the polynomials \(P_{ij}\).