Spectrum of the Sturm–Liouville operator on a curve with parameters in the boundary conditions and discontinuity conditions for solutions

For large values of the modulus of the spectral parameter, we obtain and analyze the asymptotics of solutions of the standard Sturm–Liouville equation with a piecewise integer potential on a general rectifiable curve lying in the complex plane and having a finite number of points at which the solutions and/or their derivatives have discontinuities polynomially depending on the spectral parameter. For decaying boundary conditions that also depend on the spectral parameter polynomially, we examine the spectrum of the corresponding Sturm–Liouville operator.