Uniform stability of the inverse Sturm-Liouville problem with polynomials in a boundary condition

This paper deals with the Sturm-Liouville problem with singular potential of the Sobolev space $W_2^{-1}$ and polynomials of the spectral parameter in a boundary condition. We prove the uniform boundedness and the uniform stability for the inverse spectral problem in the general non-self-adjoint case. It is remarkable that our stability estimates are valid for some cases with different degrees of the polynomials for two compared operators.