Refining Eigenvalue Estimates for a String with a Singular Weight

We consider the linear differential pencil , where is the Sturm–Liouville operator with singular potential , is the operator of multiplication by a singular weight , , and is the spectral parameter. It is assumed that the potential and the weight belong to the Sobolev space with a negative smoothness exponent, is a real-valued function, and is a complex-valued function. We prove that the estimate , where is a constant independent of , holds for the eigenvalues , , of the pencil for all .