Estimates of Approximation Numbers and Completeness of Root Vectors of A Singular Operator Generated by the Linear Part of the Korteweg‐de Vries Operator

We investigate a linear operator associated with the linear part of the Korteweg‐de Vries operator on an unbounded domain with unbounded coefficients. The existence and compactness of the resolvent are established, along with separability results ensuring maximal regularity of solutions. Two‐sided estimates are derived for the distribution function of approximation numbers, characterizing the rate of finite‐dimensional approximation of the resolvent. The completeness of the system of root vectors is also proved, which plays a key role in the analysis of related nonlinear operators.