A class of Sturm–Liouville operators and approximate calculation of the first eigenvalues

A special class $S$ of Sturm–Liouville operators with simple asymptotic properties of eigenfunctions is investigated. The analytic properties of the potentials are analyzed and the operators in this class are described in terms of the transition function of the inverse problem. The following result is established: the class $S$ is dense in the set of Sturm–Liouville operators with potentials in $L_2$. A subset of $S$ that also has the density property is effectively distinguished. Based on the properties of the operators in this subset, a method of the approximate evaluation of the first eigenvalues of a Sturm–Liouville operator through its regularized traces is proposed and substantiated.