Asymptotics of the Sturm-Liouville boundary value problem with summable potential on an interval (Q1593889)

Let \(D^2+q\) be given on a compact interval \(I\) with suitable boundary conditions at the endpoints and \(r_n\) the difference of the square roots of the \(n\)th eigenvalue to this problem and the one with \(q=0\). If \(q\) has \(m-1\) absolutely continuous derivatives and \(q^{(m)}\) is in \(L^2(I)\), then it is well-known that \(r_n\) can be expressed in terms of sums of integrals of these derivatives and an error term which is \(O(n^{-(m+1)})\). In this note, it is indicated that under the sole assumption \(q\in L^1(I)\) there is a successive approximation to \(r_n\) such that the error after the \(m\)th iteration is again \(O(n^{-(m+1)})\). The essential tool is to write the solution to the matrix differential equation \(X'=AX\) as the exponential of what is called a ``new form of the Hausdorff series'' [\textit{V. A. Vinokurov}, Sov. Math., Dokl. 44, No. 1, 273-280 (1992); translation from Dokl. Akad. Nauk SSSR 319, No. 5, 1057-1064 (1991; Zbl 0781.46029)]. It is not made clear how this series differs from those given by Baker, Dynkin, Magnus and others in this extremely well cultivated field.