On computing first eigenvalues of the Sturm-Liouville operator (Q1374975)

The authors introduce a \(\varphi (x,\lambda)\) computational technique for the asymptotic expansion of the first few eigenvalues of the Sturm-Liouville \((S-L)\) operator \[ L(y)= -q''+ qy= \lambda y; \] \(y'(\pi)+ Hy(\pi) =0\), \(y'(0)- hy(0)=0\), by using a system of regularized traces in the manner proposed by \textit{I. M. Gel'fand} and \textit{L. A. Dikij} in the middle 1950-s. See the application of similar scheme for the Mathieu equation [\textit{L. A. Dikij}, Dokl. Akad. Nauk SSSR 116, 12-14 (1957; Zbl 0091.29203)]. A system of regularized traces is an infinite system \(\Sigma_n (\lambda_n^k- A_n(k)= B(k)\), where \(A_n(k)\) is a segment of an asymptotic and absolutely convergent expansion of \(\lambda_n^k\) in powers of \(n\). However S. A. Shkarin proved that in general the Gel'fand-Dikij technique cannot be justified, and in particular cannot be applied to the first eigenvalue. The reviewer comments that the last objection is not as serious as it appears at first, since excellent numerical approximations to the first eigenvalue can be found by combining the Rayleigh-Galerkin approximations for an upper bound, combined with Weinstein approximations for a lower bound. However, the Gel'fand-Dikij technique and the authors' improvements are very important. The authors proceed as follows. The \(S-L\) operator is said to belong to the class \(S\) if the solution \(\varphi (x,\lambda)\) to the Cauchy problem \(\varphi (0,\lambda) =1\), \(\varphi'(0, \lambda) =h\), has as an asymptotic expansion as \(\lambda\to\infty\): \[ \begin{multlined} \varphi(x, \lambda) \sim\cos (x,\lambda^{1/2}) +k_i(x) \sin(x, \lambda^{1/2})/ \lambda^{1/2} +k_2(x) \cos(x, \lambda^{1/2})/ \lambda+ \cdots+ \\ +k_{2n-1} (x) \sin (x,\lambda^{1/2})/ \lambda^{(2n-1)/2} +k_{2n} (x) \cos (x,\lambda^{1/2})/ \lambda^n+ \cdots, \end{multlined} \] and such that only a finite number of coefficients \(k_j(x)\) are not identically zero on \([0,\pi]\). The following theorems are proved: Theorem 1: Operators of class \(S\) are dense in the set of \(S-L\) operators with \(L_2 [0,\pi]\) potential. Theorem 2 states that the spectrum of an operator of class \(S\) with a complete system of regularized traces completely determines the spectrum of corresponding the \(S-L\) operator. Moreover, for \(A_n(k)\), \(k=1,2, \dots,K\), by taking sufficiently many terms \(n=1,2, \dots,N\), one can approximate \(B(k)\) arbitrarily close by the series \(\Sigma_n (\lambda_n^k -A_n(k))\). (Giving an \(\varepsilon >0\), there exists \(K\) and \(N\), etc\(\dots)\) The proof is short but clever.