The computation of negative eigenvalues of singular Sturm-Liouville problems (Q2725336)

An accurate interpolation method is derived and tested for approximating the negative eigenvalues of the singular Sturm-Liouville problem NEWLINE\[NEWLINE-y''(ik, x)+ q(x) y(ik,x)= -k^2 y(ik, x),\quad 0\leq x<\infty,\tag{i}NEWLINE\]NEWLINE NEWLINE\[NEWLINEB(ik)= h_1 y(ik,0)+ h_2 y'(ik, 0)= 0,\quad h^2_1+ h^2_2\neq 0,NEWLINE\]NEWLINE where either \(\int^\infty_0 (1+ x)q(x) dx< \infty\), \(h_2= 0\) or \(\int^\infty(1+ x)q(x) dx< \infty\), \(\int^\infty_0 q(x)^2 dx< \infty\), \(h_2\neq 0\). It is proved that NEWLINE\[NEWLINEB_T(ik)- h_1+ h_2\Biggl(k+{1\over 2} \int^T_0 q(t) dt\Biggr)\tag{ii}NEWLINE\]NEWLINE is in a Hardy space of functions \(f(k)\) defined on \(\text{Re }k\geq 0\) where \(B_T(ik)\) is the boundary condition (i) associated with the truncated potential \(q_T(x)= q(x)\), \(0\leq x\leq T\), \(q_T(x)= 0\), \(x> T\). The Laplace transform of the Laguerre functions form an interpolation basis in the Hardy space and these are employed to interpolate for the zero's of \(B_T(ik)\). These converge to zero's of \(B(ik)\), \(T\to \infty\). An error analysis is given and results of computation for two test problems are presented.