On a computer assisted proof of the existence of eigenvalues below the essential spectrum of the Sturm-Liouville problem (Q1841970)

A class of self-adjoint Sturm-Liouville eigenvalue problems on \([0,\infty)\) defined by \(-y''+qy=\lambda y\), \(y(0)=0\), where \(q\) is subject to certain assumptions, is considered. In an earlier paper [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No.~1986, 2229-2234 (1999; Zbl 0934.34073)] the authors have presented a new method for proving the existence of eigenvalues below the essential spectrum. This method combines operator theory and numerical analysis including interval analysis. In this paper, the authors develop further the method to cover a greater class of problems. For various values of the positive constant \(c\), eigenvalues below the essential spectrum are calculated for \(q(x)=-c \text{ exp}(-x/4)\cos(x)\), \(q(x) = -c \text{exp}(- x^2)\), and \(q(x) = c \sin(x + 1/(1 + x^2))\).