The author considers the differential operator \(H\) acting on \(L^2(- \alpha, +\alpha)\) given by \[ Hf= -{d\over dx} \Biggl(a(x) {df\over dx}\Biggr) \] and subject to Dirichlet boundary conditions at \(-\alpha\) and \(+\alpha\), where \(a: (- \alpha, +\alpha)\to (0, +\infty)\) is measurable with \(\gamma^{- 1}\leq a(x)\leq \gamma\) for all \(x\in (- \alpha, +\alpha)\) and some \(0 0\) and on the heat kernel \(K(t; x, y)\) for all \(t> 0\), whose dependence upon \(\alpha\) is explicit. He also studies in detail the stability of \(G\) and \(K\) under perturbations of \(a\). There are two reasons for this: the first being that it is needed for his method of obtaining sharper upper bounds on \(K\) than the usual ones and the second being that it provides theoretical support for numerical methods of determining the spectrum of \(H\) when \(a\) is highly irregular. He discovers that the convergence of a sequence of such operators \(H_n\) to a limit \(H\) in the norm resolvent sense implies the convergence of the Green functions and heat kernel.