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| | [No title] |
 | | XI.6) that \begin{equation} \widehat{\psi_0} (x) = \psi_0 (x) \int_{0}^{x} p_0 (t)^{-1} \psi_0 (t)^{-2}\, dt \lb{1.24} \end{equation} is another linearly independent solution of $\tau_0 \psi =0$ which is principal near $0$ and nonprincipal near $\infty$. |
| | For subtle cancellation effects and/or divergence of energy integrals of the type $\int_a^b p_0 (x) f' (x)^2 \,dx$, $\int_a^b q (x) f (x)^2 \,dx$ for elements $f \in \clD (H_0^0)$, the Friedrichs extension of $\check{H}_0^0$, see, for instance, \cite{fglp}, \cite{kalf}, \cite{rel}, and \cite{ros}. |
| | \eqref{1.19} or \eqref{3.28}) \begin{equation} q (x) =\mu p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4} \Big(\int_{x_0}^x p_0 (y)^{-1} \psi_0 (\lam_0,y)^{-2}\,dy \Big)^{-2}, \quad \mu< 0 \lb{3.48} \end{equation} is a border--line case since it yields a logarithmically divergent expression on the right--hand--sides of \eqref{3.39}, \eqref{3.46} (and similarly in the subcritical case of $(\tau_0 -\lam_0)$ on $(a,b)$). |
| www.ma.utexas.edu /mp_arc/html/papers/96-5 (3146 words) |
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