%&amslatex %% @texfile{ %% filename="Pert.tex", %% version="1.0", %% date="JAN-1996", %% filetype="AMS-LaTeX", %% copyright="Copyright (C) M.Unal & F.Gesztsy" %% } \documentclass[reqno]{amsart} %\documentstyle{amsart} %\renewcommand{\baselinestretch}{2} \textwidth 6.4in \textheight 8in \oddsidemargin 0.25in \evensidemargin 0.25in %\setlength{\textwidth}{16.2cm} %\setlength{\textheight}{24.1cm} %%%%%%%%%THEOREMS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \newtheorem{prop}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \renewcommand{\thecor}{} \newtheorem{ques}[thm]{Question} \newtheorem{res}[thm]{Result} \newtheorem{prob}[thm]{Problem} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \newtheorem{exmp}{Example} \renewcommand{\theexmp}{} \newtheorem{ack}{Acknowledgment} \renewcommand{\theack}{} \newsymbol\gtrless 133F %%%%%%%%%%%%%%FONTS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\R}{{\Bbb R}} \newcommand{\N}{{\Bbb N}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\C}{{\Bbb C}} \newcommand{\clD}{{\mathcal{D}}} \newcommand{\clH}{{\mathcal{H}}} \newcommand{\clB}{{\mathcal{B}}} %%%%%%%%%%%%%%%GREEK%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\lam}{\lambda} \newcommand{\sig}{\sigma} \newcommand{\veps}{\varepsilon} \newcommand{\vphi}{\varphi} \newcommand{\gam}{\gamma} \newcommand{\eps}{\epsilon} %%%%%%%%%%%%%%%%%%ABBRS%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\ti}{\tilde} %\newcommand{\eps}{\epsilon} \newcommand{\bs}{\backslash} \newcommand{\bi}{\bibitem} \newcommand{\emt}{\emptyset} \newcommand{\nn}{\nonumber} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\bay}{\begin{array}} \newcommand{\eay}{\end{array}} %\newcommand{\D}{\displaystyle} \newcommand{\ol}{\overline} \newcommand{\ul}{\underline} \newcommand{\upa}{\uparrow} \newcommand{\lra}{\longrightarrow} \newcommand{\ra}{\rightarrow} \newcommand{\prl}{\parallel} \newcommand{\downa}{\downarrow} \newcommand{\Ome}{\Omega} \newcommand{\al}{\alpha} \newcommand{\ome}{\omega} \newcommand{\del}{\delta} \newcommand{\pa}{\partial} \newcommand{\Lam}{\Lambda} \newcommand{\vrho}{\varrho} \newcommand{\Del}{\Delta} \newcommand{\Gam}{\Gamma} \newcommand{\Sig}{\Sigma} %\newcommand{\sig}{\sigma} \newcommand{\Tht}{\Theta} \newcommand{\kp}{\kappa} \newcommand{\tht}{\theta} \newcommand{\vph}{\varphi} \newcommand{\supp}{\operatorname{supp}} \newcommand{\compact}{\operatorname{compact}} \newcommand{\sgn}{\operatorname{sgn}} \newcommand{\dint}{\displaystyle\int} \newcommand{\ess}{{\operatorname{ess}}} \newcommand{\loc}{{\operatorname{loc}}} \newcommand{\Ran}{{\rm Ran}} \newcommand{\lb}{\label} %%%%%%%%%%%%%%%%%%%%%%%%NUMBERING%%%%%%%%%%%%%%%%%%%%%%%% \makeatletter \def\theequation{\thesection.\@arabic\c@equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{Perturbative Oscillation Criteria and Hardy-Type Inequalities} \author{F. Gesztesy} \address{Department of Mathematics, University of Missouri, Columbia, MO 65211, USA} \email{mathfg@mizzou1.missouri.edu} \author{M. \"Unal} \address{Department of Mathematics, University of Missouri, Columbia, MO 65211, USA} \email{mathgr31@mizzou1.missouri.edu} %\keywords{Sturm-Liouville operators} % Math Subject Classifications %\subjclass{Primary ??, ??; Secondary ??, ??} \maketitle \newcounter{me} \maketitle \begin{abstract} We prove a natural generalization of Kneser's oscillation criterion and Hardy's inequality for Sturm--Liouville differential expressions. In particular, assuming $-\frac{d}{dx} p_0 (x) \frac{d}{dx} + q_0 (x)$, $x \in (a,b)$, $-\infty \leq a < b \leq \infty$ to be nonoscillatory near $a$ (or $b$), we determine conditions on $q (x)$ such that $-\frac{d}{dx} p_0 (x) \frac{d}{dx} + q_0 (x) + q (x)$ is nonoscillatory, respectively, oscillatory near $a$ (or $b$). \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction}\lb{s1} In this note we compare oscillation properties of solutions of Sturm--Liouville equations $\tau_0 \psi_0 = \lam \psi_0$ and $\tau \psi = \lam \psi$, where $\tau_0$ is of the type $\tau_0 = -\frac{d}{dx} p_0 \frac {d}{dx} + q_0 (x)$ and its perturbation $\tau$ is of the form $\tau = \tau_0 + q (x)$. More precisely, assuming \begin{equation} 0 < p_0 ^{-1} \in L_{loc}^{1} (( (a,b)),\: q_0 \in L_{loc}^{1}((a,b)) \: \text{real--valued}, \lb{1.1} \end{equation} consider the (quasi) differential expression \begin{equation} \tau_0 = -\frac{d}{dx} p_0 (x) \frac{d}{dx} +q_0 (x), \quad x \in (a,b), \: -\infty \leq a {-1/4}, \lb{1.5} \end{equation} then $(\tau -\lam_0)$ is nonoscillatory near $b$\/},\\ and\\ {\it (ii) If \begin{equation} \hspace*{-.5in} \limsup_{x\upa b}p_0 (x) \big\{ \psi_0 (\lam_0,x)^{4} \Big(\int^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}dt\Big)^{2} q (x)\big\}<{-1/4}, \lb{1.6} \end{equation} then $(\tau -\lam_0)$ is oscillatory near $b$\/}.\\ (The case $| \int^{\infty} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt| < \infty$ will be dealt with analogously.) The special case $p_0 = \psi_0 = 1$, $q_0 = \lam_0 = 0$ in \eqref{1.5} and \eqref{1.6} represents Kneser's \cite{kns} original (non)oscillation criterion dating back to 1893. In fact, we shall prove in Theorem \ref{t2.3} a scale of (non)oscillation criteria generalizing the well--known (non)oscillation criteria by Weber \cite{wbr}, p. 53--62 (see also Hartman \cite{hrt1} and Hille \cite{hl1}) which, in turn, extended Kneser's original criterion. Before we continue the description of the content of this paper, a few hints concerning the literature on oscillation theory might be in order. Since an exhaustive list of references is impossible due to the incredible amount of papers devoted to this subject, we are forced to confine ourselves to monographs and reviews. Comprehensive treatments of oscillation theory can be found, for instance, in \cite{bar}, \cite{cop}, Ch. 1, \cite{dnsch}, Sects. XIII.7--XIII.10, \cite{hrt2}, Ch. XI, \cite{hl2}, Sect. 9.4--9.6, \cite{krh}, Ch. 1, \cite{rd}, Ch. IV, \cite{swn}, Ch. 2, and \cite{wllt}. >From a historical perspective, Sturm's original memoir \cite{stm} of 1836 and B\^ocher's monograph \cite{boc} of 1917 played a premier role in the development of the field. References intimately related to the topics we discuss in this paper are Kneser's original work \cite{kns} in 1893, its generalization due to Weber \cite{wbr} from 1912 and the rediscovery of the latter by Hartman \cite{hrt1} and Hille \cite{hl1} in 1948. In connection with the special case $p_0 = \psi_0 = 1$, $q_0 = \lam_0 = 0$ and Kneser's (non)oscillation criterion \begin{equation} \lim_{x \upa b} {\mbox{\raisebox{-.7ex} {$\stackrel{\textstyle \inf} {\sup}$}}} \big\{ x^2 q (x) \big\} \gtrless {-1/4}\: \: \text{implies}\: \: \mbox{\raisebox{-.7ex}{$\stackrel{\textstyle \text{nonoscillation}} {\text{oscillation}}$}} \: \: \text{of $\tau$ near $b$} \lb{1.7} \end{equation} one should mention the possibility of factoring \begin{equation} \tau_{\mu}^{(0)} = -\frac{d^2}{dx^2} + \frac{\mu}{(x-a)^2},\quad x> R>a,\: \mu \in \R \lb{1.8} \end{equation} for $aR,\: \al \in \C. %\end{array} \lb{1.9} \end{equation} Then one verifies \begin{equation} \tau_{\mu}^{(0)} =A_{\al}^{(0) +} A_{\al}^{(0)}\: \text{if and only if}\: \mu =\al (\al-1). \lb{1.11} \end{equation} In particular, since for $\al \in \R$, $\al (\al-1) \geq {-1/4}$, $\tau_{\mu}^{(0)}$ in \eqref{1.8} admits the factorization \eqref{1.11} with $\al \in \R$ if and only if $\mu \geq {-1/4}$. Intimately connected with the existence of the factorization \eqref{1.11} for $\al \in \R$ and the associated borderline $\mu_0 = {-1/4}$ is (a special case of) Hardy's inequality (see, e.g., \cite{hdy}, \cite{hdy1}, \cite{hlp}, p. 240) \begin{equation} \int_{a}^{\infty} |\phi' (x)|^2\, dx > \frac{1}{4} \int_{a}^{\infty} \frac{|\phi (x)|^2}{(x-a)^2}\,dx,\quad 0\neq \phi \in C_{0}^{\infty} \big( (a,\infty) \big). \lb{1.12} \end{equation} In connection with the general Sturm--Liouville differential expressions \eqref{1.2} and \eqref{1.3} we shall show in Section \ref{s3} that \begin{align}\nn &\tau_{\mu} = -\frac{d}{dx} p_0 (x) \frac{d}{dx} + q_0 (x) + \mu p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4} \Big(\int^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\,dt \Big)^{-2},\\ & \hspace*{3in} x > R> a,\: \mu \in \R, \lb{1.13} \end{align} for $a0$ for $x \in (a,b)$, $|\int_{a} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2} \, dt| < \infty$, and $|\int^{b} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2} \, dt| = \infty$. Then \begin{align}\nn \int_{a}^{b} & p_0 (x) |\phi' (x)|^2 \, dx > \int_{a}^{b} \Big[ \lam_0 - q_0 (x) + 4^{-1} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4} \\ & \times \Big( \int_{a}^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \Big)^{-2} \Big] |\phi (x)|^{2}\,dx, \quad 0 \neq \phi \in \clD_0 ((a,b)), \lb{1.16} \end{align} where \begin{align} &\clD_0 ( (a,b) ) = \{ f \in L^{2} ((a,b)) | f\in AC_{loc} ( (a,b) ) ;\, \text{supp}(f) \subset (a,b) \:\text{compact};\: p_0^{1/2} f'\in L^{2} ((a,b)) \}. \lb{1.17} \end{align} (Here $AC_{loc} (\Ome)$ denotes the set of locally absolutely continuous functions on $\Ome \subset \R$.) The constant $1/4$ in \eqref{1.16} (and \eqref{1.12}) is optimal\/}. A look at \eqref{1.8} and \eqref{1.13} reveals that the role of the basic comparison potential \begin{equation} q_{\mu}^{(0)} (x) =\mu (x-a)^{-2},\quad x>a, \lb{1.18} \end{equation} in the special case where $p_0 = \psi_0 = 1, \: q_0 = \lam_0 = 0$, $\tau_0^{(0)} = -\frac{d^2}{dx^2}$, is played by \begin{equation} q_{\mu}^{0} (x) = \mu p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4} \Big( \int_{a}^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt\Big)^{-2},\quad x \in (a,b) \lb{1.19} \end{equation} in the general Sturm--Liouville case $\tau = \tau_0 +q (x)$, where $\tau_0 = -\frac{d}{dx} p_0 (x) \frac{d}{dx} + q_0 (x)$. We close this introduction with a heuristic comment on why the comparison potential \eqref{1.19} is a natural generalization of \eqref{1.18}. Taking $a =0,\: b = \infty,\: \lam_0 = 0$ for simplicity, one considers \begin{equation} \tau_0^{(0)} = -\frac{d^2}{dx^2}\quad \text{and} \quad \mu x^{-2},\quad x> 0 \lb{1.20} \end{equation} and hence gets a fundamental system of solutions \begin{equation} \psi_0 (x) =1,\quad \widehat{\psi_0} (x) = x \lb{1.21} \end{equation} of $\tau_0 \psi = 0$, where $\psi_0 (x)$ is principal near $\infty$ and nonprincipal near $0$ whereas $\widehat{\psi_0} (x)$ is principal near $0$ but nonprincipal near $\infty$. In particular, one observes that \begin{equation} \mu x^{-2} = \mu (\psi_0 (x) \widehat{\psi_0} (x))^{-2}, \quad x> 0. \lb{1.22} \end{equation} This trivial fact is no accident. Indeed, consider \begin{equation} \tau_0 = -\frac{d}{dx} p_0 (x) \frac{d}{dx} +q_0 (x), \quad x> 0 \lb{1.23} \end{equation} and assume $\psi_0 (x)> 0$ is a principal solution of $\tau_0 \psi = 0$ near $\infty$, but nonprincipal near $0$. Then it is a well--known fact (see, e.g., \cite{hrt2}, Sect. 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 $\infty$. Moreover, \begin{equation} \mu p_0 (x)^{-1} ( \psi_0 (x) \widehat{\psi_0} (x))^{-2} = \mu p_0 (x)^{-1} \psi_0 (x)^{-4} \Big( \int_{0}^{x} p_0 (t)^{-1} \psi_0 (t)^{-2}\, dt \Big)^{-2} \lb{1.25} \end{equation} is precisely of the form \eqref{1.19}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{A Generalization of Kneser's Theorem}\lb{s2} \setcounter{equation}{0} \setcounter{thm}{1} In order to set the stage we assume \begin{align} &0< p_0^{-1} \in L_{loc}^{1} ((a,b)),\: q_0, q \in L_{loc}^{1} ((a,b))\: \text{real--valued},\: -\infty \leq a 0$ near $b$ is called principal near $b$ if and only if $|\int^{b} p_0 (x)^{-1} \psi_0 (x)^{-2}\, dx|= \infty$. Every linearly independent solution $\widehat{\psi_0}$ of $\tau \psi = \lam_0 \psi$ then satisfies $|\int^{b} p_0 (x)^{-1} \widehat{\psi_0} (x)^{-2}\, dx|< \infty$ and is called nonprincipal (see, e.g., \cite{hrt2}, Sect. XI.6). Analogous definitions apply at the point $a$. In addition to \eqref{2.1}, we shall need a further spectral hypothesis in connection with $\tau_0$ and hence introduce our basic hypothesis \newline {\bf (H.2.1).} {\it Suppose $p_0$, $q_0$, and $q$ satisfy \eqref{2.1}. Moreover, assume that $(\tau_0 -\lam_0)$ is nonoscillatory near $b$ (resp. $a$) for some $\lam_0 \in \R$, and let $\psi_0 (\lam_0,x)$ denote a solution of $\tau_0 \psi =\lam_0 \psi$ which is positive in a neighborhood of $b$ (resp. $a$)\/}. Occasionally we shall consider maximal and minimal operators $H_0$, $H$ and ${\check{H}}_{0},\: \check{H}$ in $L^{2} ((a,b))$ associated with $\tau_0$ and $\tau$ defined as follows, \begin{align} &H_{(0)} f = \tau_{(0)} f,\quad f \in \clD (H_{(0)}) = \{ g \in L^{2} ((a,b)) |\: g,\: p_0 g' \in AC_{loc} ((a,b)); \tau_{(0)} g \in L^{2} ((a,b)) \}, \lb{2.4} \end{align} \begin{equation} \hspace*{-.5in} \check{H}_{(0)}f =\tau_{(0)} f,\: f \in \clD (\check{H}_{(0)}) =\{g \in \clD (H_{(0)})|\: \text{supp}(g)\, \subset \,(a,b)\,\text{compact}\}, \lb{2.5} \end{equation} where $H_{(0)}$, $\check{H}_{(0)}$ abbreviates $H_0$ or $H$, $\check{H}_0$ or $\check{H}$. If and only if $\tau_{(0)}$ is in the $l.p.$ case at $a$ and $b$ one obtains \begin{equation} \overline{\check{H}_{(0)}} = H_{(0)}, \lb{2.6} \end{equation} where $-$ denotes the operator closure in $L^{2} ((a,b))$, otherwise one has \begin{equation} \overline{\check{H}_{(0)}} %{\mbox{\raisebox{-.4ex}{$\stackrel{ \subset}{\neq}$}}} \stackrel{ \subset}{\neq} H_{(0)}. \lb{2.7} \end{equation} Given these preliminaries, we can now state our generalization of Kneser's \cite{kns} criterion. \begin{thm}\lb{t2.2} Assume Hypothesis (H.2.1) near $b$. If $|\int^{b} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx|$ $=\infty$ (i.e., if $\psi_0 (\lam_0,x)$ is principal near $b$) choose $R \in (a,b)$ in a sufficiently small neighborhood of $b$ such that $\psi_0 (\lam_0,x) > 0$ on $(R,b)$. Then, \newline (i) If \begin{equation} \liminf_{x \uparrow b} \big\{ p_0 (x) \psi_0 (\lam_0,x)^{4} \Big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\,dt\Big)^{2} q(x) \big\} > {-1/4}, \lb{2.8} \end{equation} $(\tau -\lam_0)$ is nonoscillatory near $b$.\\ (ii) If \begin{equation} \hspace*{-.1in} \limsup_{x \uparrow b}\big\{ p_0 (x) \psi_0 (\lam_0,x)^{4} \Big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\,dt\Big)^{2} q(x) \big\} < {-1/4}, \lb{2.9} \end{equation} $(\tau -\lam_0)$ is oscillatory near $b$.\\ If $|\int^{b} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx| < \infty$ (i.e., if $\psi_0 (\lam_0,x)$ is nonprincipal near b) replace $\int_R^x$ by $\int_x^b$ in \eqref{2.8} and \eqref{2.9} to reach the same conclusions. \end{thm} Instead of proving Theorem~\ref{t2.2} at this point we shall actually consider a generalization, implying a scale of new oscillation criteria. We start with a bit of notation. Define \begin{equation} \ln_0 (x) =|x|,\: \ln_{n} (x) = \ln (\ln_{n-1} (x)), \quad n \in \N,\: x \in \R\backslash \{0\}, \lb{2.10} \end{equation} \begin{align}\nn q_{1,\mu} (x) &= \mu x^{-2},\\ \nn q_{n,\mu} (x) &= x^{-2} \big[ -4^{-1}\sum_{k=0}^{n-2} \prod_{j=1}^{k} \big( \ln_{j} (x) \big)^{-2} + \mu \prod_{j=1}^{n-1} \big( \ln_{j} (x) \big)^{-2} \big],\\ & =x^{-2} [ -4^{-1} + q_{n-1,\mu} (\ln|x|)],\quad \mu \in \R,\: n \geq 2, \lb{2.11} \end{align} \begin{align}\nn & Q_{n,\mu} (x) = p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4} \\ \nn & \times\begin{cases} q_{n,\mu} (\int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt) \quad \text{if}\: |\int^b p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx| = \infty\\ q_{n,\mu} (\int_x^b p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt) \quad \text{if}\: |\int^b p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx| < \infty \end{cases},\\ & \hspace*{3.4in} n \in \N. \lb{2.12} \end{align} (As usual we denote $\prod_{j=1}^{0} (\cdot) = 1$ in \eqref{2.11}.) Explicitly, one obtains \begin{align}\nn &Q_{1,\mu} (x) = \mu p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4}\\ \nn &\times \begin{cases} \big(\int_{R}^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\,dt \big)^{-2}\quad \text{if} \:|\int^b p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx| = \infty\\ \big(\int_{x}^{b} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\,dt \big)^{-2}\quad \text{if} \:|\int^b p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx| < \infty \end{cases}, \end{align} \begin{align}\nn &Q_{2,\mu} (x) = p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4} \left\{ \begin{array}{l} \big(\int_{R}^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \big)^{-2}\\ \big(\int_{x}^{b} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \big)^{-2} \end{array} \right\}\\ &\times \left\{ -4^{-1} + \mu \left( \ln \left\{ \begin{array}{l} \Big( \int_R^x p_0(t)^{-1} \psi_0(\lam_0,t)^{-2}\,dt\Big) \\ \Big( \int_x^b p_0(t)^{-1} \psi_0(\lam_0,t)^{-2}\,dt\Big) \end{array}\right\} \right)^{-2} \right\} \begin{array}{l} \text{if}\: |\int^b p_0(x)^{-1} \psi_0(\lam_0,x)^{-2}\,dx|=\infty\\ \text{if}\: |\int^b p_0(x)^{-1} \psi_0(\lam_0,x)^{-2}\,dx|<\infty \end{array},\\ \nn & \text{etc.} \end{align} Our first major result then reads as follows. \begin{thm}\lb{t2.3} Assume Hypothesis (H.2.1) near $b$. If $| \int^{b} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx|$ $= \infty$ choose $R \in (a,b)$ sufficiently close to $b$ such that $\psi_0 (\lam_0,x) > 0$ on $(R,b)$. Suppose \begin{align}\nn &\lim_{x \uparrow b}\Big\{ p_0 (x) \psi_0 (\lam_0,x)^{4} \Big( \int_{R}^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2} dt\Big)^{2}\\ \nn & \times \prod_{j=1}^{n-2} \Big[ \ln_{j} \big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2} \, dt \big) \Big]^{2} \Big[ q (x) +4^{-1} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4}\\ & \times \Big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\,dt \Big)^{-2} \sum_{k=0}^{n-3} \prod_{j=1}^{k} \Big(\ln_{j} \big(\int_{R}^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \big) \Big)^{2} \Big]\Big\} = -1/4,\quad n \geq 3. \lb{2.14} \end{align} Then,\\ (i) If \vspace*{-2ex} \begin{align}\nn &\liminf_{x \uparrow b} \Big\{ p_0 (x) \psi_0 (\lam_0,x)^{4} \Big(\int_{R}^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}dt \Big)^{2} \prod_{j=1}^{n-1} \Big[\ln_{j} \big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2} dt \big) \Big]^{2} \Big[ q (x) \\ \nn & +4^{-1} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4} \Big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2} \,dt \Big)^{-2} \sum_{k=0}^{n-2} \prod_{j=1}^{k} \Big(\ln_{j} \big(\int_{R}^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \big) \Big)^{2} \Big]\Big\}\\ & > -1/4, \lb{2.15} \end{align} $(\tau -\lam_0)$ is nonoscillatory near $b$. \newline (ii) If \begin{align}\nn &\limsup_{x \uparrow b}\Big\{ p_0 (x) \psi_0 (\lam_0,x)^{4} \Big( \int_{R}^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2} dt \Big)^{2} \prod_{j=1}^{n-1} \Big[ \ln_{j} \big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2} \, dt \big) \Big]^{2} \Big[ q (x)\\ \nn & +4^{-1} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4} \Big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2} \,dt \Big)^{-2} \sum_{k=0}^{n-2} \prod_{j=1}^{k} \Big(\ln_{j} \big(\int_{R}^{x} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \big) \Big)^{2} \Big]\Big\}\\ & < -1/4, \lb{2.16} \end{align} $(\tau -\lam_0)$ is oscillatory near $b$. If $| \int^{b} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx|< \infty$, replace $\int_R^x$ by $\int_x^b$ in \eqref{2.14}--\eqref{2.16} in order to reach the same conclusions. \end{thm} \begin{proof} By Sturm's comparison theorem (see, e.g., \cite{dnsch}, Lemma XIII.7.35) it suffices to describe solutions of \begin{equation} \tau_{n,\mu} \psi = \lam_0 \psi,\quad \tau_{n,\mu} = \tau_0 + Q_{n,\mu} (x),\quad n \geq 2. \lb{2.17} \end{equation} Assuming $|\int^{b} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx|= \infty$, one verifies inductively that fundamental systems of solutions of \eqref{2.17} are given by\\ $(a)$ $\mu =-1/4$: \begin{align}\nn &\psi_{1} (\lam_0,x) =\psi_0 (\lam_0,x) \Big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \Big)^{1/2} \prod_{j=1}^{n-1}\Big( \ln_{j} \big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \big) \Big)^{1/2},\\ & \psi_{2} (\lam_0,x) = \psi_{1} (\lam_0,x) \ln_{n} \Big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \Big). \lb{2.18} \end{align}\\ $(b)$ $\mu > -1/4$: \begin{align}\nn & \psi_{p} (\lam_0,x) =\psi_0 (\lam_0,x) \Big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \Big)^{1/2} \prod_{j=1}^{n-1}\Big( \ln_{j} \big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \big) \Big)^{1/2} \\ &\hspace*{1.5in}\times \Big( \ln_{n-1} \big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \big) \Big)^{(-1)^{p} (\mu + 4^{-1})^{1/2}}, \quad p= 1,2. \lb{2.19} \end{align}\\ $(c)$ $\mu < -1/4$: \begin{align}\nn &\psi_{p} (\lam_0,x) =\psi_0 (\lam_0,x) \Big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \Big)^{1/2} \prod_{j=1}^{n-1}\Big( \ln_{j} \big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \big) \Big)^{1/2} \\ &\hspace*{1.3in}\times \exp \Big[ (-1)^{p} i {(-\mu -4^{-1})}^{1/2} \ln_{n} \big( \int_R^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \big) \Big], \quad p= 1,2. \lb{2.20} \end{align} This proves \eqref{2.15} and \eqref{2.16}. If $|\int^b p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx |< \infty$, consistently replacing $\int_R^x$ by $\int_x^b$ in \eqref{2.14}--\eqref{2.16} and \eqref{2.18}--\eqref{2.20} completes the proof. \end{proof} The special case $p_0 = \psi_0 =1$, $q_0 = \lam_0 =0$ in Theorem~\ref{t2.2} is Kneser's \cite{kns} original criterion, while that special case in Theorem \ref{t2.3} represents a generalization of Kneser's result originally due to Weber \cite{wbr}, p.53--62 and later rediscovered by Hartman \cite{hrt1} and Hille \cite{hl1}. Theorem \ref{t2.2} (and \ref{t2.3}) can also be proved by reducing it to Kneser's criterion \eqref{1.7} using an appropriate change of variables (Kummer--type transformations). Moreover, a variety of extensions of Kneser's original criterion \eqref{1.7} corresponding to $p_0=1$, $q_0 = \lam_0 =0$ can be generalized to our perturbative setting which relates $p_0, q_0$ and $p_0, q_0 +q$. Since a comprehensive treatment is beyond the scope of the present paper we omit further details. Next we describe a simple application of Theorem \ref{t2.2} to the case where $\tau_0$ is periodic, $(a,b)= \R$, $p_0 =1$, and $q (x) \underset{x\rightarrow {\infty}}{=} O(x^{-2})$. In addition to assuming that $q_0$ satisfies Hypothesis (H.2.1) near $\pm \infty$ we suppose that $q_0$ is periodic of period one, that is, \begin{equation} p_0 (x) =1,\quad q_0 (x+1) = q_0 (x),\quad x\in \R, \lb{2.21} \end{equation} and that $\lam_0$ is the lowest periodic eigenvalue associated with $\tau_0 |_{[0,1]}$ and $\psi_0 > 0$ the corresponding unique eigenfunction extended to all of $\R$ by periodicity, that is, \begin{equation} \psi_0 (x+1) = \psi_0 (x),\quad x \in \R. \lb{2.22} \end{equation} Introducing the operator $H_0$ in $L^2 (\R)$ associated with $\tau_0$ as in \eqref{2.4} one infers that $H_0$ is self--adjoint and bounded from below. In fact, \begin{equation} \sig (H_0) = \sig_{\ess} (H_0) \subseteq {[\lam_0,\infty)}, \lb{2.23} \end{equation} where $\sig (\cdot)$ and $\sig_{\ess} (\cdot)$ abbreviate the spectrum and essential spectrum, respectively. Concerning the perturbation $q$ we assume \begin{equation} q \in L^{\infty} (\R) \: \text{real--valued},\quad q (x) \underset{|x|\rightarrow {\infty}}{=} {c}{|x|^{-2}} [1+ o(1)],\quad c>0, \lb{2.24} \end{equation} introduce $\tau = \tau_0 +q (x)$, and define the self--adjoint operator $H$ associated with $\tau$ as in \eqref{2.4}. (One could easily admit local singularities of $q$ as long as $q$ stays locally integrable but this is not the point of our investigation below.) Since $q (x) \underset{|x|\rightarrow {\infty}}{\rightarrow}0$ one obtains \begin{equation} \sig_{\ess} (H) = \sig(H_0). \lb{2.25} \end{equation} Finally we abbreviate \begin{align} & \min_{x \in [0,1]} \psi_0 (\lam_0,x) = \psi_{0,\min},\quad \max_{x \in [0,1]} \psi_0 (\lam_0,x) = \psi_{0,\max} \lb{2.26}\\ \intertext{and} & A_0 = \int_0^1 \psi_0 (\lam_0,t)^{-2} \,dt. \lb{2.27} \end{align} Given these preliminaries one easily obtains \begin{lem}\lb{l2.4} Introduce the notation established in \eqref{2.21}--\eqref{2.27}. Then,\\ (i) If \begin{equation} c > -4^{-1} \psi_{0,\max}^{-4} A_0^{-2}\:\: (\geq -4^{-1}), \lb{2.28} \end{equation} $(\tau -\lam_0)$ is nonoscillatory near $\pm \infty$.\\ (ii) If \begin{equation} c < -4^{-1} \psi_{0,\min}^{-4} A_0^{-2}\:\: (\leq -4^{-1}), \lb{2.29} \end{equation} $(\tau -\lam_0)$ is oscillatory near $\pm \infty$.\\ Equality with $-1/4$ on the right--hand--sides of \eqref{2.28} and \eqref{2.29} holds if and only if $q_0 (x) = constant$. \end{lem} \begin{proof} Consider $x >0$. \eqref{2.28} and \eqref{2.29} immediately follow from \eqref{2.8} and \eqref{2.9} upon decomposing $x = [x] + r(x)$, $0 \leq r(x) <1$, $[x] =\inf_{n \in \N} \{n \leq x\}$, the integer part of $x>0$, and noting that \begin{equation} \int_0^x \psi_0 (\lam_0,t)^{-2}\, dt = A_0 [x] +s(x), \quad 0 \leq s(x) < A_0,\: x> 0. \lb{2.30} \end{equation} Thus \begin{equation} \lim_{x \to +\infty} \mbox{\raisebox{-.7ex}{$\stackrel{\textstyle \inf}{\sup}$}} \big\{ \psi_0(\lam_0,x)^4 \Big( \int_{0}^x \psi_0(\lam_0,t)^{-2}\, dt \Big)^2 c\: x^{-2}\big\}= c\: \psi_{0, {\mbox{\raisebox{-.7ex}{$\stackrel{ \max}{\scriptstyle \min}$}}}}^4\: A_0^2. \lb{2.31} \end{equation} The proof for $x <0$ is similar. Equality with $-1/4$ on the right--hand--sides of \eqref{2.28} and \eqref{2.29} holds if and only if $\psi_0 (\lam_0,x)$ is constant, that is, if and only if $q(x)$ is constant. \end{proof} In order to fill the gap left between \eqref{2.28} and \eqref{2.29}, that is, to determine the (non)oscillation status of $\tau = \tau_0 + c (1+|x|^2)^{-1}$ for \begin{equation} c \in [-4^{-1} \psi_{0,\min}^{-4} A_0^{-2}, -4^{-1} \psi_{0,\max}^{-4} A_0^{-2}], \lb{2.32} \end{equation} one needs to invoke more detailed information on the periodic background operator $H_0$. The result stated in \cite{khr} (without proof) requires the additional knowledge of the effective mass $m_0$ associated with $\lam_0 = \inf \sig (H_0)$ (and similar for higher spectral gaps of $H_0$). Interest in the periodic operator $H_0$ and its perturbation $H$ stems from possible applications in solid state physics. In fact, $q_0$ models one--dimensional periodic structures such as crystals and $q$ describes additional impurities or defects (responsible, e.g., for the color of crystals). Moreover, as a further illustration of \eqref{2.28} and \eqref{2.29} it should be mentioned that Rofe--Beketov \cite{fsr} proved that $H$ has finitely many eigenvalues in each spectral gap of $H_0$ whenever $q \in L^{1} (\R; (1+|x|)dx)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Hardy-Type Inequalities and Birman--Schwinger Bounds} \lb{s3} \setcounter{equation}{0} \setcounter{thm}{0} In our final section we discuss factorizations of the type \eqref{1.14}, \eqref{1.15} and use them to prove a natural generalization of Hardy's inequality \eqref{1.12} from the point of view of quadratic form perturbations. At the end we illustrate some explicit bounds on the number of bound states of $H$ below a fixed real number $\lam_0 \leq \inf \sig (H_0)$. We start with a factorization of our comparison differential expression on $(a,b)$, \begin{align}\nn &\tau_{\mu} = -\frac{d}{dx} p_0 (x) \frac{d}{dx} + q_0 (x) + \mu p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4} \Big( \int_{x_0}^x p_0 (t)^{-1} \psi_0 (\lam_0,x)^{-2}\,dt \Big)^{-2}\\ & \hspace{3in} \text{for some}\: a \leq x_0 \leq b,\: \mu \in \R. \lb{3.1} \end{align} Introducing \begin{align}\nn A_{\al} &= p_0 (x)^{1/2}\frac{d}{dx} - p_0 (x)^{1/2} \psi_0 (\lam_0,x)^{-1} \psi_0' (\lam_0,x) - \al p_0 (x)^{-1/2} \psi_0 (\lam_0,x)^{-2} \\ &\hspace*{1.5in} \times \Big( \int_{x_0}^x p_0 (t)^{-1} \psi_0 (\lam_0,x)^{-2}\,dt \Big)^{-1},\quad \al \in \C, \lb{3.2}\\ \nn A_{\al}^{+} &= -\frac{d}{dx} p_0 (x)^{1/2} - p_0 (x)^{1/2} \psi_0 (\lam_0,x)^{-1} \psi_0' (\lam_0,x) - \al p_0 (x)^{-1/2} \psi_0 (\lam_0,x)^{-2} \\ &\hspace*{2in} \times \Big( \int_{x_0}^x p_0 (t)^{-1} \psi_0 (\lam_0,x)^{-2}\,dt \Big)^{-1}, \lb{3.3} \end{align} one verifies that \begin{equation} \tau_{\mu} = A_{\al}^{+} A_{\al} + \lam_0\quad \text{if and only if}\quad \mu = \al (\al-1). \lb{3.4} \end{equation} Moreover, solutions of $\tau_{\mu} \psi = \lam_0 \psi$ are given by \begin{equation} \psi_{\mu} (\lam_0,x) = \psi_0 (\lam_0,x) \Big( \int_{x_0}^x p_0 (t)^{-1} \psi_0 (\lam_0,x)^{-2}\,dt \Big)^{\frac{1}{2} \pm (\mu +\frac{1}{4})^{\frac{1}{2}}}. \lb{3.5} \end{equation} The factorization \eqref{3.4} of $\tau$ is possible for $\al \in \R$ if and only if $\mu \geq -1/4$. In this case $A_{\al}^{+}$ is the formal adjoint of $A_{\al}$, $\al \in \R$. Our second main result now reads as follows. \begin{thm}\lb{t3.1} Assume $p_0$, $q_0$, and $\psi_0 (\lam_0)$ satisfy (H.2.1) near $a$ and $b$ and $\psi_0 (\lam_0,x)$ ${>0}$ for all $x \in (a,b)$. Moreover, suppose $|\int_{a} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx|<\infty$ and %\linebreak[4] $|\int^{b} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2}\, dx|=\infty$ (i.e., $\psi_0 (\lam_0,x)$ is nonprincipal near $a$ and principal near $b$). Then for all $0 \neq \phi \in \clD_{0} ((a,b))$, \begin{align} &\int_{a}^{b} p_0 (x)|\phi' (x)|^2\,dx > \int_a^b \big[ \lam_0 - q_0 (x) + 4^{-1} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4}(\int_a^x p_0 (t)^{-1} \psi_0 (\lam_0,t)\,dt)^{-2}\big]| \phi (x)|^{2}\,dx, \lb{3.6} \end{align} where \begin{align} &\clD_0 ((a,b)) = \{ f\in L^2 ((a,b)) |\: f\in AC_{loc} ((a,b)); \: \supp (f) \subset (a,b)\: \compact,\: p_0^{1/2} f' \in L^2 ((a,b))\: \}. \lb{3.7} \end{align} The constant $1/4$ in \eqref{3.6} is optimal. \end{thm} \begin{proof} Choose $c, \:d \in (a,b)$, $c 0$ on $(a,b)$ in Theorem \ref{t3.1} implies that \begin{equation} \ol{\check{H_0}} \geq \lam_0 \lb{3.11} \end{equation} (cf. \eqref{2.6}). Alternatively, one can prove Theorem \ref{t3.1} using \begin{equation} \int_c^d |(A_{1/2} f) (x)|^2\, dx = \eqref{3.8}, \quad x_0 = a \lb{3.12} \end{equation} (cf. \eqref{3.1}--\eqref{3.5}). In order to avoid the compact support assumption on $\phi$ in Theorem \ref{t3.1}, one needs to guarantee the vanishing of the boundary terms in \eqref{3.8} as $c \downa a$ and $d \upa b$. The following observations illustrate such a possibility. We need to introduce a bit of notation. Define in $L^2 ((a,b))$, \begin{equation} \check{H}_0^0 f = \tau_0^0 f,\quad \tau_0^0 = -\frac{d}{dx} p_0 (x) \frac{d}{dx}, \quad x \in (a,b), \lb{3.13} \end{equation} \begin{align}\nn &f \in \clD (\check{H}_0^0) = \{ g \in L^2 ((a,b))\,|\: g,\: p_0 g'\in AC_{loc} ((a,b))\,; \supp (g)\: \subset\: (a,b)\; \text{compact}\; ; \tau_0^0 g \in L^2 ((a,b)) \}, \end{align} \begin{align} (A_0^0 f) (x) = p_0 (x)^{1/2} f' (x),\: f\in \clD (A_0^0) = \{ & g\in L^2((a,b))\, |\: g \in AC_{loc} ((a,b));\: p_0^{1/2} g' \in L^2((a,b)) \}. \lb{3.14} \end{align} Then one infers \begin{equation} \clD (\check{H}_0^0)\: \subset\: \clD (A_0^0) \lb{3.15} \end{equation} since \begin{equation} p_0^{1/2} f' = p_0^{-1/2} (p_0 f') \in L^2 ((a,b)),\quad f \in \clD (\check{H}_0^0) \lb{3.16} \end{equation} using $p_0^{1/2} \in L_{loc}^{2} ((a,b))$ and $p_0^{1/2} f' \in L^{p} ((a,b)) \cap AC_{loc} ((a,b))$, $1 \leq p \leq \infty$ since $\supp (f) \subset (a,b)$ is compact. We also introduce \begin{equation} (\check{A}_0^0 f) (x) = p_0 (x)^{1/2} f' (x),\quad f \in \clD (\check{A}_0^0) = \clD (\check{H}_0^0). \lb{1.17a} \end{equation} Next we define the form $\check{Q}_0^0$ in $L^2 ((a,b))$, \begin{equation} \check{Q}_0^0 (f,g) = (f,\check{H}_0^0 g),\quad \clD (\check{Q}_0^0) = \clD (\check{H}_0^0). \lb{3.17} \end{equation} One verifies, \begin{align}\nn \check{Q}_0^0 (f,g) &= -\lim_{\mbox{\raisebox{-.7ex}{$\stackrel{c \downa a} {\scriptstyle {d \upa b} }$}}} \int_c^d \ol{f (x)} (p_0 (x) g' (x))'\, dx \: = -\lim_{\mbox{\raisebox{-.7ex}{$\stackrel{c \downa a} {\scriptstyle {d \upa b} }$}}} \ol{f (x)} p_0 (x) g' (x) |_{x=c}^d \\ &+ \lim_{\mbox{\raisebox{-.7ex}{$\stackrel{c \downa a} {\scriptstyle {d \upa b} }$}}} \int_c^d p_0 (x) \ol{f' (x)} g' (x)\, dx\: = (\check{A}_0^0 f,\check{A}_0^0 g),\quad f,\: g\, \in \clD (\check{H}_0^0). \lb{3.18} \end{align} Since $\check{H}_0^0 \geq 0$, the form $\check{Q}_0^0$ is closable (\cite{kato}, p. 318) and we denote its closure by $Q_0^0$, that is, $Q_0^0 = \ol{\check{Q}_0^0}$. On the other hand, a form $Q_{S}$ in some Hilbert space $\clH$ of the type \begin{equation} Q_{S} (f,g) = (S f,S g),\quad \clD (Q_{S}) =\clD (S) \lb{3.19} \end{equation} is closable (closed) if and only if $S$ is closable (closed). In particular, if $Q_S$ is closable, its closure is given by \begin{equation} \ol{Q_S} (f,g) = (\ol{S} f,\ol{S} g),\quad \clD (\ol{Q_{S}}) =\clD (\ol{S}), \lb{3.20} \end{equation} with $\ol{S}$ the operator closure of $S$ in $\clH$ (see \cite{kato}, p. 311, 317). Applied to \eqref{3.18} this yields that $\check{A}_0^0$ is closable and that the Friedrichs extension $H_0^0$ of $\check{H}_0^0$, that is, the unique self--adjoint nonnegative operator associated with $Q_0^0 = \ol{\check{Q}_0^0}$, \begin{equation} Q_0^0 (f,g) = (\ol{\check{A}_0^0} f,\ol{\check{A}_0^0} g),\quad f, g \in \clD (\ol{\check{A}_0^0}) \lb{3.21} \end{equation} ( $\ol{\check{A}_0^0}$ the operator closure of ${\check{A}_0^0}$ ), is given by \begin{equation} H_0^0 = \Big(\ol{\check{A}_0^0} \Big)^{\ast}\, \ol{\check{A}_0^0} \lb{3.22} \end{equation} (cf. \cite{fgzz}, Theorem 2.7, \cite{kato}, p. 322--326, \cite{wsch}). We note that \begin{equation} \clD (\ol{\check{A}_0^0}) \subseteq \{ g \in L^2 ((a,b))\,| \: g \in AC_{loc} ((a,b));\: p_0^{1/2} g' \in L^2 ((a,b)) \} = \clD (A_0^0) \lb{3.23} \end{equation} since $A_0^0$ is a closed operator in $L^2 ((a,b))$, that is, \begin{equation} \ol{A_0^0} = A_0^0. \lb{3.24a} \end{equation} \eqref{3.24a} directly proves that $\check{A}_0^0$ is closable. Even though \eqref{3.24a} might seem obvious, we prefer to provide a straightforward proof. Pick $\{ f_{n} \}_{n \in \N} \subset \clD (A_0^0)$ and assume \begin{equation} f_{n} \underset{n \rightarrow {\infty}} {\stackrel{s}{\lra}} f \in L^2 ((a,b)) \quad \text{and} \quad p_0^{1/2} f_n' \underset{n \rightarrow {\infty}} {\stackrel{s}{\lra}} g \in L^2 ((a,b)). \lb{3,25a} \end{equation} In order to prove \eqref{3.24a} one needs to show that \begin{equation} f \in \clD (A_0^0) \quad \text{and} \quad g = p_0^{1/2} f'. \lb{3.26a} \end{equation} Since $f_n \in AC_{loc} ((a,b))$, we may write \begin{equation} f_n (x) = \int_c^x f_n' (t)\, dt + f_n (c),\quad c,\, x \in (a,b). \lb{3.27a} \end{equation} Next, since $g \in L^2 ((a,b))$ and $p_0^{-1/2} \in L_{loc}^2 ((a,b))$ by \eqref{2.1}, one infers $p_0^{-1/2} g$ $\in L_{loc}^1 ((a,b))$ and hence $\int_c^x p_0 (t)^{-1/2} g (t)\,dt, \quad c, x \in (a,b)$ is well--defined. One obtains by Cauchy's inequality, \begin{align}\nn \Big| \int_c^x \big[ f_n' (t) - & p_0 (t)^{-1/2} g (t) \big] \,dt \Big| \leq \left( \int_c^x p_0 (t)^{-1}\, dt \right)^{1/2} \left( \int_c^x | p_0 (t)^{1/2} f_n' (t) - g(t) |^2\, dt \right)^{1/2}\\ & \leq \left( \int_c^x p_0 (t)^{-1}\, dt \right)^{1/2} \prl p_0^{1/2} f_n' - g \prl_{2} \underset{n \ra {\infty}}{\lra} 0,\quad x \in (a,b). \lb{3.28a} \end{align} Next, using \eqref{3.27a}, one estimates \begin{equation} | f_n (c) - f_m (c) |^2 \leq 2 | f_n (x) - f_m (x) |^2 +2 \left( \int_c^x | f_n' (t) - f_m' (t) |\, dt\right)^2, \lb{3.29a} \end{equation} and choosing $\eps > 0$ such that $(c-\eps, c+\eps) \subset (a,b)$, integrating \eqref{3.29a} from $c-\eps$ to $c+\eps$ yields \begin{align}\nn &2\eps |f_n (c)-f_m (c)|^2 \\ \nn &\leq 2\prl f_n - f_m {\prl}_2^2 +2 \int_{c-\eps}^{c+\eps} \left( \int_c^x p_0 (t)^{-1}\,dt \right)\left( \int_c^x |p_0 (s)^{\frac12} f_n' (s)- p_0 (s)^{\frac12} f_m' (s) |^2\,ds\right)\,dx \\ &\leq 2 \prl f_n - f_m {\prl}_2^2 +2 \prl p_0^{1/2} f_n' - p_0^{1/2} f_m' \prl_2^2 \int_{c-\eps}^{c+\eps} \left( \int_c^x p_0 (t)^{-1}\,dt \right)\,dx \underset{{n,m} \ra {\infty}}{\lra} 0. \lb{3.30a} \end{align} Thus $\{ f_n (c) \}_{n \in \N}$ is a Cauchy sequence and \begin{equation} f_n (c) \underset{n \ra {\infty}}{\lra} F \in \C \lb{3.31a} \end{equation} implying \begin{equation} f_n (x) \underset{n \ra {\infty}}{\lra} \int_c^x p_0 (t)^{-1/2} g (t) \,dt + F\quad \text{pointwise} \lb{3.32a} \end{equation} for all $x \in (a,b)$. Since $f_{n} \underset{n \rightarrow {\infty}} {\stackrel{s}{\lra}} f$ in $L^2 ((a,b))$ guarantees the existence of a subsequence $\{ f_{n_{k}} (x) \}_{k \in \N} \subset \{ f_n (x) \}_{n \in \N}$ which converges $a.e.$ to $f (x)$ on $(a,b)$, that is, \begin{equation} f_{n_{k}} (x) \underset{ k \ra \infty}{\lra} f (x)\quad \text{for a.e.}\: x\in (a,b), \lb{3.33a} \end{equation} uniqueness of the limit yields \begin{equation} f (x) = \int_c^x p_0 (t)^{-1/2} g (t) \,dt + F \quad \text{for a.e.}\: x \in (a,b). \lb{3.34a} \end{equation} In particular, this proves \begin{equation} f \in AC_{loc} ((a,b)),\: f' = p_0^{-1/2} g \quad \text{a.e. on}\: (a,b),\: \text{and}\: p_0^{1/2} f' \in L^2((a,b)) \lb{3.35a} \end{equation} and hence \eqref{3.26a}. The inclusion \eqref{3.23} yields \begin{equation} Q_0^0 (f,g) = \int_a^b p_0 (x) \ol{f' (x)} g' (x) \,dx,\quad f,\: g \in \clD (\ol{\check{A}_0^0}) = \clD (Q_0^0). \lb{3.36a} \end{equation} Finally we need to introduce the following closed forms in $L^2 ((a,b))$, \begin{equation} Q_{q_0} (f,g) = ( |q_0|^{1/2} f, \sgn (q_0) |q_0|^{1/2} g ),\quad \clD (Q_{q_0}) = \clD (|q_0|^{1/2}), \lb{3.24} \end{equation} \begin{equation} Q_{q_{0,\pm}} (f,g) = ( |q_{0,\pm}|^{1/2} f, \sgn (q_{0,\pm}) |q_{0,\pm}|^{1/2} g ),\quad \clD (Q_{q_{0,\pm}}) = \clD (|q_{0,\pm}|^{1/2}), \lb{3.25} \end{equation} where \begin{equation} q_{0,\pm} (x) =\big[ |q_0 (x)| \pm q_0 (x) \big] /2, \lb{3.26} \end{equation} and \begin{equation} Q_{q_{\mu}^{0}} (f,g) = ( |q_{\mu}^{0}|^{1/2} f, \sgn (q_{\mu}^{0}) |q_{\mu}^{0}|^{1/2} g ),\quad \clD (Q_{q_{\mu}^{0}}) = \clD (|q_{\mu}^{0}|^{1/2}), \quad \mu \in \R, \lb{3.27} \end{equation} where \begin{equation} q_{\mu}^{0} (x) = \mu p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4} \left( \int_a^x p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \right)^{-2}. \lb{3.28} \end{equation} Given these preliminaries we finally obtain the following generalization of Theorem \ref{t3.1}. \begin{thm}\lb{t3.2} Assume $p_0,\: q_0$ and $\psi_0 (\lam_0)$ satisfy (H.2.1) near $a$ and $b$, $\psi_0 (\lam_0,x)$ %\linebreak[4] $> 0$ for all $x \in (a,b)$, and suppose $|\int_a p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2} \,dx| < \infty$ and %\linebreak[4] $|\int^b p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-2} \,dx| = \infty$. In addition, assume that $\clD (|q_{0,+}|^{1/2}) \supseteq \clD (Q_0^0)$. Then for all $\phi \in \clD (Q_0^0) = \clD (\ol{\check{A}_0^0})$, \begin{align}\nn \int_a^b p_0 (x) |\phi' (x)|^2 \,dx \geq \int_a^b &\Big[ \lam_0 - q_0 (x) +4^{-1} p_0 (x)^{-1} \psi_0 (\lam_0,x)^{-4}\\ & \times \Big( \int_a^b p_0 (x)^{-1} \psi_0 (\lam_0,t)^{-2}\, dt \Big)^{-2} \Big] |\phi (x)|^2 \,dx. \lb{3.29} \end{align} Moreover, if $\psi_0 (\lam_0,\cdot) \Big( \int_a^{\bullet} p_0 (t)^{-1} \psi_0 (\lam_0,t)^{-2} \,dt \Big)^{1/2}\not\in \clD (\ol{\check{A}_0^0})$, one obtains strict inequality in \eqref{3.29} for $\phi \neq 0$. \end{thm} \begin{proof} By \eqref{3.6} and \eqref{3.15} we have \begin{align} &\int_a^b p_0 (x) |\phi' (x)|^2 \,dx \geq \int_a^b \big[ \lam_0 - q_{0,+} (x) +q_{0,-} (x) + |q_{-1/4}^0 (x)| \big] |\phi (x)|^2 \,dx,\quad \phi \in \clD (\check{Q}_0^0) = \clD (\check{H}_0^0). \lb{3.30} \end{align} Since $\clD (q_{0,+}^{1/2}) \supseteq \clD (\ol{\check{A}_0^0})$ implies \begin{equation} \prl q_{0,+}^{1/2} f \prl \leq C \prl \ol{\check{A}_0^0} f \prl + D \prl f \prl,\quad f \in \clD (\ol{\check{A}_0^0}) \lb{3.31} \end{equation} (cf. \cite{kato}, p. 191), and $\clD (\check{H}_0^0)$ is a form core of $Q_0^0$, \eqref{3.30} extends to all $\phi \in \clD (Q_0^0)$. \eqref{3.36a} then yields \eqref{3.29}. The last part is clear from the proof of Theorem \ref{t3.1}. \end{proof} Applied to the special case where $q_0 (x) = \lam_0 = 0$, $p_0 (x) = \psi_0 (0,x) = 1$, \eqref{3.29} shows that \eqref{1.12} extends to all \begin{align} &\phi \in \{ g \in L^2 ((a,\infty)) |\: g \in AC_{loc} ([a,R]),\: R >a;\: \lim_{\eps \downa 0} g (a+\eps) =0;\: g'\in L^2 ((a,\infty)) \}, \lb{3.32} \end{align} which is of course well--known (cf. \cite{hdy}, \cite{hdy1}, \cite{hlp}, p. 240). A detailed history of Hardy's inequality can be found in \cite{hkjw}. For generalizations of Hardy's inequality in different directions and a detailed bibliography see, for instance, \cite{fg}, \cite{fglp}, \cite{hkjw}, \cite{opic} and the references therein. We note that a condition of the type \eqref{3.31} is necessary to ensure that each individual term of the right--hand--side of \eqref{3.29} is finite for $\phi \in \clD (\ol{\check{A}_0^0})$. 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}. Our final result below indicates how to bound the number of eigenvalues of the operator $H$ in $L^2 ((a,b))$ associated with $\tau= \tau_0+ q (x), \: x\in (a,b)$ below the infimum $\lam_0$ of the spectrum of $H_0$ (associated with $\tau_0$). We shall assume that $\tau_0$ is in the l.p. case at $a$ and $b$. We start with a few preparations concerning Green's functions associated with $\tau_0$ on $(a,b)$ and $(x_0,b)$, $(a,x_0)$, $x_0 \in (a,b)$, respectively. First, assume that $(\tau_0 -\lam_0)$ is subcritical on $(a,b)$ (cf. \cite{fgzz}) and denote by ${\psi_{0,\pm} (\lam_0,x) > 0}$ the corresponding principal solutions of $\tau_0 \psi= \lam_0 \psi$ near $a$ and $b$, that is, \begin{equation} \Big|\int_a p_0 (x)^{-1} \psi_{0,-} (\lam_0,x)^{-2} \, dx\Big| = \infty = \Big|\int^b p_0 (x)^{-1} \psi_{0,+} (\lam_0,x)^{-2} \, dx\Big|. \lb{3.33} \end{equation} Since $\tau_0$ is assumed to be in the l.p. case at $a$ and $b$, denote by $G_0 (z,x,x')$ the Green's function of the uniquely associated operator $H_0$ (cf. \eqref{2.4}), that is, the integral kernel of $(H_0 -z)^{-1}$. For $z= \lam_0$ one obtains \begin{align} &G_0 (\lam_0,x,x') = \begin{cases} \psi_{0,+} (\lam_0,x) \psi_{0,+} (\lam_0,x') \int_{a}^{\min (x,x')} p_0 (y)^{-1} \psi_{0,+} (\lam_0,y)^{-2}\,dy\\ \psi_{0,-} (\lam_0,x) \psi_{0,-} (\lam_0,x') \int_{\max (x,x')}^{b} p_0 (y)^{-1} \psi_{0,-} (\lam_0,y)^{-2}\, dy \end{cases} \hspace*{-3mm},\quad x , \: x' \in (a,b). \lb{3.34} \end{align} Next, assume that $(\tau_0 -\lam_0)$ is critical on $(a,b)$. Then $G_0 (\lam_0,x,x')$ does not exist, however, the restrictions $(\tau_0 -\lam_0)|_ {\mbox{\raisebox{-.7ex}{$\stackrel{(x_0,b)} {\scriptstyle {(a,x_0)}}$}}}$ are subcritical and the corresponding Green's functions $G_{0,\pm}^D (\lam_0,x,x')$ associated with the Dirichlet operators $H_{0,+}^D$ (resp. $H_{0,-}^D$) in $L^2 \big( (x_0,b) \big)$ (resp. $L^2 ((a,x_0))$), \begin{equation} H_{0,\pm}^D f = \tau_0 f, \lb{3.35} \end{equation} $$ f\in \clD (H_{0,\pm}^D) := \Big\{ g\in L^2 \big({\mbox{\raisebox{-.7ex} {$\stackrel{(x_0,b)}{\scriptstyle {(a,x_0)}}$}}}\big) |\: g, p_0 g' \in AC \big( {\mbox{\raisebox{-.7ex} {$\stackrel{[x_0,R]} {\scriptstyle {[R,x_0]}}$}}} \big),\:R \in {\mbox{\raisebox{-.7ex} {$\stackrel{(x_0,b)}{\scriptstyle {(a,x_0)}}$}}},\: g(x_{0,\pm})= 0, \: \tau_0 g\in L^2 \big({\mbox{\raisebox{-.7ex} {$\stackrel{(x_0,b)}{\scriptstyle {(a,x_0)}}$}}}\big) \Big\}, $$ then read \begin{align} & G_{0,+}^D (\lam_0,x,x') = \psi_{0,+} (\lam_0,x) \psi_{0,+} (\lam_0,x') \int_{x_0}^{\min(x,x')} p_0 (y)^{-1} \psi_{0,+} (\lam_0,y)^{-2}\,dy,\quad x,\: x' \in (x_0,b), \lb{3.36}\\ & G_{0,-}^D (\lam_0,x,x') = \psi_{0,-} (\lam_0,x) \psi_{0,-} (\lam_0,x') \int_{\max(x,x')}^{x_0} p_0 (y)^{-1} \psi_{0,-} (\lam_0,y)^{-2}\,dy,\quad x,\: x' \in (a,x_0). \lb{3.37} \end{align} Next, consider $q \in L_{loc}^1 (\R)$ real--valued with \begin{equation} \int_a^b \psi_{0,\pm} (\lam_0,x)^2 \Big| \int_{\mbox{\raisebox{-.7ex} {$\stackrel{a}{\scriptstyle {b}}$}}}^x p_0 (y)^{-1} \psi_{0,\pm} (\lam_0,y)^{-2}\, dy \Big| q_{-} (x) \, dx < \infty \lb{3.38} \end{equation} if $(\tau_0 -\lam_0)$ is subcritical and \begin{equation} \int_a^b \psi_{0,\pm} (\lam_0,x)^2 \Big| \int_{x_0}^x p_0 (y)^{-1} \psi_{0,\pm} (\lam_0,y)^{-2}\, dy \Big| q_{-} (x) \, dx < \infty \lb{3.39} \end{equation} if $(\tau -\lam_0)$ is critical. Here we abbreviated, as usual, \begin{equation} q_{\pm} (x) = \big[ \: |q (x)| \pm q (x) \big]/2. \lb{3.40} \end{equation} Introducing \begin{equation} \tau = \tau_0 + q (x),\quad x \in (a,b), \lb{3.41} \end{equation} one concludes that $\tau$ is in the l.p. case at $a$ and $b$ because of \eqref{3.38} and \eqref{3.39}. This follows from \eqref{3.38} in the subcritical case since \begin{equation} q_{-}^{1/2} \big( H_0 -\lam_0 \big)^{-1/2} \in \clB_{2} \big( L^2((a,b)) \big) \lb{3.42} \end{equation} and hence \begin{equation} q_{-}^{1/2} \big( H_0 -\lam_0 \big)^{-1} q_{-}^{1/2} \in \clB_{1} \big( L^2((a,b)) \big) \lb{3.43} \end{equation} as a consequence of the fact that \begin{align} &\|q_{-}^{1/2} \big( H_0 -\lam_0 \big)^{-1/2} \|_{2} ^2 = Tr \big[ q_{-}^{1/2} \big( H_0 -\lam_0 \big)^{-1} q_{-}^{1/2} \big]=\int_a^b G_0 (\lam_0,x,x) q_{-} (x) \, dx < \infty. \lb{3.44} \end{align} Here $\clB_{p} \big( L^2((a,b) \big)$, $p= 1,2$ denote the usual trace class and Hilbert--Schmidt ideals and $\|\cdot\|_{2}$ abbreviates the Hilbert--Schmidt norm. In the critical case one replaces $H_0$ by $H_{0,\pm}^D$ and uses \eqref{3.39} instead. Given these preliminaries, a standard application of Birman--Schwinger techniques (see, e.g., \cite{fgbs}, \cite{mkl}, \cite{resm4}, Sect. XIII.3, \cite{simo}, Ch. III, \cite{sim}) yields \begin{thm}\lb{t3.3} Suppose $p_0, q_0, q $ satisfy (H.2.1) near $a$ and $b$ and $q$ satisfies \eqref{3.38} in the subcritical case and \eqref{3.39} in the critical case. In addition, assume that $\quad \lam_0 \leq \inf\sig (H_0)$, and let $H$ be the unique operator associated with $\tau$ in $L^2 ((a,b))$ according to \eqref{2.4}. If $E_H (\Ome)$, $\Ome \subseteq \R$ denotes the family of spectral projections of $H$, then \begin{align} &\dim \Ran\big\{ E_H \big( (-\infty,\lam_0] \big)\big\} \leq \int_a^b \psi_{0,\pm} (\lam_0,x)^2 \Big| \int_{\mp \infty}^x p_0 (y)^{-1} \psi_{0,\pm} (\lam_0,y)^{-2}\, dy \Big| q_{-}(x)\, dx \lb{3.45} \end{align} if $(\tau_0 -\lam_0)$ is subcritical on $(a,b)$, and \begin{align} &\dim \Ran\big\{ E_H \big( (-\infty,\lam_0] \big) \big\} \leq 1+ \int_a^b \psi_{0,\pm} (\lam_0,x)^2 \Big| \int_{x_0}^x p_0 (y)^{-1} \psi_0 (\lam_0,y)^{-2}\,dy \Big| q_{-}(x)\, dx \lb{3.46} \end{align} if $(\tau_0 -\lam_0)$ is critical on $(a,b)$. \end{thm} \begin{proof} If $(\tau_0 -\lam_0)$ is subcritical it suffices to consider the trace of $q_{-}^{1/2}( H_0 - \lam_0)^{-1} q_{-}^{1/2}$ (cf. \eqref{3.44}). If $(\tau_0 -\lam_0)$ is critical one considers the half--line operators $q_{-}^{1/2} (H_{0,\pm}^D$ $-\lam_0 )^{-1} q_{-}^{1/2}$ and then notes that \begin{equation} \big( H_0 -z\big)^{-1} -\big[ \big( H_{0,-}^D -z \big)^{-1} \oplus \big( H_{0,+}^D -z \big)^{-1} \big] \lb{3.47} \end{equation} is a rank--one operator producing the 1 on the right--hand--side of \eqref{3.46}. The estimates are first obtained for $E_H \big( (-\infty,\lam_0) \big)$ and then extended to $E_H \big( (-\infty,\lam_0] \big)$ by a standard limiting argument (see, e.g., \cite{simo}, Ch. 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