Content-Type: multipart/mixed; boundary="-------------1203090842880" This is a multi-part message in MIME format. ---------------1203090842880 Content-Type: text/plain; name="12-32.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="12-32.keywords" Dirichlet Laplacian, cusped regions, eigenvalue estimates ---------------1203090842880 Content-Type: application/x-tex; name="curvedcusp120309.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="curvedcusp120309.tex" \documentclass[10pt]{article} %\usepackage[cp866]{inputenc} %\usepackage[russian]{babel} \usepackage{amsmath, amssymb, amsthm, mathrsfs} \usepackage{enumerate} \usepackage{colordvi} %\usepackage{epsfig} %\usepackage{picture} %\overfullrule5pt \newtheorem{claim}{Claim}[section] \newtheorem{theorem}[claim]{Theorem} %\newtheorem{proposition}[claim]{Proposition} %\newtheorem{lemma}[claim]{Lemma} \newtheorem{remark}[claim]{Remark} %\newtheorem{remarks}[claim]{Remarks} %\newtheorem{example}[claim]{Example} \newtheorem{corollary}[claim]{Corollary} \textwidth=125 mm \textheight=195 mm \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \begin{document} \begin{center} {\Large{\textbf{Spectral estimates for Dirichlet Laplacians \\ [.01em] and Schr\"odinger operators on geometrically \\ [.3em] nontrivial cusps}}} \bigskip {\large{Pavel Exner and Diana Barseghyan}\footnote{The research was partially supported by the Czech Science Foundation within the project P203/11/0701.}} \bigskip \emph{To Elliott Lieb on the occasion of his 80th birthday.} \end{center} \bigskip \textbf{Abstract.} The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schr\"odinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds. \bigskip \textbf{Mathematical Subject Classification (2010).} 35P15, 81Q10. \bigskip \textbf{Keywords.} Dirichlet Laplacian, cusped regions, eigenvalue estimates %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{s: intro} \setcounter{equation}{0} The object of our interest in this paper will be Schr\"odinger type operators % ------------- % \begin{equation} \label{cusp-schroed} H_\Omega=-\Delta_D^\Omega-V \end{equation} % ------------- % with a bounded measurable potential $V\ge 0$ on $L^2(\Omega)$, where $-\Delta_D^\Omega$ is the Dirichlet Laplacian on a region $\Omega\subset \mathbb{R}^d$. We will be particularly interested is situations where $\Omega$ is unbounded but $H_\Omega$ still has a purely discrete spectrum. This is not the case, of course, for most open regions; a necessary condition is the quasi-boundedness of $\Omega$ which means the requirement \cite{AF03} % ------------- % $$ \lim_{\stackrel{x\in\Omega}{|x|\rightarrow\infty}} \mathrm{dist}(x,\partial\Omega)=0\,. $$ % ------------- % Nevertheless, it is well known --- see \cite{Si83} or \cite{GW11} and references therein --- that for some unbounded regions the spectrum may be purely discrete; typically it happens if $\Omega$ has cusps. The negative spectrum of $H_\Omega$ consists of a finite number of eigenvalues counted with their multiplicities. In this situation one can ask about bounds on the negative spectrum moments in terms of their geometrical properties, in the spirit the seminal work of of Lieb and Thirring \cite{LT76}, or in the present context referring to Berezin, Lieb, Li and Yau \cite{Be72a, Be72b, Li73, LY83}. Estimates of this type have been derived recently in \cite{GW11} for various cusped regions; a typical example is $\Omega = \{ (x,y)\in\mathbb{R}^2:\: |xy|<1 \}$ with hyperbolic ends. Our aim is the present paper is to investigate situations when such infinite cusps of $\Omega$ are geometrically nontrivial being either curved or twisted and to find in which way does the geometry influence the spectral estimates. First we note that if $\Omega$ can be regarded as a union of subsets of a different geometrical nature it could be useful to make a decomposition and to find estimates from those for separate parts using, say, bracketing technique. Motivated by this observation we will consider in this paper always a single or double cusp-shaped region $\Omega$. We will start from discussing the simplest case of a curved planar cusp and derive estimates on negative spectrum moments which include a curvature-induced potential describing the effective attractive interaction coming from the region geometry. After this motivating considerations we are going to proceed to generalization to curved cusps in $\mathbb{R}^d,\: d\ge 2$; before doing that we shall present in Section~\ref{s: BLY} an example showing that for regions with finitely cut cusps the obtained inequality can be at intermediate energies much stronger than the usual estimate using phase-space volume. In Section~\ref{s: twisted} we will consider cusps of a non-circular cross section in $\mathbb{R}^3$ which are straight but twisted. The geometry of the region will be again involved in the obtained eigenvalue estimates, now in a different way than for curved cusps, because the effective interaction associated with twisting is repulsive rather than attractive. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{A warm-up: curved planar cusps} \label{s: twodim} \setcounter{equation}{0} We start with an unbounded cusp-shaped $\Omega\subset \mathbb{R}^2$ assuming that its boundary is sufficiently smooth, to be specified below. To describe the region $\Omega$ we fix first a curve regarded as its axis and employ the natural locally orthogonal coordinates in its vicinity, in analogy with the theory of quantum waveguides \cite{ES89}, which will be used to ``straighten'' the cusp translating its geometric properties into those of the coefficients of the resulting operator. To be specific, we characterize our region by three functions: sufficiently smooth $a,b:\: \mathbb{R} \to \mathbb{R}^2$ and a positive continuous $f:\: \mathbb{R} \to \mathbb{R}^+$, in such a way that % ------------- % \begin{equation} \label{Omega-2} \Omega :=\{ (a(s)-u \dot b(s),b(s)+u \dot a(s)):\: s\in\mathbb{R},\, |u|0$, and to look for its negative spectrum. Consider first again the region (\ref{Omega-2}) satisfying the condition (\ref{Omega-cusp}) together with % ------------- % \begin{eqnarray} \|f\gamma\|_\infty < c < \frac{-\pi-1+\sqrt{(\pi+1)^2 +4\pi}}{2} \approx 0.655\,, \label{fgamma} \\[.5em] \max\{ \|f\dot\gamma\|_\infty,\, \|f\ddot\gamma\|_\infty\} < 1\,. \label{fgammader} \phantom{AAAAAAA} \end{eqnarray} % ------------- % Put $W_\Lambda^-(s):=W^-(s)+\Lambda$. In view of the assumption (\ref{fgamma}) and Theorem~\ref{thm: 2D bound} we can estimate $\mathrm{tr}\,(H_\Omega)_-^\sigma$ for any $\sigma\ge3/2$ from above by % ------------- % \begin{eqnarray*} \lefteqn{ \left\|1+f|\gamma|\right\|_\infty^{-2\sigma}\, L_{\sigma,1}^{\mathrm{cl}}\int_{\mathbb{R}}\sum_{j=1}^\infty \left(-\left(\frac{\pi\,j}{2f(s)}\right)^2 +\left\|1+f|\gamma| \right\|_\infty^2W_\Lambda^-(s)\right)_+^{\sigma+1/2}\,\mathrm{d}s} \\ && \le\left\|1+f|\gamma|\right\|_\infty\,L_{\sigma,1}^{\mathrm{cl}} \int_{f(s)\ge\frac{\pi}{2(1+c)}W_\Lambda^-(s)^{-1/2}} \sum_{j=1}^{\left[2(1+c)f(s) W_\Lambda^-(s)^{1/2}/\pi\right]} W_\Lambda^-(s)^{\sigma+1/2}\,\mathrm{d}s \\ && \le\frac{2(1+c)}{\pi}\,\left\|1+f|\gamma|\right\|_\infty\,L_{\sigma,1}^{\mathrm{cl}} \int_{f(s)\ge\frac{\pi}{2(1+c)} W_\Lambda^-(s)^{-1/2}} W_\Lambda^-(s)^{\sigma+1}\,f(s)\,\mathrm{d}s \\ && \le\frac{8}{\pi}\,L_{\sigma,1}^{\mathrm{cl}}\int_{ f(s)\ge\pi\left(\left(\frac{1+c}{1-c}\right)^2\gamma(s)^2+ 4(1+c)^2\Lambda\right)^{-1/2}} W_\Lambda^-(s)^{\sigma+1}\,f(s)\,\mathrm{d}s\,. \end{eqnarray*} % ------------- % Notice that for a fixed $f$ the right-hand side of the last inequality reaches its maximum if $\gamma(s)f(s)=c$. Consequently, setting $\alpha_c^2:= \frac{\pi^2-c^2(1+c)^2/(1-c)^2}{4(1+c)^2}$ which is a positive number under the assumption (\ref{fgamma}) we get % ------------- % \begin{eqnarray} \lefteqn{\mathrm{tr}\,(H_\Omega)_-^\sigma\le\frac{8}{\pi}\, L_{\sigma,1}^{\mathrm{cl}} \int_{f(s)\ge\alpha_c\Lambda^{-1/2}} \left(\frac{c^2}{4(1-c)^2f^2(s)}+\Lambda\right)^{\sigma+1}\,f(s)\,\mathrm{d}s} \nonumber \\ && \label{eq2.11} \le\frac{8}{\pi}\,\left(\frac{c^2}{4(1-c)^2\alpha_c^2}+1\right)^{\sigma+1}\, L_{\sigma,1}^{\mathrm{cl}}\,\Lambda^{\sigma+1} \int_{f(s)\ge\alpha_c\Lambda^{-1/2}} f(s)\,\mathrm{d}s\,; \phantom{AAA} \end{eqnarray} % ------------- % note that the curvature is present in this estimate through the constant $c$ only. Let us now show that such an estimate can be stronger than the phase-space bound mentioned above which says that the operator $H_{\Omega'}$ defined by (\ref{cusp-schroed}) on an open bounded region $\Omega'$ with constant potential $V=\Lambda,\,\Lambda>0$ satisfies % ------------- % \begin{equation} \label{eq2.12} \mathrm{tr}\left(H_{\Omega'}\right)_-^\sigma\le L_{\sigma,1}^{\mathrm{cl}}\Lambda^{\sigma+1}\, \mathrm{vol}\,(\Omega')\,,\quad \sigma\ge1\,. \end{equation} % ------------- % First we shall construct an unbounded cusped region $\Omega$ determined by functions $\gamma$ and $f$ satisfying the conditions (\ref{Omega-cusp}), (\ref{fgamma}) and (\ref{fgammader}), then we will pass to cut-off regions $\Omega'\subset\Omega$ such that % ------------- % \begin{equation} \label{eq2.13} \mathrm{tr}\left(H_\Omega'\right)_-^\sigma\le \mathrm{tr}\left(H_\Omega\right)_-^\sigma\,,\quad \sigma\ge0\,. \end{equation} % ------------- % We choose an arbitrary positive number $\alpha$ and a natural number $N$ and set % ------------- % \begin{eqnarray} f_{\alpha,N}(x): =\frac{\pi}{2}\:x^{-1-\alpha} \qquad\text{for}\quad |x|>N\,, \\ f_{\alpha,N}(x): =\frac{\pi}{2}\:N^{-1-\alpha}\qquad\text{for}\quad |x|\le N\,. \end{eqnarray} % ------------- % Since we have mentioned that the curvature does not play a substantial role in the estimate (\ref{eq2.11}) we put it equal to zero and consider the straight region $\Omega_{\alpha,N} :=\{x\in \mathbb{R}:\: |y|0$ and $A\subset \mathbb{R}^2$. Using (\ref{eq4.1}) we define a straight cusped region determined by $\omega_o$ and the function $f$ as $\Omega_0:= \left\{ (s,x,y):\: s\in\mathbb{R},\,(x,y)\in\omega_s\right\}$ with $\omega_s$. In the next step we twist the region. We fix a $C^1$-smooth function $\theta:\mathbb{R}\rightarrow\mathbb{R}$ with bounded derivative, $\|\dot \theta\|_\infty <\infty$, and introduce the region $\Omega_\theta$ as the image % ------------- % \begin{equation} \label{eq4.2} \Omega_\theta:=\mathfrak{L}_\theta(\Omega_0)\,, \end{equation} % ------------- % where the map $\mathfrak{L}_\theta:\,\mathbb{R}^3\to \mathbb{R}^3$ is given by % ------------- % \begin{equation} \label{eq4.3} \mathfrak{L}_\theta(s,x,y):= \left(s,x\cos\theta(s)+y\sin\theta(s),-x\sin\theta(s)+y\cos\theta(s)\right)\,. \end{equation} % ------------- % We are interested primarily in the situation when the region is twisted, that is % ------------- % \begin{enumerate}[(i)] \setlength{\itemsep}{0pt} \item the function $\theta$ is not constant\,, \item $\omega_0$ is not rotationally symmetric with respect to the origin in $\mathbb{R}^2$. \end{enumerate} If the first condition is not valid we have a straight region with the cross section rotated by a fixed angle, $\omega_{0,\theta}:= \left\{x\cos\theta+y\sin\theta ,-x\sin\theta+y\cos\theta\,:\,(x,y)\in\omega_0 \right\}$, while if $\omega_0$ is rotationally symmetric (i.e. $\omega_{0,\theta} = \omega_0$ up a set of zero capacity for any $\theta\in(0,2\pi)$) the choice of $\theta$ does not matter. To formulate the result of this section, we need a few more preliminaries. First of all, we introduce $\varrho := \sup_{(x,y)\in\omega_0}\sqrt{x^2+y^2}$ and assume that % ------------- % \begin{equation} \label{eq4.4} \varrho\|f\dot{\theta}\|_\infty<1\,. \end{equation} % ------------- % Next we we set $\widetilde{V}(s,x,y) := V(\mathfrak{L}_\theta(s,x,y))$ in analogy with the corresponding definitions in the previous sections, and finally, we introduce the operator % ------------- % $$ L_\mathrm{trans} := -i\left(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right)\,,\quad \mathrm{Dom} \big(L_\mathrm{trans} \big) = \mathcal{H}^1_0 (\omega_0)\,, $$ % ------------- % of the angular momentum component canonically associated with rotations in the transverse plane. Now we are ready to state the result. % ------------- % \begin{theorem} \label{twist bound} Let $H_{\Omega_\theta}$ be the operator (\ref{cusp-schroed}) referring to the region $\Omega_\theta$ defined by (\ref{eq4.2}) and (\ref{eq4.3}) with a potential $V\ge 0$ which is bounded and measurable. Under the assumption (\ref{eq4.4}) the negative spectrum of $H_{\Omega_\theta}$ the inequality % ------------- % $$ \mathrm{tr}\big(H_D^{\Omega_\theta}\big)_-^\sigma \le L_{\sigma,1}^{\mathrm{cl}} \left(1-\varrho\|f\dot{\theta}\|_{\infty}\right)^\sigma \int_{\mathbb{R}}\sum_{j=1}^\infty \left(-\frac{\lambda_{0,j}(s)}{f^2(s)} +\frac{\|\widetilde{V}(s,\cdot)\|_\infty}{1-\varrho\|f\dot{\theta}\|_\infty} \right)_+^{\sigma+1/2}\,\mathrm{d}s $$ % ------------- % holds true for $\sigma\ge3/2$, where $L_{\sigma,1}^{\mathrm{cl}}$ is the constant (\ref{LTconstant}) and $\lambda_{0,j}(s),\, j=1,2,\ldots,\,$ are the eigenvalues of the operator % ------------- % $$ H_{f,\theta}(s):=-\Delta_D^{\omega_0}+f^2(s)\dot{\theta}^2(s) L_\mathrm{trans}^2 $$ % ------------- % defined on the domain $\mathcal{H}^2_0 (\omega_0)$ in $L^2(\omega_0)$. \end{theorem} % ------------- % \begin{proof}[Proof:] As before we employ suitable curvilinear coordinates, this time to ``untwist'' the region. We define a unitary operator from $L^2(\Omega_\theta)$ to $L^2(\Omega_0)$ by $U_\theta\psi:= \psi\circ\mathfrak{L}_\theta$ which allows us to pass from $H_{\Omega_\theta}^D$ to the operator % ------------- % $$ %\begin{equation} \label{eq4.5} H_0:=U_\theta\left(H_{\Omega_\theta}^D\right)U^{-1}_\theta $$ %\end{equation} % ------------- % in $L^2(\Omega_0)$. From \cite{KZ11} we know that $H_0$ is the self-adjoint operator associated with the quadratic form % ------------- % $$ Q_0:\: Q_0[\psi]:=\|\partial_s\psi+i\dot{\theta} L_\mathrm{trans} \psi\|^2 +\|\nabla_\mathrm{trans}\psi\|^2 -\int_{\Omega_0} \big(\widetilde{V}|\psi|^2\big) (s,x,y)\,\mathrm{d}s\,\mathrm{d}x\,\mathrm{d}y $$ % ------------- % defined on $\mathcal{H}_0^1$, where $\nabla_\mathrm{trans}:= \left( \partial_x,\partial_y\right)$ and the norms refer to $L^2(\Omega_0)$. In order to estimate the form we note that % ------------- % $$ |L_\mathrm{trans}\phi|\le \varrho f(s)|\nabla_\mathrm{trans}\phi| $$ % ------------- % holds for any function $\phi\in\mathcal{H}_0^1(\omega_s)$, hence using Cauchy-Schwarz we get % ------------- % $$ 2\,\mathrm{Re}\left|\int_{\Omega_0} \dot{\theta}(s) \big(\partial_s\psi\, \overline{L_\mathrm{trans}\psi}\big) (s,x,y) \,\mathrm{d}s\,\mathrm{d}x\,\mathrm{d}y\right| \le \varrho\|f\dot{\theta}\|_\infty(\|\partial_s\psi\|^2 +\|\nabla'\psi\|^2)\,, $$ % ------------- % where the last two norm refer again to $L^2(\Omega_0)$, which in turn yields % ------------- % \begin{eqnarray} \lefteqn{Q_0[\psi] \ge \big(1-\varrho\|f\dot{\theta}\|_{\infty}\big) \big(\|\nabla\psi\|^2+ \|\dot{\theta}L_\mathrm{trans}\psi\|^2 \big)} \nonumber \\[.5em] && \label{eq4.5} \hspace{1em} -\int_{\Omega_0}\|\widetilde{V}(s,\cdot)\|_\infty|\psi (s,x,y)|^2\, \mathrm{d}s\,\mathrm{d}x\,\mathrm{d}y\,. \end{eqnarray} % ------------- % Introducing thus the operator % ------------- % $$ H^-_0=-\Delta_D^{\Omega_0}+\dot{\theta}^2(s) L_\mathrm{trans}^2 -\frac{1}{1-\varrho\|f\dot{\theta}\| _\infty}\|\widetilde{V}(s,\cdot)\|_\infty $$ % ------------- % with the domain $\mathcal{H}_0^2(\Omega_0)$, we get from (\ref{eq4.5}) the following lower bound, % ------------- % \begin{equation} \label{eq4.6} H_0\ge \big(1-\varrho\|f\dot{\theta}\|_\infty \big)H_0^-\,, \end{equation} % ------------- % which makes sense in view of the condition (\ref{eq4.4}); by minimax principle it is then enough to establish a bound to the negative spectrum of the operator $H_0^-$. For any $u\in\,C_0^\infty(\Omega_0)$ we can write % ------------- % \begin{eqnarray*} \lefteqn{\|\nabla\,u\|^2+\|\dot{\theta}L_\mathrm{trans}u\|^2- \frac{1}{1-\varrho\|f\dot{\theta}\|_\infty}\int_{\Omega_0} \|\widetilde{V}(s,\cdot)\|_\infty|u(s,x,y)|^2\,\mathrm{d}s\, \mathrm{d}x\,\mathrm{d}y} \\ && = \int_{\Omega_0}|\partial_s u(s,x,y)|^2\, \mathrm{d}s\,\mathrm{d}x \,\mathrm{d}y +\int_{\mathbb{R}}\, \mathrm{d}s\int_{\omega_s} \bigg(|\partial_x u(s,x,y)|^2 +|\partial_y u(s,x,y)|^2 \\ && \hspace{1em} +\dot{\theta}^2(s) \big|(L_\mathrm{trans}u)(s,x,y)\big|^2- \frac{1}{1-\varrho\|f\dot{\theta}\|_\infty} \|\widetilde{V}(s, \cdot)\|_\infty|u(s,x,y)|^2\bigg)\,\mathrm{d}x\, \mathrm{d}y \\ && \ge\int_{\Omega_0}|\partial_s u(s,x,y)|^2\,\mathrm{d}s\,\mathrm{d}x\, \mathrm{d}y+\int_{\mathbb{R}}\left\langle\,H(s,\widetilde{V})\,u(s,\cdot),\, u(s,\cdot)\right\rangle_{L^2(\omega_s)}\,\mathrm{d}\,s\,, \end{eqnarray*} % ------------- % where the norm without a label refer again to $L^2(\Omega_0)$ and $H(s,\widetilde{V})$ is the negative part of two-dimensional Schr\"{o}dinger operator % ------------- % \begin{equation} \label{eq4.7} -\Delta_D^{\omega_s}+\dot{\theta}^2(s) L_\mathrm{trans}^2 -\frac{1}{1-\varrho\|f\dot{\theta}\|_\infty}\|\widetilde{V}(s, \cdot)\|_\infty \end{equation} % ------------- % defined on $\mathcal{H}_0^2(\omega_s)$. The next step is analogous to what we did in the proofs of Theorems~\ref{thm: 2D bound} and \ref{thm: gen bound}: we extend the operator $H(s,\widetilde{V})$ to the whole $\mathbb{R}^2$ by regarding it as a direct sum with zero component in $C_0^\infty\left(\mathbb{R}^2\setminus \overline{\omega_s}\right)$. For a function $g=u+v$ with $u\in\,C_0^\infty(\Omega_0)$ and $v\in\,C_0^\infty(\widehat{\Omega}_0)$ extended by zero in the complement regions in $\mathbb{R}^3$ we have thus the inequality % ------------- % \begin{eqnarray*} \lefteqn{\|\nabla\,u\|^2 + \|\nabla\,v\|^2_{L^2(\widehat{\Omega}_0)} +\|\dot{\theta}L_\mathrm{trans}u\|^2} \\ && \hspace{1.5em} -\frac{1}{1-\varrho\|f\dot{\theta}\|_\infty}\int_{\Omega_0} \|\widetilde{V}(s,\cdot)\|_\infty|u(s,x,y)|^2\,\mathrm{d}s\, \mathrm{d}x\,\mathrm{d}y \\ && \ge\int_{\Omega_0}|\partial_s g(s,x,y)|^2\,\mathrm{d}s\,\mathrm{d}x\, \mathrm{d}y+\int_{\mathbb{R}}\left\langle\,H(s,\widetilde{V})\,u(s,\cdot),\, u(s,\cdot)\right\rangle_{L^2(\mathbb{R}^2)}\,\mathrm{d}s\,, \end{eqnarray*} % ------------- % valid for all $g\in\,C_0^\infty\left(\mathbb{R}^3 \backslash \partial\Omega_0\right)$. Its left-hand side of is the form associated with the operator $H_0^-\oplus \big(-\Delta_D^{\widehat{\Omega}_0} \big)$ while the right-hand side is associated with the operator $-\partial^2_s\otimes\, I_{L^2(\mathbb{R}^2)}+H(s,\widetilde{V})$ defined on the enlarged domain $\mathcal{H}^1\left(\mathbb{R}, L^2(\mathbb{R}^2) \right)$. Using the positivity of $-\Delta_D^{\widehat{\Omega}_0}$ we get % ------------- % $$ \mathrm{tr}\,\left(H^-_0\right)_-^\sigma \le \,\mathrm{tr}\left(-\partial^2_s\otimes\,I_{L^2(\mathbb{R}^2)}+ H(s,\widetilde{V})\right)_-^\sigma\,,\quad \sigma\ge 0\,, $$ % ------------- % hence the Lieb-Thirring inequality for operator-valued potentials yields % ------------- % \begin{equation} \label{eq4.10} \mathrm{tr}\,\left(H^-_0\right)_-^\sigma\le\,L_{\sigma,1}^{\mathrm{cl}} \int_{\mathbb{R}} \mathrm{tr} \,H(s,\widetilde{V})_-^{\sigma+1/2}\,\mathrm{d}s\,,\quad \sigma\ge3/2\,, \end{equation} % ------------- % with the semiclassical constant $L_{\sigma,1}^{\mathrm{cl}}$ given by (\ref{LTconstant}). Combining thus the unitary equivalence of $H_D^{\Omega_\theta}$ and $H_0$ with the inequalities (\ref{eq4.6}), (\ref{eq4.10}) and the condition (\ref{eq4.4}) we get % ------------- % \begin{equation} \label{eq4.11} \mathrm{tr}\left(H_D^{\Omega_\theta}\right)_-^\sigma \le\,L_{\sigma,1}^{\mathrm{cl}} \left(1-\varrho\|f\dot{\theta}\|_{\infty}\right)^\sigma \int_{\mathbb{R}} \mathrm{tr}\,H(s,\widetilde{V})_-^{\sigma+1/2}\, \mathrm{d}s \quad\mathrm{for}\quad \sigma\ge3/2\,. \end{equation} % ------------- % It remains to determine the eigenvalues of the operator (\ref{eq4.7}). Since the potential in it is independent of the transverse variables, it is easy to see that they are % ------------- % $$ \frac{\lambda_{0,j}(s)}{f^2(s)}-\frac{1}{1-\varrho\|f\dot{\theta}\|_\infty} \|\widetilde{V}(s,\cdot)\|_\infty\,,\quad j=1,2,\ldots\,, $$ % ------------- % where $\lambda_{0,j}(s),\,j=1,2,\ldots$ are the eigenvalues of the operator $H_{f,\theta}(s)$ defined in the theorem. Combining this result with (\ref{eq4.11}) we conclude the proof. \end{proof} Once more, the result of Theorem\ref{twist bound} can be easily extended to twisted regions with one-sided cusps in analogy with the claim of Corollary~\ref{2cor}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{10} \bibitem[AF03]{AF03} R.A.Adams, J.F.Fournier, \textit{Sobolev spaces}, 2nd ed., Academic Press, New York 2003. \bibitem[Be72a]{Be72a} F.A.~Berezin, Covariant and contravariant symbols of operators, \emph{Izv. Akad. Nauk SSSR Ser. Mat.} \textbf{36} (1972), 1134--167. \bibitem[Be72b]{Be72b} F.A.~Berezin, Convex functions of operators, \emph{Mat. Sb. (NS)} \textbf{36} (130) (1972), 268--276. \bibitem[BS72]{BS72} M.S.~Berger, M.~Schechter, Embedding theorems and quasi-linear elliptic boundary value problems for unbounded domain, \emph{Trans. Am. Math. Soc.} \textbf{172} (1972), 261--278. \bibitem[BEK96]{BEK96} M.~Bordag, E.~Elizalde, K.~Kirsten, Heat-kernel coefficients of the Laplace operator on the $D$-dimensional ball, \emph{J. Math. Phys.} \textbf{37} (1996), 895--916. \bibitem[CDFK05]{CDFK05} B.~Chenaud, P.~Duclos, P.~Freitas, D.~Krejcirik \textit{Geometrically induced discrete spectrum in curved tubes}, Differential Geom. Appl. \textbf{23} (2005), 95--105. \bibitem[E\v{S}89]{ES89} P.Exner, P.~\v{S}eba, Bound states in curved quantum wavequides, \emph{J. Math. Phys.} \textbf{30} (1989), 2574--2580. \bibitem[GW11]{GW11} L.~Geisinger, T.~Weidl, Sharp spectral estimates in domain of infinite volume, \emph{Rev. Math. Phys.} \textbf{23} (2011), 615--641. \bibitem[KZ11]{KZ11} D.~Krej\v{c}i\v{r}\'{\i}k, E.~Zuazua, The Hardy inequality and the heat equation in the twisted tubes, \emph{J. Diff. Eqs} \textbf{250} (2011), 2334--2346. \bibitem[LW00]{LW00} A.~Laptev, T.~Weidl, Sharp Lieb-Thirring inequalities in high dimensions, \emph{Acta Math.} \textbf{184} (2000), 87--100. \bibitem[LY83]{LY83} P.~Li, S.T.~Yau, On the Schr\"odinger equation and the eigenvalue problem, \emph{Comm. Math. Phys.} \textbf{88} (1983), 309--318. \bibitem[Li73]{Li73} E.H.~Lieb, The classical limit of quantum spin systems, \emph{Commun. Math. Phys.} \textbf{31} (1973), 327-–340. \bibitem[LT76]{LT76} E.H. Lieb, W.~Thirring, Inequalities for the moments of the eigenvalues of the Schr\"odinger Hamiltonian and their relation to Sobolev inequalities, in \emph{Studies in Math. Phys., Essays in Honor of Valentine Bargmann} (E.~Lieb, B.~Simon and A.S.~Wightman, eds.); Princeton Univ. Press, Princeton, 1976; pp.~269--330. \bibitem[Si83]{Si83} B.~Simon, Some quantum operators with discrete spectrum but classically continuous spectrum, \emph{Ann. Phys.}, Amer. Math. Soc. Transl. \textbf{146} (1983), 209--220. \bibitem[Wei08]{Wei08} T.~Weidl, Improved Berezin-Li-Yau inequalities with a remainder term, in \emph{Spectral Theory of Differential Operators}, Amer. Math. Soc. Transl. \textbf{225} (2008), 253--263. \end{thebibliography} \bigskip \begin{flushleft} Pavel Exner and Diana Barseghyan \smallskip Doppler Institute for Mathematical Physics and Applied Mathematics \\ B\v{r}ehov\'{a} 7, 11519 Prague \\ and Nuclear Physics Institute ASCR \\ 25068 \v{R}e\v{z} near Prague, Czechia \smallskip Email: exner@ujf.cas.cz, dianabar@ujf.cas.cz \end{flushleft} \end{document} ---------------1203090842880--