% % AMS-LATEX % % \documentstyle[12pt,amssymb,amstex]{amsart} \sloppy \newcounter{const}[section] \newcommand{\co}{c_{\thesection .\refstepcounter{const}\arabic{const}}} \newcommand{\bl}{|\hspace{-1mm}|\hspace{-1mm}|} \newtheorem{tm}{\\ $\ \ \ \$ Theorem} \newtheorem{lm}{\\ $\ \ \ \$ Lemma} \newtheorem{cl}{\\ $\ \ \ \$ Corollary} \newtheorem{pr}{\\ $\ \ \ \$ Proposition} \newtheorem{df}{\\ $\ \ \ \$ Definition} \newcounter{rem} \newcommand{\rema}{{\bf Remark \refstepcounter{rem}\arabic{rem}.\ }} \begin{document} \title[On Lieb-Thirring inequalities]{On Lieb-Thirring inequalities for higher order operators with critical and subcritical powers.} \author{Y. Netrusov and T. Weidl} \date{\today} \address{ School of Mathematical Sciences, University of Sussex at Brighton, BN1 9QH Brighton Falmers, UK (phone int+44-(0)1273-607555, fax int+44-(0)1273-678097). T. Weidl also Royal Institute of Technology, Department of Mathematics, S-10044 Stockholm,Sweden (phone: int+46-(0)8-7906194, fax: int+46-(0)8-7231788). Y. Netrusov also POMI, Fontanka 27, 191011 St. Petersburg, Russia. } \email{weidl@@math.kth.se,mmfd3@@sussex.ac.uk} \thanks{The first author was supported by the EPSRC grant GR/J 32084. The second author was supported by {\em Deutsche Forschungsgemeinschaft} grant We 1964-1.} \maketitle \begin{abstract} Let $\varkappa_i(H_l(V))$ denote the negative eigenvalues of the operator $H_l(V)u:=(-\Delta)^lu-V(x)u,\ V\geq 0,\ x\in{\Bbb R}^d$ on $L_2({\Bbb R}^d).$ We prove the two-sided estimate % $\tilde{{\frak L}}(d,l)\int_{{{\Bbb R}}^d} V(x)dx\leq \sum_k|\varkappa_k(H_l(V))|^{1-\kappa} \leq {\frak L}(d,l,1-\kappa)\int_{{{\Bbb R}}^d} V(x)dx,\ \ \kappa=d/2l<1.$ We discuss bounds on the Riesz means $\sum_k|\varkappa_k(H_l(V))|^\mu$ if $0<\mu<1-\kappa.$ \end{abstract} \section{Introduction} \subsection{} We consider the quadratic form % ${\bold h}_l(V)[u,u]:=\int_{{\Bbb R}^d}|\nabla^l u|^2dx - \int_{{\Bbb R}^d}V|u|^2dx, \ \ 0\leq V\in L_1^{\mbox{loc}}({\Bbb R}^d),\ \ l\in{\Bbb N}_+,$ % defined on functions $u\in C_0^\infty({\Bbb R}^d).$ If the function $V$ vanishes properly at infinity, this form can be closed. Its closure generates the self-adjoint operator % \begin{equation}\label{DL} H_l(V):=(-\Delta)^l- V(x) \end{equation} % on $L_2({\Bbb R}^d),$ the negative spectrum of which is discrete and bounded from below. Let $\{\varkappa_k(H_l(V))\}$ stand for the non-decreasing, finite or infinite sequence of the negative eigenvalues of the operator $H_l(V).$ Estimates on the negative spectrum of operators $H_l(V)$ in terms of the potential $V$ have been studied for many years, see e.g. \cite{Barg}, \cite{B}, \cite{Schwinger}, \cite{Rbl}, \cite{C}, \cite{[L]}, \cite{LiYau}, \cite{EK}. For given $d, l$ define % \begin{equation}\label{nukappa} \kappa=\kappa(d,l):=\frac{d}{2l}, \ \ \ \ \nu=\nu(d,l):=1-\frac{d}{2l}. \end{equation} % In \cite{LT} Lieb and Thirring proved the inequalities % \begin{equation}\label{LiTh} S_{l,\mu}(V):=\sum_k|\varkappa_k(H_l(V))|^\mu\leq {\frak L}(d,l,\mu)\int_{{{\Bbb R}}^d} V^{\mu+\kappa}(x)dx, \end{equation} % for potentials $0\leq V\in L_{\mu+\kappa}({\Bbb R}^d)$ with $\mu>\max\{0,\nu\}$ in the case $l=1.$ Their argument can easily be extended to arbitrary $l\in{\Bbb N}_+.$ In \cite{Rbl}, \cite{C} and \cite{[L]} the respective inequality was shown for $\mu=0$ if $\nu<0.$ On the other hand it is known, that \eqref{LiTh} fails for $0\leq\mu<\nu$ if $\nu>0$ and for $\mu=0$ if $\nu=0.$ In \cite{W} the author verified \eqref{LiTh} for $l=d=1$ and $\mu=\nu(1,1)=1/2,$ where in fact the two-sided estimate % \begin{equation}\label{Weidl} \frac{1}{4}\int_{{\Bbb R}}Vdx\leq S_{1,1/2}(V) \leq 1.005 \int_{{\Bbb R}}Vdx,\ \ d=l=1,\ 0\leq V\in L_1({\Bbb R}), \end{equation} % holds, cf. \cite{GGM}. In this note we prove \eqref{LiTh} for the remaining case of a positive critical power $\mu=\nu(d,l)>0$ for arbitrary $d,\ l\in{\Bbb N}_+,$ such that $2l>d.$ In analogy to \eqref{Weidl} we find a two-sided estimate % \begin{equation}\label{WeiNe} \tilde{{\frak L}}(d,l)\int_{{{\Bbb R}}^d} V(x)dx\leq S_{l,\nu}(V)\leq {\frak L}(d,l,\nu)\int_{{{\Bbb R}}^d} V(x)dx, \end{equation} % which holds for all sumable, non-negative potentials $0\leq V\in L_1({\Bbb R}^d)$ with certain constants $0<\tilde{\frak L}(d,l)\leq{\frak L}(d,l,\nu)<\infty.$ It is well-known that \eqref{LiTh} is of sharp order in the limit of large potentials. This follows from the Weyl type asymptotical formula % \begin{equation}\label{Wl} S_{l,\mu}(\alpha V)=\alpha {\frak L}^{\mbox{cl}}(d,l,\mu)\int_{{\Bbb R}^d}V^{\mu+\kappa}dx + o(\alpha) \ \ \mbox{as}\ \alpha\to\infty, \end{equation} % \begin{equation}\label{Wclas} {\frak L}^{\mbox{cl}}(d,l,\mu)= \frac{\mu\Gamma(\mu)\Gamma(\kappa+1)} {2^d\pi^{d/2}\Gamma(\frac{d}{2}+1)\Gamma(\kappa+\mu+1)}, \ \ \mu\geq 0, \end{equation} % which can be obtained for sufficiently regular non-negative potentials, and which can be closed to all potentials $0\leq V\in L_{\mu+\kappa}({\Bbb R}^d)$ if \eqref{LiTh} holds. On the other hand for $\nu>0$ the operator $H_l(\alpha V),\ 0\leq V, 0\not\equiv V$ has negative spectrum for arbitrary small $\alpha>0,$ and for sufficiently regular, non-negative potentials the asymptotics % \begin{equation}\label{ato0} S_{l,\mu}(\alpha V)=\bigl(\alpha {\frak L}^{\mbox{0}}(d,l,\nu) \int_{{\Bbb R}^d}Vdx\bigr)^{\mu/\nu} + o(\alpha^{\mu/\nu} )\ \ \mbox{as}\ \ \alpha\to 0, \ \mu>0, \end{equation} % ${\frak L}^{\mbox{0}}(d,l,\nu)=\frac{\pi\kappa} {\sin\pi\kappa} {\frak L}^{\mbox{cl}}(d,l,0),$ % can be calculated. \footnote{We attach the proof of \eqref{ato0} in the Appendix.} In the case of a positive critical power $\mu=\nu>0$ this asymptotics is of the same type as \eqref{WeiNe}, and we can close \eqref{ato0} with $\mu/\nu=1$ to all potentials $0\leq V\in L_1({\Bbb R}^d).$ Comparing \eqref{ato0} and \eqref{Wl} we see, that a two-sided estimate can hold only in the critical case. Naturally formula \eqref{Wl} agrees with the estimate \eqref{LiTh} for supercritical powers $0<\nu<\mu .$ However, in the scale of subcritical powers $0<\mu<\nu$ we find $\mu/\nu<\mu+\kappa,$ and \eqref{Wl} disproves \eqref{LiTh}. Hence a proper substitute of \eqref{LiTh} for positive subcritical powers should contain two terms on the right hand side: one of homogeneity order $\mu/\nu$ serving for small coupling constants, and one of Weyl type order $\mu+\kappa,$ serving as $\alpha\to\infty.$ In the final section of this paper we shall prove such estimates. The authors are grateful to M. Birman, B. Davies, D. Edmunds, A. Laptev, G. Rosenblyum and D. Vassiliev for many stimulating discussions. \subsection{Notations.} Below ${\Bbb Q}^d=\{x=(x_1,...,x_d)\in{\Bbb R}^d:|x_j|\leq 1/2, j=1,...,d\}.$ Moreover ${\Bbb N}=\{n\in{\Bbb Z}:n\geq 0\},$ while ${\Bbb N}_+={\Bbb N}\backslash\{0\}.$ For a multiindex ${\boldsymbol{\iota}}\in{\Bbb N}^d$ we use the notations $|\boldsymbol{\iota}|=\sum_{j=1}^d\iota_j$ and $\boldsymbol{\iota}!=\prod_{j=1}^d\iota_j!.$ The vector $\nabla^k$ consists of the elements $\sqrt{\frac{k!}{{\boldsymbol{\iota}}!}} \frac{\partial^{\boldsymbol{\iota}}} {\partial x^{\boldsymbol{\iota}}}$ with $|{\boldsymbol{\iota}}|=k.$ Further $\Omega_{d,k}$ stands for the $\binom{k+d}{d}$-dimensional lineal of all polynomials over ${\Bbb R}^d,$ the order of which does not exceed $k.$ Throughout the paper $\kappa$ and $\nu$ are defined as in \eqref{nukappa}. Finally, if the self-adjoint operator $T$ is semi-bounded from below and its lower portion of the spectrum is discrete, then $\{\varkappa_k(T)\}$ denotes the non-decreasing sequence of the respective eigenvalues (according to their multiplicity). \section{The Lieb-Thirring inequality for positive critical powers} \subsection{Main result.} In this section we shall prove \begin{tm}\label{Hdtm} Assume $d,l\in{\Bbb N}_+$ and $\nu=1-d/2l>0.$ Then for all potentials $V(x)\geq 0,\ V\in L_1({\Bbb R}^d),$ the inequality % \begin{equation}\label{highD} \tilde{{\frak L}}(d,l)\int_{{\Bbb R}^d}V dx\leq S_{l,\nu}(V) \leq {\frak L}(d,l,\nu) \int_{{\Bbb R}^d}V dx \end{equation} % holds. \end{tm} \subsection{Two covering Lemmata.} We introduce \begin{df} Let $0\leq V(x)\in L_1({\Bbb R}^d)$ have compact support. A family ${\bold Q}=\{{\cal Q}_\tau\}$ of cubes ${\cal Q}_\tau=x_\tau+a_\tau {\Bbb Q}^d, \ x_\tau\in{\Bbb R}^d,\ a_\tau>0,$ is called a $A$-proper covering of {\normalshape $\mbox{supp}\ V$} of multiplicity $\Xi({\bold Q}),$ if {\normalshape $\mbox{supp} V\subseteq \cup_\tau {\cal Q}_\tau ,$ } % \begin{equation}\label{13} a_\tau^{2l-d}\int_{{\cal Q}_\tau} Vdx =A, \ \ A>0,\ \ \mbox{and} \ \ \Xi({\bold Q}):=\sup_{x\in{\Bbb R}^d} \sum_{\tau:x\in\mbox{{\normalshape int}}\ {\cal Q}_\tau} 1 \ <\ \infty. \end{equation} % \end{df} The following result dates back to Besikovic \cite{Bes}. For the convenience of the reader we give its proof and follow the argument of de Guzman \cite{Guzman}. \begin{lm} For each non-trivial potential $0\leq V\in L_1({\Bbb R}^d)$ of compact support and any fixed $A>0$ there exists some finite $A$-proper covering ${\bold Q}(V)$ of {\normalshape $\mbox{supp}\ V$} of multiplicity $\Xi({\bold Q}(V))\leq 2^d.$ \end{lm} {\bf Proof.} We can assume $V\not\equiv 0.$ Then for each $x\in{\Bbb R}^d$ there exists a unique $a(x)>0,$ such that for ${\cal Q}_x=x+a(x){\Bbb Q}^d$ the equality % $a^{2l-d}(x)\int_{{\cal Q}_x} Vdx =A$ % holds. The function $a:{\Bbb R}^d\to {\Bbb R}_+$ is continuous and bounded from below by % \begin{equation}\label{bfbel} a(x)\geq \Bigl(A^{-1}\int_{{\Bbb R}^d} V dx\Bigr)^{\frac{1}{d-2l}}>0 . \end{equation} % Choose $\tilde{{\bold Q}}=\{{\cal Q}_x:x\in\mbox{supp}\ V\}.$ We shall select the sought finite proper covering as an appropriate subset from $\tilde{{\bold Q}}.$ Assume we have already chosen the points $x_i,\ i=1,...,m$ and the respective cubes ${\cal Q}_{x_i}.$ Then let $x_{m+1}$ be one of the points $x$, where the continuous function $a(x)$ achieves its maximum value on the compact set $x\in \mbox{supp} V\backslash \cup_{i=1}^m\mbox{int}{\cal Q}_{x_i}.$ Since the interiors of the cubes $x_i+\frac{a(x_i)}{2}{\Bbb Q}^d$ do not intersect each other, by \eqref{bfbel} this process stops after a finite number of iterations, and we put ${\bold Q}(V)=\{{\cal Q}_{x_i}\}.$ Evidently $\mbox{supp} V \subseteq \cup_i{\cal Q}_{x_i}.$ Let us show, that $\Xi({\bold Q}(V))\leq 2^d.$ Each of the points $x_i$ does not belong to the interior of any other cube than ${\cal Q}_{x_i}.$ Fix some point $y\in{\Bbb R}^d,\ y\neq x_i.$ Assume $y\in \cap_{k=1}^r \mbox{int}{\cal Q}_{x_{i_r}}$ with $x_{i_p}\neq x_{i_q}$ for all $1\leq p\neq q\leq r$ and $r>2^d.$ Let $\vec{\boldsymbol{\iota}}=(\iota_1,...,\iota_d)$ denote vectors of the type $\iota_k\in\{0,1\},\ k=1,...,d.$ Then one of the $2^d$ sectors $\Sigma_{y,\vec{\boldsymbol{\iota}}} :=y+\otimes_{k=1}^d [0,(-1)^{\iota_k}\infty)$ should contain more then one of the points $x_{i_p},\ p=1,...,r.$ On the other hand, if $x_{i_p}, x_{i_q}\in \Sigma_{y,\vec{\boldsymbol{\iota}}},\ |y-x_{i_p}|\leq |y-x_{i_q}|, p\neq q,$ and $y\in\mbox{int} {\cal Q}_{x_{i_p}}\cap\mbox{int} {\cal Q}_{x_{i_q}},$ then $x_{i_p}\in\mbox{int}{\cal Q}_{x_{i_q}},$ what contradicts to the construction. Thus $r\leq 2^d.\ \Box$ \vspace{2mm} We supplement Lemma 1 by \begin{lm}\label{extract} Assume $2l>d.$ Then there exists a positive constant $\tilde{c}(d,l)$ such, that from each finite $A$-proper covering ${\bold Q}(V)=\{{\cal Q}_i\}_{i=1}^n, \ {\cal Q}_i=x_i+a_i{\Bbb Q}$ of {\normalshape $\mbox{supp}\ V$} of multiplicity $\Xi({\bold Q}(V))\leq 2^d$ for a non-trivial potential $0\leq V\in L_1({\Bbb R}^d)$ of compact support one can extract a subset ${\bold Q}^\sharp(V)= \{{\cal Q}_i\}_{i\in I},\ I\subseteq\{1,...,n\}$ with the properties % \begin{equation}\label{nonintersect} x_i+2a_i{\Bbb Q}\cap x_j+2a_j{\Bbb Q}=\emptyset \ \ \ \mbox{for all}\ \ i\neq j,\ i,j\in I, \end{equation} % \begin{equation}\label{suminI} \sum_{i\in I}\int_{{\cal Q}_i} Vdx\geq \tilde{c}(d,l)\int_{{\Bbb R}^d}Vdx. \end{equation} % \end{lm} {\bf Proof.} Put $I_0=J_0=\emptyset,\ M_0=\{1,...,n\}.$ Assume the sets $I_k, J_k, M_k$ have already been constructed. If $M_k=\emptyset$ we abbreviate the process and take $I=I_k.$ Otherwise choose $i_{k+1}$ such that $a_{i_{k+1}}=\min_{j\in M_k} a_j,$ and take % $I_{k+1}=I_k\cup \{i_{k+1}\}, \ \ \ \ J_{k+1}=\{j\in M_k: x_j+2a_j{\Bbb Q}\cap x_{i_{k+1}}+2a_{i_{k+1}}{\Bbb Q} \neq\emptyset\}\backslash\{i_{k+1}\},$ % $M_{k+1}=M_k\backslash\Bigl(J_{k+1}\cup\{i_{k+1}\}\Bigr).$ % Obviously $x_{i^\prime}+2a_{i^\prime}{\Bbb Q}\cap x_{i^{\prime\prime}}+2a_{i^{\prime\prime}}{\Bbb Q}=\emptyset$ for all $i^\prime, i^{\prime\prime}\in I.$ Moreover notice, that $a_j\geq a_{i_k}$ for $j\in J_k.$ Thus we can decompose $J_k$ as % $J_k=\cup_{m\in{\Bbb N}} J_k^m,\ \ \ J_k^m=\{j\in J_k:2^m a_{i_k}\leq a_{j}<2^{m+1}a_{i_k}\}.$ % If $j\in J_k^m,$ then ${\cal Q}_j\subset x_{i_k}+(1+3\cdot 2^{m})a_{i_k}{\Bbb Q}.$ Since $\Xi({\bold Q}(V))\leq 2^d$ and $\mbox{vol}\ {\cal Q}_j\geq 2^{md}a_{i_k}^d$ we find % $\mbox{card}\ J_k^m\leq \frac{(1+3\cdot 2^{m})^d2^d}{2^{md}} \leq 8^d.$ % Moreover % $\sum_{j\in J_k} \int_{{\cal Q}_j}Vdx= \sum_{m\in{\Bbb N}}\sum_{j\in J_k^m} Aa_j^{d-2l}\leq 8^d\sum_{m\in{\Bbb N}} 2^{m(d-2l)}Aa_{i_k}^{d-2l}= \frac{8^d}{1-2^{d-2l}}\int_{{\cal Q}_{i_k}} Vdx.$ % Since $2l>d$ we conclude % $\int_{{\Bbb R}^d} Vdx\leq\sum_{i=1}^n\int_{{\cal Q}_i}Vdx =\sum_k^{card I}\Bigl( \int_{{\cal Q}_{i_k}}Vdx+\sum_{j\in J_k}\int_{{\cal Q}_j}Vdx \Bigr) \leq\frac{1}{\tilde{c}(d,l)} \sum_k\int_{{\cal Q}_{i_k}} Vdx$ % with $\tilde{c}(d,l)=(1-2^{d-2l})/(1-2^{d-2l}+8^d)>0.\ \Box$ \subsection{The negative spectrum of the Neumann'' problem on the cube.} In what follows put ${\cal Q}=a{\Bbb Q}^d$ for some $a>0.$ Let $H^N_{l,{\cal Q}}(V)$ be the self-adjoint operator on $L_2({\Bbb Q})$, corresponding to the closure of the hermitian form % ${\bold h}^N_{l,{\cal Q}}(V)[u,u] :=\int_{\cal Q}|\nabla^l u|^2dx-\int_{\cal Q}V|u|^2dx, \ \ 0\leq V\in L_1({\cal Q}),\ \ u\in C^\infty({\cal Q}).$ % For the negative spectrum of this operator the following standard fact holds. \begin{lm} Assume $2l>d,\ l, d\in{\Bbb N}_+.$ Then there exists a positive finite constant $\hat{c}(d,l)$ such, that for all potentials $0\leq V\in L_1({\cal Q})$ with % \begin{equation}\label{12} \hat{c}(d,l)a^{2l-d}\int_{\cal Q}Vdx\leq 1,\ \ {\cal Q}=a{\Bbb Q}^d, \ \ a>0, \end{equation} % the operator $H^N_{l,{\cal Q}}(V)$ has not more than $\binom{l+d-1}{d}$ negative eigenvalues. \end{lm} {\bf Proof.} By homogeneity we can take $\hat{c}(d,l)$ as the sharp constant in the inequality % \begin{equation}\label{ineq} |u(x)|^2\leq a^{2l-d}\hat{c}(d,l)\int_{{\cal Q}}|\nabla^l u(x)|^2, \ \ {\cal Q}=a{\Bbb Q}^d,\ \ a>0, \ \ u\in W^{l}_2({\cal Q})\ominus_{L_2({\cal Q})} \Omega_{d,l-1}|_{\cal Q}, \end{equation} % which holds in view of the Sobolev embedding for $2l>d$ and the Theorem on equivalent norms. Because of \eqref{12} and \eqref{ineq} the form ${\bold h}^N_{l,{\cal Q}}[u,u]$ is non-negative on $u\in C_0^\infty({\cal Q})\ominus_{L_2({\cal Q})}\Omega_{d,l-1}.$ This subspace is of codimension $\binom{l+d-1}{d}$ in $L_2({\cal Q}),$ what by Glazmanns Lemma completes the proof. $\Box$ \subsection{The Birman-Schwinger principle for $H^N_{l,{\cal Q}}(V)$} If $2l>d$ the resolvent of the unperturbed operator $H^N_{l,{\cal Q}}(0)$ % $\bigl((H^N_{l,{\cal Q}}(0)-\varkappa)^{-1}u\bigr)(x)= \int_{\cal Q} G_{{\cal Q}}(x,z,\varkappa) u(z)dz$ % is an integral operator with a bounded continuous kernel $G_{{\cal Q}}(x,z,\varkappa)\in C({\cal Q}\times{\cal Q})$ for any $\varkappa<0,$ see \cite{Agm}. The Green function $G_{{\cal Q}}(x,z,\varkappa)$ obeys the homogeneity property % \begin{equation}\label{Dhom} G_{\cal Q}(x,z,\varkappa)= a^{2l-d}G_{\Bbb Q}^d(a^{-1}x,a^{-1}z,a^{2l}\varkappa), \ \ \ {\cal Q}=a{\Bbb Q}^d,\ \ a>0,\ \varkappa<0. \end{equation} % >From Hilberts resolvent identity one immediately concludes, that % ${\cal G}_{\cal Q}(\varkappa) :=\max_{x\in{\cal Q}} G_{\cal Q}(x,x,\varkappa)$ % is a continuous, strongly increasing function in $\varkappa<0.$ Moreover % ${\cal G}_{\cal Q}(\varkappa)\to 0 \ \ \mbox{as}\ \ \varkappa\to - \infty, \ \ \ \ \ {\cal G}_{\cal Q}(\varkappa)\to +\infty \ \ \mbox{as}\ \ \varkappa\to -0,$ % while \eqref{Dhom} implies % \begin{equation}\label{Dhom1} {\cal G}_{\cal Q}(\varkappa)= a^{2l-d}{\cal G}_{\Bbb Q}^d(a^{2l}\varkappa), \ \ \ {\cal Q}=a{\Bbb Q}^d,\ \ a>0. \end{equation} % Now let $\{\varkappa_k(H^N_{l,{\cal Q}}(V))\}_k$ denote the non-decreasing sequence of eigenvalues of $H^N_{l,{\cal Q}}(V).$ Consider the counting function % $N(\varkappa,H^N_{l,{\cal Q}}(V)):=\sum 1: \{k: \varkappa_k(H^N_{l,{\cal Q}}(V))<\varkappa\},\ \ \varkappa<0,$ % for the common multiplicity of the spectrum of $H^N_{l,{\cal Q}}(V)$ below $\varkappa<0.$ According to the Birman-Schwinger principle \cite{B}, \cite{Schwinger} this quantity can be estimated by % \begin{equation}\label{BirSchw}\begin{split} N(\varkappa,H^N_{l,{\cal Q}}(V))&\leq \mbox{Tr}\ \Bigr\{V^{1/2}(x)\int_{\cal Q}G_{\cal Q}(x,z,\varkappa)V^{1/2}(z) \cdot dz\Bigr\}\\ &\leq {\cal G}_{\cal Q}(\varkappa) \int_{\cal Q}V(x)dx= a^{2l-d}{\cal G}_{\Bbb Q}^d(a^{2l}\varkappa)\int_{\cal Q}V(x)dx. \end{split} \end{equation} % If we put $\varkappa=\varkappa_1(H^N_{l,{\cal Q}}(V))+0$, we find % $1/{\cal G}_{{\Bbb Q}^d}(a^{2l}\varkappa_1(H^N_{l,{\cal Q}}(V)))\leq a^{2l-d} \int_{\cal Q} Vdx.$ % The monotone decreasing continuous function $1/{\cal G}_{\Bbb Q}^d:{\Bbb R}_-^0\to{\Bbb R}_+^0$ has the monotone decreasing inverse ${\cal F}:{\Bbb R}_+^0\to{\Bbb R}_-^0.$ Thus for the lowest eigenvalue the estimate % \begin{equation}\label{endl} |\varkappa_1(H^N_{l,{\cal Q}}(V))|^\nu\leq a^{d-2l} \Bigl|{\cal F}\Bigl(a^{2l-d}\int_{\cal Q} Vdx\Bigr)\Bigr|^\nu, \ \ \ \nu=1-\frac{d}{2l}, \end{equation} % holds. \subsection{Proof of Theorem \ref{Hdtm} - The estimate from above.} We start with potentials $0\leq V\in L_1({\Bbb R}^d)$ with compact support. Let ${\bold Q}(V)=\{{\cal Q}_{x_1},...,{\cal Q}_{x_m}\}$ be a $A-$proper finite covering of $\mbox{supp}\ V$ with multiplicity $\Xi({\bold Q}(V))\leq 2^d$ and $A=2^{-d}/\hat{c}(d,l).$ According to \eqref{12}, \eqref{endl} and \eqref{13} each of the operators $H^N_{l,{\cal Q}_{x_i}}(2^dV)$ has not more than $\binom{l+d-1}{d}$ negative eigenvalues $\varkappa_j(H^N_{l,{\cal Q}_{x_i}}(2^dV)).$ Put $J(i)=\{j:\varkappa_j(H^N_{l,{\cal Q}_{x_i}}(2^dV))<0\}.$ Then % \begin{equation}\label{14} \sum_{j\in J(i)} |\varkappa_j(H^N_{l,{\cal Q}_{x_i}}(2^dV))|^\nu\leq 2^d \binom{l+d-1}{d}\hat{c}(d,l) |{\cal F}(\hat{c}^{-1}(d,l))|^\nu \int_{{\cal Q}_{x_i}}Vdx. \end{equation} % Using the variational principle and the estimate on the multiplicity of the covering it is easy to verify, that % \begin{equation}\label{varia} \varkappa_k(H_l(V)) \geq \varkappa_k(\hat{H}) \ \ \ \mbox{for all}\ \ k:\ \varkappa_k(H_l(V))<0, \ \ \ \hat{H}:=\oplus_i H^N_{l,{\cal Q}_{x_i}}(2^dV), \end{equation} % where $\hat{H}$ acts on $\oplus_i L_2({\cal Q}_{x_i}).$ The negative eigenvalues $\{\varkappa_k(\hat{H})\}$ of $\hat{H}$ coincides as set and in its multiplicity with the union of the sets $\{\varkappa_j(H^N_{l,{\cal Q}_{x_i}}(V))<0\}.$ For the sum of powers of negative eigenvalues of $H_l$ this implies % $\sum_k|\varkappa_k(H_l(V))|^\nu\leq \sum_{k:\varkappa_k(\hat{H})<0}|\varkappa_k(\hat{H})|^\nu =\sum_{i,j\in J(i)} |\varkappa_j(H^N_{l,{\cal Q}_{x_i}}(2^dV))|^\nu \leq {\frak L}(d,l,\nu)\int_{{\Bbb R}^d}Vdx.$ % The constant on the r.h.s. does not depend on the support of $V.$ A standard argument allows one to close this inequality to all potentials $0\leq V\in L_1({\Bbb R}^d). \Box$ \subsection{Proof of Theorem \ref{Hdtm} - The estimate from below.} Let $\hat{\cal Q}$ be some cube in ${\Bbb R}^d$ and let $H^D_{l,\hat{\cal Q}}(V)$ be the self-adjoint operator on $L_2(\hat{\cal Q})$, corresponding to the closure of the hermitian form % ${\bold h}^D_{l,\hat{\cal Q}}(V)[u,u] :=\int_{\hat{\cal Q}}|\nabla^l u|^2dx-\int_{\hat{\cal Q}}V|u|^2dx, \ \ 0\leq V\in L_1(\hat{\cal Q}),\ \ u\in C^\infty_0(\hat{\cal Q}).$ % Below $\{\varkappa_k(H^D_{l,\hat{\cal Q}}(V))\}_k$ denotes the non-decreasing sequence of eigenvalues of $H^D_{l,\hat{\cal Q}}(V).$ Fix a function $\psi\in C_0^\infty(2{\Bbb Q}),$ such that $\psi\equiv 1$ on ${\Bbb Q}.$ Put % $\varsigma:=\int_{2{\Bbb Q}}|\nabla^l\psi|^2dx, \ \ \ \vartheta:=\int_{2{\Bbb Q}}|\psi|^2dx.$ % For the lowest eigenvalue of $H^D_{l,\hat{\cal Q}}(V)$ with ${\cal Q}=a{\Bbb Q}+y,\ \hat{\cal Q}=2a{\Bbb Q}+y,\ a>0,\ y\in{\Bbb R}^d$ the variational estimate % \begin{equation} \label{turns} \varkappa_1(H^D_{l,\hat{\cal Q}}(V))\leq \frac{\int_{\hat{\cal Q}}|\nabla^l\psi(a^{-1}(x-y))|dx- \int_{\hat{\cal Q}} V|\psi(a^{-1}(x-y))|^2dx}{\int_{\hat{\cal Q}}|\psi(a^{-1}(x-y)|^2dx} \leq \frac{a^{d-2l}\varsigma-\int_{{\cal Q}}Vdx}{a^{d}\vartheta} \end{equation} % holds. For potentials $0\leq V\in L_1({\Bbb R}^d)$ of compact support we choose a finite $\kappa^{-1}\varsigma$-proper covering of the support of $V,$ and according to Lemma \ref{extract} extract the subset % ${\bold Q}^\sharp(V)=\{{\cal Q}_i\}_{i\in I}, \ \ \ {\cal Q}_i=x_i+a_i{\Bbb Q},$ % with the properties \eqref{nonintersect}, \eqref{suminI}. >From the variational principle we find, that % \begin{equation}\label{varia1} \varkappa_k(H_l(V))\leq \varkappa_k(\tilde{H}) \ \ \ \mbox{for all}\ \ k:\ \varkappa_k(H_l(V))<0, \ \ \ \tilde{H}:=\oplus_{i\in I} H^D_{l,\tilde{{\cal Q}}_{x_i}}(V), \end{equation} % where $\tilde{H}$ acts on $L_2(\cup_{i\in I}\tilde{{\cal Q}}_{x_i})$ with $\tilde{{\cal Q}}_i=x_i+2a_i{\Bbb Q}$ as $i\in I.$ For $\hat{\cal Q}=\tilde{\cal Q}_i$ \eqref{turns} turns into % $\varkappa_1(H^D_{l,\tilde{\cal Q}_i}(V))\leq -\vartheta^{-1}\nu\kappa^{\kappa/\nu} \varsigma^{-\kappa/\nu} \Bigl(\int_{{\cal Q}_i}Vdx\Bigr)^{1/\nu}.$ % The quantity on the r.h.s. is negative, thus $\varkappa_1(H^D_{l,\tilde{\cal Q}_i}(V))<0$ and % \begin{equation}\label{into} |\varkappa_1(H^D_{l,\tilde{\cal Q}_i}(V))|^{\nu}\geq \vartheta^{-\nu}\nu^{\nu} \kappa^\kappa\varsigma^{-\kappa} \int_{{\cal Q}_i}Vdx. \end{equation} % Hence from \eqref{into} and \eqref{suminI} we conclude % \begin{equation}\begin{split}\notag \sum_k|\varkappa_k(H_l(V))|^\nu &\geq \sum_{k:\varkappa_k(\tilde{H})<0}|\varkappa_k(\tilde{H})|^\nu \geq\sum_{i\in I} |\varkappa_1(H^D_{l,\tilde{{\cal Q}}_{x_i}}(V))|^\nu\\ &\geq \vartheta^{-\nu}\nu^{\nu} \kappa^\kappa\varsigma^{-\kappa} \sum_{i\in I} \int_{{\cal Q}_i}Vdx\geq \tilde{{\frak L}}(d,l)\int_{{\Bbb R}^d}Vdx, \end{split} \end{equation} % with % $\tilde{{\frak L}}(d,l)= \tilde{c}(d,l)\vartheta^{-\nu}\nu^{\nu} \kappa^\kappa\varsigma^{-\kappa}>0.$ % Closing this estimate to all $0\leq V\in L_1({\Bbb R}^d)$, we complete the proof of Theorem \ref{Hdtm}. $\Box$ \subsection{Positive supercritical powers.} Following an argument of Lieb and Aizenmann one can easily show, that Theorem \ref{Hdtm} implies % $S_{l,\mu}(V):=\sum_k|\varkappa_k(H_l(V))|^\mu\leq {\frak L}(d,l,\mu)\int_{{{\Bbb R}}^d} V^{\mu+\kappa}(x)dx$ % for all powers $\mu\geq \nu>0.$ As usual the condition $V\geq 0$ in the r.h.s. of Theorem \ref{Hdtm} can be dropped, if we substitute $V$ by $\max\{V(x),0\}$ in the integral in the r.h.s. of \eqref{highD}. Then % \begin{equation}\label{aiz1} S_{\mu,l}=\frac{1}{B(\mu-\nu,\nu+1)} \sum_m\int_0^\infty \lambda^{\mu-\nu-1}(|\varkappa_m|-\lambda)_+^\nu d\lambda \end{equation} % $\leq{\frak L}(d,l,\nu)\int_0^\infty \frac{d\lambda}{\lambda}\lambda^{\mu-\nu} \int_0^\infty dx (V(x)-\lambda)_+= \frac{{\frak L}(d,l,\nu)B(\mu-\nu,2)}{B(\mu-\nu,\nu+1)} \int V^{\mu+\frac{d}{2l}}(x)dx\ .$ % Thus ${\frak L}(d,l,\mu)$ is finite for all $\mu\geq\nu. \Box$ \subsection{Asymptotics for small coupling constants.} \begin{tm}\label{2ld$and assume the potential$0\leq V\in L_1({\Bbb R}^d)$has compact support and is not identically zero. Then there exist exactly$\binom{l+[\frac{d}{2}]}{d}$negative eigenvalues for the operator$H_l(\alpha V)$for all sufficiently small coupling constants$0<\alpha<\alpha_0(V).$\end{lm} \begin{lm}\label{aux1} Suppose$2l>d$and$0\leq V\in L_1({\Bbb R}^d).$Then the bottom eigenvalue$\varkappa_1(H_l(\alpha V))$of$H_l(\alpha V)$obeys the asymptotical formula % \begin{equation}\label{0asy} |\varkappa_1(H_l(\alpha V))|^{\nu} =\alpha {\frak L}^{\mbox{0}}(d,l,\nu) \int_{{\Bbb R}^d}Vdx + o(\alpha)\ \ \mbox{as}\ \ \alpha\to 0. \end{equation} % If$l+[\frac{d}{2}]>d$and$V$is of compact support, for the subsequent negative eigenvalues the asymptotical estimates % \begin{equation}\label{l=2} |\varkappa_j(H_l(\alpha V))|=o(|\varkappa_1(H_l(\alpha V))|) \ \ \mbox{as}\ \ \alpha\to 0, \ \ \ j\geq 2, \end{equation} % hold. \end{lm} \rema The asymptotical formula \eqref{0asy} is accompanied by the well-known estimate % \begin{equation}\label{accom} |\varkappa_1(H_l(\alpha V))|^{\nu} \leq \alpha {\frak L}^{\mbox{0}}(d,l,\nu) \int_{{\Bbb R}^d}Vdx\ , \end{equation} % which holds for all$\alpha>0$and$0\leq V\in L_1({\Bbb R}^d).$\vspace{2mm} {\bf Proof of Theorem \ref{2ld,$ remains unresolved. \vspace{2mm} \rema If $2l>d$ for compactly supported potentials $0\leq V\in L_1({\Bbb R}^d)$ the asymptotics % \begin{equation}\label{ato00} S_{l,\mu}(\alpha V)=\bigl(\alpha {\frak L}^{\mbox{0}}(d,l,\nu) \int_{{\Bbb R}^d}Vdx\bigr)^{\mu/\nu} + o(\alpha^{\mu/\nu} )\ \ \mbox{as}\ \ \alpha\to 0,\ \mu>0, \end{equation} % holds. \section{Lieb-Thirring type inequalities for subcritical powers} \subsection{Main result.} In this section we discuss substitutes for \eqref{LiTh}, if $0<\mu<\nu.$ Below ${\bold E}$ denotes the sequence of shifted unit cubes % $\{{\cal E}_{\vec{j}}\}_{\vec{j}\in{\Bbb Z}^d}:= \{{\Bbb Q}^d+\vec{j}\}_{\vec{j}\in{\Bbb Z}^d}.$ % Moreover ${\bold F}$ stands for the sequence $\{{\cal F}_j\}_{j\in{\Bbb N}}$ with ${\cal F}_1={\Bbb Q}^d$ and ${\cal F}_j:=2^j{\Bbb Q}^d\backslash 2^{j-1}{\Bbb Q}^d,\ j=2,3,...$ For a locally sumable potential we introduce the notations $\boldsymbol{\beta}^{{\bold E}}(V):= \{\beta^{{\bold E}}_{\vec{j}}(V)\}_{\vec{j}\in{\Bbb Z}^d}$ and $\boldsymbol{\beta}^{{\bold F}}(V):= \{\beta^{{\bold F}}_j(V)\}_{j\in{\Bbb N}}$ with % $\beta^{{\bold E}}_{\vec{j}}(V):=\int_{{\cal E}_{\vec{j}}}|V|dx \ \ \ \mbox{and} \ \ \ \beta^{{\bold F}}_j(V):=\int_{{\cal F}_j}|V|dx\ .$ % Norms of such sequences have been used by Birman and Solomyak \cite{BS2} to give estimates on the number of negative bound states for the operator $H_l(V)$ if $2l>d.$ We shall prove \begin{tm}\label{subLiTh} Assume that for $0\leq V\in L_1^{\mbox{{\normalshape loc}}}({\Bbb R}^d)$ the sequence $\boldsymbol{\beta}^{{\bold E}}(V)$ belongs to $\ell_{\mu/\nu}, \ 0<\mu<\nu=1-\kappa,\ \kappa=d/2l.$ Then the estimate % \begin{equation}\label{sub} S_{l,\mu}(V)\leq C(d,l,\mu)\Bigl( \|\boldsymbol{\beta}^{{\bold E}}(V) \|_{\ell_{\mu/\nu}}^{\mu/\nu}+ \|\boldsymbol{\beta}^{{\bold E}}(V) \|_{\ell_{\mu+\kappa}}^{\mu+\kappa} \Bigr) \end{equation} % holds. \end{tm} \begin{tm}\label{Cl} Assume that for $0\leq V\in L_1^{\mbox{{\normalshape loc}}}({\Bbb R}^d)$ the sequence $\boldsymbol{\beta}^{{\bold F}}((1+|x|)^{\sigma}V(x))$ belongs to $\ell_{\mu+\kappa}, \ \sigma:=d(\nu-\mu)/(\mu+\kappa), \ 0<\mu<\nu=1-\kappa,\ \kappa=d/2l.$ Put $\theta(t):=t^{\mu/\nu}+t^{\mu+\kappa}$ for all $t\geq 0.$ Then the estimate % \begin{equation}\label{subb} S_{l,\mu}(V)\leq c(d,l,\mu) \theta\Bigl(\bigl\|\boldsymbol{\beta}^{\bold F}\bigl( (1+|x|)^{\sigma}V(x)\bigr) \bigr\|_{\ell_{\mu+\kappa}}\Bigr) \end{equation} % holds. \end{tm} \subsection{Proof of Theorem \ref{subLiTh}.} First we consider potentials $0\leq V\in L_1({\Bbb R}^d)$ of compact support. Let ${\bold Q}(V)=\{{\cal Q}_{x_i}\}_{i=1}^m$ be a finite $A$-proper covering of supp $V, \ A=2^{-d}/\hat{c}(d,l).$ Combining \eqref{varia} and \eqref{14} as in the proof of Theorem \ref{Hdtm} one finds % \begin{equation}\label{hlv} S_{l,\mu}(V)=\sum_k|\varkappa_k(H_l(V))|^\mu \leq \co(d,l,\mu)\sum_i\Bigl(\int_{{\cal Q}_{x_i}}Vdx\Bigr)^{\mu/\nu}\ . \end{equation} % Put ${\cal P}_{i,\vec{j}}={\cal Q}_{x_i}\cap{\cal E}_{\vec{j}}$ and $I(\vec{j}):= \{i:\mbox{int}\ {\cal P}_{i,\vec{j}}\neq\emptyset\}, \ \ N(\vec{j})=\mbox{card}\ I(\vec{j}).$ Then % \begin{equation}\label{hhllvv}\begin{split} \sum_i\Bigl(\int_{{\cal Q}_{x_i}}Vdx\Bigr)^{\mu/\nu}&\leq \sum_{i,\vec{j}}\Bigl(\int_{{\cal P}_{i,\vec{j}}}Vdx\Bigr)^{\mu/\nu} \leq \sum_{\vec{j}}(N(\vec{j}))^{1-\frac{\mu}{\nu}} \Bigl(\sum_{i\in I(\vec{j})} \int_{{\cal P}_{i,\vec{j}}}Vdx\Bigr)^{\mu/\nu}\\ &\leq 2^{d\mu/\nu}\sum_{\vec{j}}(N(\vec{j}))^{1-\frac{\mu}{\nu}} \Bigl( \int_{{\cal E}_{\vec{j}}}Vdx\Bigr)^{\mu/\nu}\ . \end{split} \end{equation} % Next we estimate the value of $N(\vec{j}).$ Therefore we split the index set $I(\vec{j})$ into % \begin{equation}\label{p} I^\prime(\vec{j}):=\{i\in I(\vec{j}): \mbox{vol}\ {\cal Q}_{x_i}>1\}, \ \ \ I^{\prime\prime}(\vec{j})=I(\vec{j}) \backslash I^\prime(\vec{j})\ . \end{equation} % If $i\in I^\prime(\vec{j})$ then the interior of ${\cal Q}_{x_i}$ contains at least one of the corners of ${\cal E}_{\vec{j}}.$ Since the proper covering ${\bold Q}(V)$ is of a multiplicity $\Xi({\bold Q}(V))\leq 2^d,$ we have $\mbox{card}\ I^\prime(V)\leq 2^{2d}.$ On the other hand $i\in I^{\prime\prime}(\vec{j})$ implies ${\cal Q}_{x_i}\subset \vec{j}+3{\Bbb Q}^d.$ Thus from $\Xi({\bold Q}(V))\leq 2^d$ we obtain % \begin{equation}\label{1of2} \sum_{i\in I^{\prime\prime}(\vec{j})} \ \mbox{vol}\ {\cal Q}_{x_i}\leq 6^d, \end{equation} % while from \eqref{13} we deduce % \begin{equation}\label{2of2} \sum_{i\in I^{\prime\prime}(\vec{j})} \ \bigl(\mbox{vol}\ {\cal Q}_{x_i}\bigr)^{1-\kappa^{-1}}\leq 4^d\hat{c}(d,l)\int_{\vec{j}+3{\Bbb Q}^d}Vdx\ . \end{equation} % Together \eqref{1of2} and \eqref{2of2} imply % \begin{equation}\label{3of2} \bigl(\mbox{card}\ I^{\prime\prime}(\vec{j})\bigr)^{\kappa^{-1}} \leq \bigl(\sum_{i\in I^{\prime\prime}(\vec{j})} \ \mbox{vol}\ {\cal Q}_{x_i}\bigr)^{\kappa^{-1}-1} \sum_{i\in I^{\prime\prime}(\vec{j})}\bigl( \mbox{vol}\ {\cal Q}_{x_i}\bigr)^{1-\kappa^{-1}} \leq \co(d,l)\int_{\vec{j}+3{\Bbb Q}^d}Vdx\ , \end{equation} % thus % \begin{equation}\label{ppp}N(\vec{j})=\co(d,l)+\co(d,l) \Bigl(\int_{\vec{j}+3{\Bbb Q}^d} Vdx\Bigr)^\kappa\ . \end{equation} % Inserting this estimate into \eqref{hlv} and \eqref{hhllvv} we arrive at % $S_{l,\mu}(V)\leq \co(d,l,\mu) \|\boldsymbol{\beta}^{\bold E}(V)\|_{\ell_{\mu/\nu}}^{\mu/\nu} +\co(d,l,\mu) \Bigl(\sum_{\vec{j}}\int_{\vec{j}+3{\Bbb Q}^d}Vdx\Bigr)^{\mu+\kappa},$ % what is equivalent to \eqref{sub}. Since the constant in this estimate does not depend on $V,$ we can close the bound to all potentials $0\leq V$ with $\boldsymbol{\beta}(V)\in\ell_{\mu/\nu}.\ \Box$ \vspace{2mm} \subsection{Proof of Theorem \ref{Cl}.} We consider potentials of compact support and choose a $A$-finite proper covering ${\bold Q}(V)$ of multiplicity $\Xi({\bold Q}(V))\leq 2^d$ of the support of $V$ with $A=2^{-d}/\hat{c}(d,l).$ We put ${\cal P}_{i,j}={\cal Q}_{x_i}\cap{\cal F}_{j},\ j\in{\Bbb N},$ and $I(j):=\{i:\mbox{int}\ {\cal P}_{i,j}\neq\emptyset\}$ is of cardinality $N(j)=\mbox{card}\ I(j).$ In analogy to the previous proof we find % \begin{equation}\label{put} S_{l,\mu}(V)\leq \co(d,l,\mu) \sum_{j}(N(j))^{1-\frac{\mu}{\nu}} \Bigl( \int_{{\cal F}_{j}}Vdx\Bigr)^{\mu/\nu}\ . \end{equation} % Choose the decomposition % $I^\prime(j):=\{i\in I(j): \mbox{vol}\ {\cal Q}_{x_i}>\max\{2^{-d},2^{d(j-3)}\}\}, \ \ \ I^{\prime\prime}(j)=I(j) \backslash I^\prime(j)\ .$ % If $i\in I^{\prime\prime}(j)$ then % ${\cal Q}_{x_i}\subset{\cal M}_j :=\bigcup_{s=\max\{1,j-1\}}^{j+1}{\cal F}_s\ ,$ % and estimates similar to \eqref{1of2}, \eqref{2of2}, \eqref{3of2} give % \begin{equation}\label{4of2} \mbox{card}\ I^{\prime\prime}(j) \leq \co(d,l) \Bigl(\mbox{vol}\ {\cal M}_j\Bigr)^\nu \Bigl(\int_{{\cal M}_j}Vdx\Bigr)^\kappa\ . \end{equation} % A simply geometrical argument shows, that in view of $\Xi({\bold Q}(V))\leq 2^d$ the estimate % \begin{equation}\label{l} \mbox{card}\ I^{\prime}(j)\leq \co(d) \end{equation} % holds. Inserting $N(j)=\mbox{card}\ I^\prime(j)+ \mbox{card}\ I^{\prime\prime}(j)$ with \eqref{4of2} and \eqref{l} into \eqref{put}, we claim % \begin{equation}\label{scd} S_{l,\mu}(V)\leq \co(d,l,\mu)\Bigl\{ \sum_{j}\Bigl( \int_{{\cal M}_{j}}Vdx\Bigr)^{\mu/\nu} + \sum_{j}\Bigl(\mbox{vol}\ {\cal M}_j\Bigr)^{\nu-\mu} \Bigl( \int_{{\cal M}_{j}}Vdx\Bigr)^{\mu+\kappa}\Bigr\}\ . \end{equation} % Notice that $\mbox{vol}\ {\cal M}_j\asymp (1+|x|)^d$ on $x\in{\cal M}_j.$ Thus the second sum on the r.h.s. of \eqref{scd} is bounded from above by $\co(d,l,\mu)\bigl\|\boldsymbol{\beta}^{\bold F} \bigl((1+|x|)^\sigma V(x)\bigr)\bigr\|_{\ell_{\mu+\kappa}}^{\mu+\kappa}.$ The first sum can be estimated by % $\sum_{j}\Bigl( \int_{{\cal M}_{j}}Vdx\Bigr)^{\mu/\nu}\leq \co(d,l,\mu)\bigl\|\boldsymbol{\beta}^{\bold F} \bigl((1+|x|)^\sigma V(x)\bigr)\bigr\|_{\ell_{\mu+\kappa}}^{\mu/\nu} \Bigl(\sum_j \bigl(\mbox{vol}\ {\cal M}_j\bigr)^{-\frac{\mu\sigma q}{\nu d}} \Bigr)^{q^{-1}},$ % where we applied H\"olders inequality with the powers $p=\nu(\mu+\kappa)/\mu>1,\ q^{-1}=1-p^{-1}.$ The sum of the negative powers of vol ${\cal M}_j$ converges, what completes the proof. $\Box$ \vspace{2mm} \subsection{Remark.} >From the proofs of Theorems \ref{subLiTh} and \ref{Cl} we see, that in the respective bounds the term of homogeneity $\mu/\nu$ corresponds to large cubes ${\cal Q}_{x_i}\in{\bold Q}(V),$ that means areas of low density of the potential, while the term of homogeneity $\mu+\kappa$ corresponds to small cubes ${\cal Q}_{x_i}\in{\bold Q}(V),$ that means areas of high density of the potential. This agrees with the fact, that under the conditions of these theorems we have $S_{l,\mu}(\alpha V)\asymp \alpha^{\mu/\nu}$ as $\alpha\to 0,$ but $S_{l,\mu}(\alpha V)\asymp \alpha^{\mu+\kappa}$ as $\alpha\to\infty.$ \section{Appendix} In the appendix we outline the proof of Lemmata \ref{aux0} and \ref{aux1}. \begin{lm}\label{aauuxxii} Assume $2l>d.$ Put $B_r=\{x\in{\Bbb R}^d:|x|From \eqref{alpha} and \eqref{hurra} we conclude \eqref{hur}.$\Box$\subsection{Proof of Lemma \ref{aux0}.} {\bf 1.} Take$r>0$such, that$\mbox{supp} V\subset \{x\in{\Bbb R}^d:|x|2$and$0\leq \psi\leq 1$for$1\leq t\leq 2.$Define$\psi_\epsilon\in C_0^\infty({\Bbb R}^d)$by$\psi_\epsilon(x):=\psi(\epsilon\ln|x|),\ \epsilon>0.$A calculation shows (cf. \cite{Sol}, p. 123), that % $\int |\nabla^l \psi_\epsilon p |^2 dx<\epsilon \bl p\bl M(d,l,\psi), \ \ p\in\Omega_{d,[l-\frac{d}{2}]},\ \ 0<\epsilon<1,$ % while % $\int V|\psi_\epsilon p(x)|^2dx=\int V|p(x)|^2dx\geq\bl p\bl m(d,l,V),$ % for sufficiently small$\epsilon_0(V)>\epsilon>0$and suitable constants$00.\ \Box$\vspace{3mm} \subsection{Proof of Lemma \ref{aux1}.} Let$<\cdot,\cdot>$denote the standard scalar product in${\Bbb R}^d.$For$V\geq 0$we put$W(x)=\sqrt{V(x)}$and % $\bigl(X_\varkappa(V) u\bigr)(x):= W(x)\int_{{\Bbb R}^d}\int_{{\Bbb R}^d}\frac{e^{i<\xi,x-y>}W(y)u(y)d\xi dy}{(2\pi)^d(|\xi|^{2l}-\varkappa)},\ \ \varkappa<0.$ % For$V\in L_1({\Bbb R}^d)$and$2l>d$this positive integral operator acts as a Hilbert-Schmidt on$L_2({\Bbb R}^d).$Let$\{\lambda_n(X_\varkappa(V))\}$denote the non-increasing sequence of the eigenvalues of$X_\varkappa(V).$Moreover$\{\varkappa_n(\alpha)\}:=\{\varkappa_n(H_l(\alpha V))\}$denotes the non-decreasing sequence of negative eigenvalues of$H_l(\alpha V).$According to the Birman-Schwinger principle the identities % \begin{equation}\label{BirSch} \lambda_k(X_{\varkappa_k(\alpha)}(V))=\alpha^{-1}, \ \ \ k\in{\Bbb N}, \end{equation} % hold. In particular one finds % $\alpha^{-1}= \|X_{\varkappa_1(\alpha)}(V)\|\leq\ \mbox{Tr} X_{\varkappa_1(\alpha)}(V)=|\varkappa_1(\alpha)|^{-\nu} {\frak L}^0(d,l,\nu)\int_{{\Bbb R}^d} V(x)dx\ ,$ % what turns into \eqref{accom}. Assume now, that$0\leq V(x)\in L_1({\Bbb R}^d)$is of compact support. We decompose the operator$X_\varkappa(V)$as % $X_\varkappa(V):=\tilde{X}_\varkappa(V) + \hat{X}_\varkappa(V) + \dot{X}_{\varkappa}(V),$ % \begin{equation}\begin{split}\notag \bigl(\tilde{X}_\varkappa(V)u\bigr)(x)&:= W(x)\int_{|\xi|\geq 1}\int_{y\in{\Bbb R}^d} \frac{e^{i<\xi,x-y>}W(y)u(y)d\xi dy}{(2\pi)^d(|\xi|^{2l}-\varkappa)},\\ \bigl(\hat{X}_\varkappa(V)u\bigr)(x)&:= W(x)\int_{|\xi|<1}\int_{y\in{\Bbb R}^d} \frac{W(y)u(y)d\xi dy}{(2\pi)^d(|\xi|^{2l}-\varkappa)},\\ \bigl(\dot{X}_\varkappa(V)u\bigr)(x)&:= W(x)\int_{|\xi|<1}\int_{y\in{\Bbb R}^d} \frac{(e^{i<\xi,x-y>}-1)W(y)u(y)d\xi dy} {(2\pi)^d(|\xi|^{2l}-\varkappa)}. \end{split} \end{equation} % Evaluating the respective Hilbert-Schmidt norms we find % $\|\tilde{X}_\varkappa(V)\|\leq \co(V),$ % $\|\dot{X}_\varkappa(V)\|\leq \Biggl\{ \begin{array}{cc} \co(V) |\ln (e+|\varkappa|^{-1})|\ \ \mbox{as}\ 2l=d+1,\\ \co(V) |\varkappa|^{\frac{d+1}{2l}-1}\ \ \mbox{as}\ 2l>d+1 \end{array} ,\ \ \ |\varkappa|<1.$ % Finally we represent$\hat{X}_\varkappa(V)$as % $\hat{X}_{\varkappa}(V)=\hat{X}^0_{\varkappa}(V)+\hat{X}_{\varkappa}^1(V),$ % $\bigl(\hat{X}_\varkappa^0(V)u\bigr)(x):= W(x)\int_{\xi\in{\Bbb R}^d} \int_{y\in{\Bbb R}^d} \frac{W(y)u(y)d\xi dy} {(2\pi)^d(|\xi|^{2l}-\varkappa)}.$ % Obviously % $\hat{X}_{\varkappa}^0(V)=|\varkappa|^{\frac{d}{2l}-1} \hat{X}_{-1}^0(V),$ % and % $\|\hat{X}_{\varkappa}^1(V)\|\leq \co(V),\ \ \ \varkappa<0.$ % We underline, that the constants$c_{4.8},\cdots,c_{4.11}$do not depend on$\varkappa<0.$>From standard perturbation theory we conclude, that the operator % $X_\varkappa(V)=|\varkappa|^{\frac{d}{2l}-1}\hat{X}_{-1}^0(V) + Y_\varkappa(V), \ \ \ Y_\varkappa(V):=\hat{H}_\varkappa^1(V)+\tilde{H}_\varkappa(V)+ \dot{H}_\varkappa(V)$ % has not more than$\mbox{rank} X_\varkappa^0(V)=1$eigenvalue larger than$\|Y_\varkappa(V)\|$, or % $\lambda_k(X_\varkappa(V))\leq \co(V)\max\{|\varkappa|^{\frac{d+1}{2l}-1}, \ln(e+|\varkappa|^{-1}),1\}\ \ \mbox{as}\ \ |\varkappa|<1, \ \ k\geq 2.$ % >From \eqref{accom} and \eqref{BirSch} we conclude, that for compactly supported potentials$0\leq V\in L_1({\Bbb R}^d)$the asymptotical estimates % $|\varkappa_k(\alpha)|=o(\alpha^{1/\nu})\ \ \mbox{as} \ \ \alpha\to 0,\ \ k\geq 2,$ % hold. On the other hand for the leading eigenvalue we have % $|\varkappa|^{\frac{d}{2l}-1} \lambda_1(\hat{X}^0_{-1}(V))-\|Y_\varkappa(V)\|\leq \lambda_1(X_\varkappa(V))\leq |\varkappa|^{\frac{d}{2l}-1} \lambda_1(\hat{X}^0_{-1}(V))+\|Y_\varkappa(V)\|,$ % which mounts into % $\lambda_1(X_\varkappa(V))=|\varkappa|^{\frac{d}{2l}-1}\mbox{Tr} \hat{X}_{-1}^0(V)+O\bigl(\max\{|\varkappa|^{\frac{d+2}{2l}-1}, |\ln(e+|\varkappa|^{-1})|\}\bigr) \ \ \mbox{as}\ \varkappa\to -0.$ % Then \eqref{accom} and \eqref{BirSch} imply % \begin{equation}\label{cclloo} |\varkappa_1(\alpha)|^{\nu}= \alpha{\frak L}^0(d,l,\nu)\int V(x) dx + o(\alpha)\ \ \mbox{as}\ \alpha\to 0. \end{equation} % In view of \eqref{accom} we can close \eqref{cclloo} to all potentials$0\leq V\in L_1({\Bbb R}^d).\ \Box$\vspace{2mm} \rema The technique of the extraction of a diverging operator of finite rank is well-known. It can be applied to the case of non-signdefined potentials and the asymptotics of the subsequent eigenvalues can also be calculated. In particular one can show that \eqref{cclloo} remains true for compactly supported non-signdefined potentials$V\in L_1({\Bbb R}^d),$if only$\int V dx>0.$For the related results on the weakly coupled one- or twodimensional Schr\"odinger operator we refer to \cite{Si}. \begin{thebibliography}{99} \bibitem{Agm} Agmon, Sh.: On kernels, eigenvalues and eigenfunctions of operators related to elliptic problems. {\em Comm. Pure and Appl. Math.}, vol. {\bf 18}, 627-663, (1965) % % \bibitem{AiL} Aizenmann M., Lieb, E.: On semi-classical bounds for eigenvalues of Schr\"odinger operators. Phys. Lett. {\bf 66A}, 427-429 (1978) % % \bibitem{Barg} Bargmann, V.: On the number of bound states in a central field of force. Proc. Nat. Acad. Sci. USA {\bf 38}, 961-966 (1952) % % \bibitem{Bergl} Bergh J., L\"ofstr\"om, J.: Interpolation spaces. An Introduction. 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