\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 60, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/60\hfil Resonances generated by analytic singularities] {Resonances generated by analytic singularities on the density of states measure for perturbed periodic Schr\"odinger operators} \author[H. Baklouti, M. Mnif\hfil EJDE-2006/60\hfilneg] {Hamadi Baklouti, Maher Mnif} % in alphabetical order \address{Hamadi Baklouti \newline D\'epartement de Maths Facult\'e des Sciences de Sfax 3038 Sfax Tunisie} \email{h\_baklouti@yahoo.com} \address{Maher Mnif \newline D\'epartement de Maths I.P.E.I. Sfax B.P. 805 Sfax 3000 Tunisie} \email{maher.mnif@ipeis.rnu.tn} \date{} \thanks{Submitted March 16, 2006. Published May 4, 2006.} \subjclass[2000]{35B34, 35B20, 35P15, 35J10} \keywords{Resonances; perturbations; periodic Schr\"odinger operators} \begin{abstract} We consider a perturbation of a periodic Shr\"odinger operator $P_0$ by a potential $W(hx)$, $(h\searrow 0)$. We study singularities of the density of states measure and we obtain lower bound for the counting function of resonances. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem {theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} In this paper we present a lower bound for the counting function of resonances for the perturbed periodic Shr\"odinger operator $$ P(h) = P_0 + W(hy) \mbox{ , } P_0 = -\bigtriangleup + V \quad (h\searrow 0). $$ Here $V$ is $C^\infty$ function, real valued and $\Gamma$-periodic with respect to a lattice $\Gamma = \oplus_{i=1}^n \mathbb{Z} e_i$ in $\mathbb{R}^n$. The potential $W$ is real valued and satisfies the hypothesis \begin{itemize} \item[(H1)] There exist positive constants $a$ and $C$ such that $W$ extends analytically to $$ \Gamma(a):=\{z\in \mathbf{C}^n: \vert \Im(z)\vert \le a \langle \Re(z) \rangle \}$$ and \begin{equation}\label{eq1} \vert W(z)\vert \le C\langle z\rangle^{-N},\text{ uniformly on } z\in \Gamma(a),\, N>n, \end{equation} where $\langle z\rangle=(1+\vert z\vert ^{2})^{1/2}$. Here $\Re(z)$, $\Im(z)$ denote respectively the real part and the imaginary part of $z$. \end{itemize} Let $ \Gamma^*=\oplus_{i=1}^n {\bf Z} e_i^*$ be the dual lattice of $\Gamma$, where $ \{e_j^*\}_{j=1}^n$ is the basis satisfying $ (e_j,e_k^*)=2\pi \delta_{jk}$. Set $ E=\{x=\sum_{j=1}^n t_je_j, \, t_j\in [ -1/2,1/2[\}$, and $E^*=\{x=\sum_{j=1}^n t_je_j^*,\, t_j\in [-1/2,1/2[\}$. We use the usual flat metrics on $ {\bf T}:=\mathbb{R}^n/\Gamma$ and $ {\bf T}^*:=\mathbb{R}^n/\Gamma^*$, when we integrate or do local considerations we identify ${\bf T} $ (resp. ${\bf T}^*$) with $E$ (resp. $E^*$). For $k\in \mathbb{R}^n$, we define the operator $P_{k}$ on $L^2({\bf T})$ by $$ P_{k}:=(D_{y}+k)^2+V(y). $$ Let $\lambda_{1}(k)\le \lambda_{2}(k)\le \dots $ be the Floquet eigenvalues of $P_k$ (enumerated according to their multiplicities). It is well known (see \cite{r1}) that $\lambda_p(k)$ are continuous functions of $k$ for any fixed $p$. Moreover $\lambda_p(k)$ is an analytic function in $k$ near any point $k_0 \in T^*$, where $\lambda_p(k_0)$ is a simple eigenvalue of $P_{k_0}$. We consider the self-adjoint operator $P_0 =-\Delta +V(y)$ on $L^{2}(\mathbb{R}^n)$ with domain $H^{2}(\mathbb{R}^n)$. By Bloch-Floquet theory, it is known that $$ \sigma(P_0)=\sigma_{\rm ac}(P_0)= \cup_{p\geq 1}\Lambda_{p}, \quad \mbox{where } \Lambda_p = \lambda_p({\bf T}^*). $$ Let us introduce the density of states measure \begin{equation}\label{eq2} \rho(\lambda):={1\over (2\pi)^n}\sum_{p\geq 1} \int_{\{k\in E^*;\ \lambda_p(k) \leq \lambda \}} dk. \end{equation} Since the spectrum of $P_0$ is absolutely continuous, the measure $\rho$ is absolutely continuous with respect to the Lebesgue measure $d\lambda$. Therefore, the density of states ${d \rho \over d \lambda }$ of $P_0$, is locally integrable. For $f\in C^\infty_0({\bf R})$, we set \begin{gather}\label{eq3} {\langle \mu,f\rangle=\int [f(W(x)) -f(0)]dx,} \\ \label{eq4}\langle \omega,f\rangle= {1 \over (2\pi)^n}\sum_j\int_{E^*}\int_{\mathbb{R}^n_x} [f(W(x)+\lambda_j(k)) -f(\lambda_j(k))]dx\,dk, \end{gather} \begin{proposition}[\cite{d1}]\label{pro1} The functionals operators $\omega$ and $\mu$ are distributions of order $\le 1$. Moreover, in $\mathcal{D}'(\mathbb{R})$, we have \begin{equation}\label{eq5}\omega=d\rho*\mu.\end{equation} \end{proposition} \begin{definition}\label{df1} \rm We say that $\lambda \in \sigma(P_0)$ is a simple energy level if it is a simple eigenvalue of $P_k$, for every $k \in F(\lambda):=\{k \in {\bf T}^*; \lambda \in \sigma(P_k) \}$. \end{definition} We use also the following hypothesis \begin{itemize} \item[(H2)] There exists an open bounded interval $I$ such that for all $\lambda \in I$ and all $k_0 \in \mathbb{R}^n/\Gamma^*$ with $\lambda_p(k_0) = \lambda$, the eigenvalue $\lambda_p(k_0)$ is simple and $d_k\lambda_p(k_0) \neq 0$. \end{itemize} We use $\mathop{\rm sing\,supp}_a(\omega)$ for analytic singular support of $\omega$. Under assumptions (H1) and (H2) in \cite{d2} it was proved that if $E \in \mathop{\rm sing\,supp}_a(\omega) \cap I$ then for every $h$-independent complex neighborhood $ \Omega$ of $E$, there exist $h_0=h( \Omega)>0$ sufficiently small and $C = C( \Omega) >0$ large enough such that for $ h \in ]0,h_0[$, $$ \#\{z\in \Omega; z\in \mathop{\rm Res}P(h)\} \ge C_\Omega h^{-n}. $$ This result is based on the trace formula in the periodic case \cite{d2,s1}. Since (\ref{eq5}) for $\omega$, the analytic singular support of $\omega$ depends on both $\mathop{\rm sing\,supp}_a(\mu)$ and $\mathop{\rm sing\,supp}_a(d\rho)$. The question is to find some criteria to determine if $e_0 = \lambda_p(k_0)$ belongs to the $\mathop{\rm sing\,supp}_a(d\rho)$. If $e_0 = \lambda_p(k_0)$ is a simple eigenvalue in a neighborhood of $k_0$ then $\lambda_p(k)$ is a smooth function there. Moreover if $e_0$ is non critical then $e_0$ is not in the analytic singular support of $\rho$ (see Lemma \ref{lem1}). The distribution $\rho$ can be singular for a variety of reasons. If $e_0 =\lambda_p(k_0)$ is a critical value, we expect in general that $e_0$ will belong to the analytic singular support of $\rho$. Multiple eigenvalues can also give rise to analytic singularities of $\rho$. We recall that, the case when $e_0 =\lambda_p(k_0)$ is a non-degenerate extremum was studied by Dimassi and Mnif in \cite{d1}. They studied also the case of bands crossing when $n = 2$. In this paper we are interested to more general situations. We first study the case when $e_0 =\lambda_p(k_0)$ is a non-degenerate critical point and we prove that in this situation $e_0$ belongs to the analytic singular support of $\rho$. We note that this result generalizes the case when $e_0$ is a non-degenerate extremum point established in \cite[Theorem 1]{d1}. In the case when $e_0$ is a degenerate critical point one gives a positive answer to the question if $e_0$ is an extremum. This result encloses the case of finite number of extremum at the same level. Finally we look for resonances near a singularity of $\rho$ generated by bands crossing at $e_0$. This study is devoted to the case $n=3$. The paper is presented as follows: Section 2: Lower bound of the number of resonances near a critical non-degenerate point. Section 3: Lower bound of the number of resonances near a degenerate critical point. Section 4: Lower bound of the number of resonances near a conic singularity of the density of states. \section{Lower bound of the number of resonances near a critical non-degenerate point} Let $O$ be an open bounded set in $\mathbb{R}^n$ with analytic boundary almost every where, and let $U$ be a complex neighborhood of $O$. Let $x\to \varphi(x)$ be analytic on $U$ and real valued for all $x$ in $ O$. Let us introduce the real function $$ I(e):=\int_{\{x\in{O},\,\varphi(x)\le e\}} dx. $$ \begin{lemma}[\cite{n1}] \label{lem1} If $\nabla\varphi(x)\not =0$ near every $x\in \Sigma_{e_0}:=\{x\in O : \varphi(x)=e_0\}$ and if the sets $\partial O$ and $\Sigma_{e_0}$ intersect transversely, then $I(e)$ is analytic near $e_0$. \end{lemma} The next lemma generalizes the result in \cite[Lemma 2]{d1}, where the authors consider the case of non-degenerate extremum. \begin{lemma}\label{lem2} If $\varphi $ has a non-degenerate critical point at $x_0$ with $\varphi (x_0) = e_0$ and if $\nabla \varphi (x) \neq 0$ for all $x \in \Sigma_{e_0} \backslash \{x_0\}$, then there exists an open interval $J$ neighborhood of $e_0$, such that $I(e)$ is analytic on $J\backslash \{e_0\}$ and has a $C^2$ singularity at $e_0$. \end{lemma} \begin{proof} Under the assumption $\nabla\varphi(x) \neq 0$ for all $x \in \Sigma_{e_0} \backslash \{x_0\}$ and since $\varphi$ has a non-degenerate critical point at $x_0$ there exists an open interval $J$ neighborhood of $e_0$ such that for all $e \in J\backslash \{e_0\}$ we have $\nabla\varphi(x) \neq 0$ near every $x \in \Sigma_{e}:=\{x\in O : \varphi(x)=e\}$. Hence by Lemma \ref{lem1} $I(e)$ is analytic on $ J\backslash \{e_0\}$. One now studies the behavior of $I$ at $e_0$. Let $(k,n-k)$ be the signature of the hessian form of $\varphi$ at $x_0$. The case $k=0$ or $k=n$ corresponds to $e_0$ non-degenerate extremum which is studied in \cite{d1}. Here we focus our study on the case of saddle point. By Morse lemma, for all $\epsilon >0$ small enough, there exist a neighborhood $\Omega$ of $x_0$ and a local analytic diffeomorphism $ D : \Omega \to B(0, \epsilon)$ such that $$ I_\epsilon (e) := \int_{\{x \in \Omega ,\, \varphi(x) \leq e \}} dx \\ = \int_{\{x \in B(0,\epsilon) ,\, \sum_{i=1}^k x_i^2 - \sum_{i=k+1}^n x_i^2 \leq e-e_0\}} \mathop{\rm Jac}(D^{-1} (x)) dx . $$ We introduce the notation: $x = (X_+, X_-)$ with $X_+ = (x_1, \dots,x_k)$ and $X_- = (x_{k+1},\dots,x_n)$. $B_{k, \epsilon} = \{ X \in \mathbb{R}^k : \Vert X \Vert < \epsilon \}$. Up to an analytic correction of $I_\epsilon (e)$, we can suppose that $$ I_\epsilon (e) = \int_{\{x= (X_+,X_-) \in B_{k,\epsilon} \times B_{n-k,\epsilon},\, \sum_{i=1}^k x_i^2 - \sum_{i=k+1}^n x_i^2 \leq e-e_0\}} \mathop{\rm Jac}(D^{-1} (x)) dx . $$ Let $x= \epsilon y$ and $E = (e-e_0)/\epsilon^2$, we have \begin{align*} I_\epsilon (e) &= \epsilon^n J(\epsilon, E)\\ &:= \epsilon^n \int_{\{y= (Y_+,Y_-) \in B_{k,1} \times B_{n-k,1} ,\, \sum_{i=1}^k y_i^2 - \sum_{i=k+1}^n y_i^2 \leq E\}} \mathop{\rm Jac}(D^{-1} (\epsilon y)) dy . \end{align*} To prove that $I_\epsilon$ has a $C^2$ singularity at $e_0$ we prove that $J(\epsilon,.)$ has a $C^2$ singularity at $E=0$. On the other hand, we can see that for $E$ small enough, $J(.,E)$ is analytic near $\epsilon=0$. Therefore, it is sufficient to prove that $E=0$ is a $C^2$ singularity for $J(0,.)$. We have $$ J(0, E) ={ 2^{n/2} \over \sqrt{\vert \det(\mathop{\rm Hess}(\varphi)(x_0)\vert }} \int_{\{y= (Y_+,Y_-) \in B_{k,1} \times B_{n-k,1} ,\, \sum_{i=1}^k y_i^2 - \sum_{i=k+1}^n y_i^2 \leq E\}} dy. $$ Using polar coordinates we get $$ J(0, E)= C_n\int_{\{0 \leq r_1 \leq 1,\, 0 \leq r_2 \leq 1 ,\, r_1^2 - r_2^2 \leq E\}} r_1^{k-1 } r_2^{n-k-1} dr_1dr_2, $$ where $$ C_n = { 2^{n \over 2} \mathop{\rm Vol}(S^{k-1})\mathop{\rm Vol}(S^{n-k-1})\over \sqrt{\vert \det(\mathop{\rm Hess}(\varphi)(x_0)\vert }} $$ For $E>0$, \begin{align*} J(0, E) &:= f_r(E) \\ &= C_n [ \int_0^{\sqrt {E}} \int_0^1 r_1^{k-1} r_2^{ n-k-1} dr_2dr_1 + \int_{\sqrt {E}}^1 \int_{\sqrt{r_1^2 - E}}^1 r_1^{k-1} r_2^{ n-k-1} dr_2dr_1] \\ & = C_n[ {1\over k (n-k)} + \int^{\sqrt {E}}_1 { (r_1^2 -E )^{n-k \over 2} \over n-k} r_1^{k-1} dr_1]. \end{align*} For $E<0$, we write $$ J(0, E) :=f_l(E)= C_n \int_{\{0 \leq r_1 \leq 1 \mbox { , }0 \leq r_2 \leq 1 \mbox{ ; } r_2^2 \geq r_1^2 -E \}} r_1^{k-1 } r_2^{n-k-1} dr_1dr_2\,. $$ In the same way as above we obtain $$ f_l(E)=-C_n \int^{\sqrt {-E}}_1 { (r_2^2 +E )^{k \over 2} \over k} r_2^{n-k-1} dr_2. $$ Computing the second derivatives, we get for $n >4$: If $n-k \neq 2$, then $$ {d^2 f_r \over dE^2}(0) = - C_n {n-k-2 \over 4 (n-4)}. $$ If $k \neq 2$, then $$ {d^2 f_l \over dE^2}(0) = C_n {k-2 \over 4 (n-4)}. $$ If $n-k = 2$, then $$ {d^2 f_r \over dE^2}(0) = 0 \quad \mbox{and}\quad {d^2 f_l \over dE^2}(0) = {C_n \over 4}. $$ If $k = 2$, then $$ {d^2 f_r \over dE^2}(0) = -{C_n \over 4}\quad \mbox{ and}\quad {d^2 f_l \over dE^2}(0) = 0. $$ So, for all $n> 4$, we have $$ {d^2 f_r \over dE^2}(0)\neq {d^2 f_l \over dE^2}(0). $$ On the other hand, for $n\leq 4$: If $n-k \neq 2$, then $$ \lim_{ E \to 0^+}{d^2 f_r \over dE^2}(E)= \infty\,. $$ If $k \neq 2$, then $$ \lim_{ E \to 0^-}{d^2 f_l \over dE^2}(E)= \infty\,. $$ If $k = 2$ and $n-k = 2$, then $$ {d^2 f_r \over dE^2}(0) = -{1 \over 4}\quad \mbox{and}\quad {d^2 f_l \over dE^2}(0) = {1 \over 4}. $$ Hence, for all $n$, $J(0,.)$ has a $C^2$ singularity at $0$. \end{proof} The following result is a consequence of Lemma \ref{lem1}, Lemma \ref{lem2} and the representation (\ref{eq2}) of $\rho$. \begin{lemma}\label{lem3} Let $e_0$ be a simple eigenvalue of $P_0$. We assume that: \begin{itemize} \item[(i)] There exist $i_0$ and $k_0$ such that $\lambda_{i_0} (k_0) = e_0$, $\nabla \lambda_{i_0} (k_0) =0$. \item[(ii)] $\nabla \lambda_{i_0} (k) \neq 0$, for all $k \in \lambda_{i_0}^{-1}(\{e_0 \}) $, $k \neq k_0$ and $\nabla \lambda_{i} (k) \neq 0$ for all $k \in \lambda_i^{-1}(\{e_0\})$, $i \neq i_0$. \end{itemize} Then there exists an open interval $J$ neighborhood of $e_0$ such that the density of states measure $\rho$ is analytic on $J\backslash \{e_0\}$ and has a $C^2$ singularity at $e_0$. \end{lemma} Therefore, by \cite[Theorem 1.6]{d2}, we obtain the following result. \begin{theorem}\label{theo1} Let $e_0$ and $J$ be as in Lemma \ref{lem3}, $I$ satisfying (H2) and let $E \in ( e_0 + \mathop{\rm sing\,supp}_{\rm a}(\mu))\cap I $ be such that $ ( E - \mathop{\rm supp}(\mu)) \subset J$. Then for all $h$-independent complex neighborhood $\Omega$ of $E$, there exist $h_0=h(\Omega)>0$ sufficiently small and $C = C( \Omega) >0$ such that for $ h \in ]0,h_0[$, $$ \#\{z\in \Omega; z\in {\rm Res}P(h)\} \ge C_\Omega h^{-n}. $$ \end{theorem} \section{Lower bound for the number of resonances near a degenerate critical point} Let $K$ be a compact set in $\mathbb{R}^n$, we consider $C(K,\mathbb{R})$ the space of continuous real functions on $K$, with the norm $ \Vert \varphi \Vert_\infty = \sup_{x \in K} \vert \varphi (x) \vert $. Let us introduce the real valued function $ \mathcal{H}_e : C(K,\mathbb{R}) \to \mathbb{R}$, $$ \varphi \mapsto \int_{\{x \in K ,\, \varphi (x) \leq e\}} dx\,. $$ \begin{lemma}\label{lem4} Let $\varphi \in C(K,\mathbb{R})$ such that $ \varphi^{-1}(\{e\})$ is a finite set. $\mathcal{H}_e$ is continuous at $\varphi$. \end{lemma} \begin{proof} Without loss of generality, we can take $\varphi^{-1}(\{e\})$ reduced to $\{x_0\}$. Let $\epsilon >0$, by the continuity of $\varphi$ on $K$ and the fact that $\varphi(x) \neq e$ for all $ x \in K_\epsilon = K \backslash B(x_0, \epsilon)$ which is a compact set, we have the statement: \begin{equation}\label{eq6} \mbox{There exists $\alpha(\epsilon) >0$ such that $\vert \varphi (x) - e \vert >\alpha(\epsilon)$, for all $x \in K_\epsilon$}. \end{equation} Let $\psi \in C(K,\mathbb{R})$ be such that \begin{equation}\label{eq7} \Vert \varphi - \psi \Vert_\infty < {\alpha(\epsilon) \over 2}. \end{equation} We denote: \begin{gather*} K_{-,-} = \{ x \in K : \varphi (x) \leq e \} \cap \{ x \in K : \psi (x) \leq e \} \\ K_{-,+} = \{ x \in K : \varphi (x) \leq e \} \cap \{ x \in K : \psi (x) > e \} \\ K_{+,-} = \{ x \in K : \varphi (x) > e \} \cap \{ x \in K : \psi (x) \leq e \} . \end{gather*} We have: \begin{gather*} \mathcal{H}_e(\varphi ) = \mathop{\rm Vol}(K_{-,-} )+ \mathop{\rm Vol}(K_{-,+}) \\ \mathcal{H}_e( \psi ) = \mathop{\rm Vol}(K_{-,-} )+ \mathop{\rm Vol}(K_{+,-} ). \end{gather*} Then $$ \mathcal{H}_e( \varphi ) - \mathcal{H}_e( \psi) = \mathop{\rm Vol}(K_{-,+}) - \mathop{\rm Vol}(K_{+,-} ). $$ By (\ref{eq6}) and (\ref{eq7}), we have $$ K_{-,+} \cap K_\epsilon = \emptyset \quad\mbox{and}\quad K_{+,-} \cap K_\epsilon = \emptyset , $$ hence $$ K_{-,+} \subset B(x_0,\epsilon) \quad\mbox {and}\quad K_{+,-} \subset B(x_0,\epsilon). $$ Then $$ \mathop{\rm Vol}(K_{-,+} ) \leq 2 \epsilon \quad\mbox {and}\quad \mathop{\rm Vol}(K_{+,-} ) \leq 2 \epsilon . $$ Finally we get $\vert \mathcal{H}_e( \varphi ) - \mathcal{H}_e( \psi) \vert \leq 4 \epsilon$. \end{proof} \begin{definition}\label{df2} \rm Let $O$ be an open bounded set in $\mathbb{R}^n$, and let $\varphi$ a function in $ C^\infty (O, \mathbb{R})$. We say that $ \varphi$ has an isolated local minimum (resp. maximum) of order $p \in \mathbb{N}^*$ at $x_0 \in O$, if the Taylor expansion of $\varphi$ near $x_0$ is as follows $$ \varphi (x+x_0) = \varphi(x_0) + \sum_{i=1}^n \alpha_i x_i^{2p} + \sum_{\sigma \in (\mathbb{N})^n; \vert \sigma \vert =p} a_\sigma x^{2\sigma} +\mathcal{O}(\vert x \vert^{2p +1}) $$ with $\alpha_i >0$, $a_\sigma \geq 0$ (resp. $\alpha_i <0$, $a_\sigma \leq 0$). $\sigma =(\sigma_1, \dots , \sigma_n) \in (\mathbb{N})^n$, $x^{2\sigma} $ denotes $x_1^{2\sigma_1} \dots x_n^{2\sigma_n}$ and $\vert \sigma \vert =\sigma_1 + \dots +\sigma_n$. \end{definition} We now return to the real valued function $I(e):=\int_{\{x\in{O} : \varphi(x)\le e\}} dx$ introduced in section 2. Let $H$ denote the Heaviside function. \begin{lemma}\label{lem5} Suppose that $\varphi $ has an isolated local extremum of order $p \in \mathbb{N}^* $ at $x_0$. If $\nabla \varphi (x) \neq 0$ for all $ x \in \Sigma_{e_0} \backslash \{x_0\}$, then \begin{itemize} \item[(i)] If $e_0$ is a minimum, \begin{equation} \label {eq8} I(e) = g(e-e_0) + H(e-e_0) ( e-e_0)^{n \over 2p} ( C + R(e)) \end{equation} with $C>0$, $\lim_{e \to e_0} R(e) =0$ and $g$ analytic function. \item[(ii)] If $e_0$ is a maximum, \begin{equation} \label {eq9} I(e) = g(e-e_0) + H(e_0-e) ( e_0-e)^{n \over 2p} ( C + R(e)) \end{equation} with $C>0$, $\lim_{ e \to e_0} R(e) =0$ and $g$ analytic function. \end{itemize} \end{lemma} \begin{proof} (i) We note that if $e_0$ is a minimum for $\varphi$ then there exists $\epsilon >0$ such that $\varphi (x+x_0) \geq e_0$ for all $x \in B(0,\epsilon)$. We write $$ I(e) = \int_{\{x \in B(0,\epsilon),\, \varphi(x) \leq e\}}dx + \int_{\{x \in O \backslash B(0,\epsilon),\, \varphi(x) \leq e\}}dx. $$ By Lemma \ref{lem1}, the second term in the right-hand side is analytic near $e_0$. Let: $$ I _\epsilon(e) := \int_{\{x \in B(0,\epsilon),\, \varphi(x) \leq e\}}dx . $$ For $ee_0$, we can write $$ \varphi(x_0+x) = e_0 + D_{2p}(x) + \mathcal{O}(\vert x \vert^{2p +1}) $$ with , $$ D_{2p}(x) = \sum_{i=1}^n \alpha_i x_i^{2p} + \sum_{\sigma \in (\mathbb{N})^n; \vert \sigma \vert =p} a_\sigma x ^{2 \sigma}. $$ Up to $\epsilon >0$, we have for all $ x \in B(0, \epsilon)$, $$ \vert \mathcal{O}(\vert x \vert^{2p +1}) \vert \leq {1 \over 2} D_{2p}(x). $$ Hence $$ J_e := \{ x \in B(0, \epsilon) : \varphi (x+x_0) \leq e\} \subset \{ x \in B(0, \epsilon) : D_{2p}(x) \leq 2(e-e_0)\} $$ Since $a_\sigma \geq 0$ for all $\sigma$, we have $$ J_e \subset \{ x \in B(0, \epsilon) : \sum_{i=1}^n \alpha_ix_i^{2p} \leq 2(e-e_0)\} \subset B(0, c (e-e_0)^{1 \over 2p}) $$ with $c>0$. Therefore, $$ I_\epsilon(e) = \int_{\{x \in B(0,\epsilon) \cap B(0,c(e-e_0)^{1 \over 2p}) : \varphi (x_0+x) \leq e\}} dx. $$ Up to reduce $e-e_0$, we can suppose that $c(e-e_0)^{1 \over 2p} <\epsilon$. Then we get $$ I_\epsilon(e) = \int_{\{x \in B(0,c(e-e_0)^{1 \over 2p} ) : \varphi (x_0+x) \leq e\}}dx. $$ By the scaling $ x = (e-e_0)^{1 \over 2p}y$, we get $$ I_\epsilon(e) = (e-e_0)^{n \over 2p} \int_{\{y \in B(0,c ) :D_{2p}(y) + (e-e_0)^{1 \over 2p} \Psi_e(y) \leq 1\}} dy, $$ with $ \Psi_e$ bounded on $B(0,c)$ uniformly on $e$ near $e_0$. By Lemma \ref{lem4}, we get, for $e>e_0 $, $$ I_\epsilon(e) = (e-e_0)^{n \over 2p} (\int_{\{y \in B(0,c ) : D_{2p}(y) \leq 1\}} dy + R(e)) $$ with $\lim_{ e \to e_0} R(e) = 0$. \end{proof} By Lemma \ref{lem5} and the representation (\ref{eq2}) of $\rho$ we obtain the following result. \begin{lemma}\label{lem6} Let $e_0$ be a simple eigenvalue of $P_0$. We assume that \begin{itemize} \item[(i)] There exist $i_0$ and $k_0$ such that $\lambda_{i_0} (k_0) = e_0$. \item[(ii)] $e_0$ is an isolated local extremum of order $p$ for $\lambda_{i_0}$. \item[(iii)] $\nabla \lambda_{i_0}(k) \neq 0$, for all $k \in \lambda_{i_0}^{-1}(e_0)$, $k \neq k_0$. Moreover $\nabla \lambda_i(k) \neq0$, for all $k \in \lambda_i^{-1}(\{e_0\})$, $ i\neq i_0$. \end{itemize} Then there exists an open interval $J$ such that the density of states measures has the representation \eqref{eq8}, \eqref{eq9} in lemma \ref{lem5}. \end{lemma} Therefore, by \cite[Theorem 1.6]{d2}, we have the following result. \begin{theorem}\label{theo2} Let $e_0$ and $J$ be as in Lemma \ref{lem6}, $I$ satisfying (H2) and let $E \in ( e_0 + \mathop{\rm sing\,supp}_{\rm a}(\mu)) \cap I$ be such that $ ( E - \mathop{\rm supp}(\mu)) \subset J$. Then for all $h$-independent complex neighborhood $\Omega$ of $E$, there exist $h_0=h(\Omega)>0$ sufficiently small and $C = C( \Omega) >0$ such that for $ h \in ]0,h_0[$, $$ \#\{z\in \Omega; z\in {\rm Res}P(h)\} \ge C_\Omega h^{-n}. $$ \end{theorem} \begin{remark} The hypothesis (iii) in Lemma \ref{lem6}, implies that the $(\lambda_i)$ have no more critical point at the $e_0$ level other than $\lambda_{i_0}$'s one at $k_0$. In the following lemmas we consider the case of finite number of extrema at the same level. For simplicity we state these lemmas for only two extrema. \end{remark} By Lemma \ref{lem5} and the representation (2) of $\rho$, we have the following result. \begin {lemma}\label{lem7} Let $e_0$ be a simple eigenvalue of $P_0$. We assume that: \begin{itemize} \item[(i)] There exist $i_1$ and $k_1$, $i_2$ and $k_2$ such that $\lambda_{i_1} (k_1) = \lambda_{i_2} (k_2)= e_0$. \item[(ii)] $\lambda_{i_1} $ (resp. $\lambda_{i_2} $) has an isolated local minimum at the $e_0$ level of order $p_1$ (resp. $p_2$) at $k_1$ (resp. $k_2$). \item[(iii)] The $\lambda_i$ have no more critical points at the $e_0$ level other than $\lambda_{i_1}$'s one at $k_1$ and $\lambda_{i_2}$'s one at $k_2$. \end{itemize} Then there exists an open interval $J$ such that the density of states measures has the representation $$ \rho(e)= g(e-e_0) + H(e-e_0) ( e-e_0)^{n \over 2p} ( C + R(e)), $$ with $C>0$, $\lim_{ e \to e_0} R(e) = 0$, $g$ analytic function and $p=\max(p_1,p_2)$. \end{lemma} \begin {lemma}\label{lem8} Let $e_0$ be a simple eigenvalue of $P_0$. We assume that: \begin{itemize} \item[(i)] There exist $i_1$ and $k_1$, $i_2$ and $k_2$ such that $\lambda_{i_1} (k_1) = \lambda_{i_2} (k_2)= e_0$. \item[(ii)] $\lambda_{i_1} $ (resp. $\lambda_{i_2} $) has an isolated local minimum (resp. maximum) at the $e_0$ level of order $p_1$ (resp. $p_2$) at $k_1$ (resp. $k_2$). Moreover if $p_1=p_2$ then we assume that ${n \over 2 p_1 } \notin \mathbb{N}$. \item[(iii)] The $\lambda_i$ have no more critical points in the $e_0$ level other than $\lambda_{i_1}$'s one at $k_1$ and $\lambda_{i_2}$'s one at $k_2$. \end{itemize} Then there exists an open interval $J$ such that the density of states measures has the representation $$ \rho(e) = g(e-e_0) + H(e-e_0) (e-e_0)^{n\over 2p_1} (C_1 + R_1(e)) + H(e_0-e) (e_0-e)^{n\over 2p_2} (C_2 + R_2(e)) $$ with $C_1>0$, $C_2 >0$ , $ \lim_{ e\to e_0}R_1(e)=\lim_{ e\to e_0}R_2(e) =0$ and $g$ analytic function. \end{lemma} Therefore, by \cite[Theorem 1.6]{d2}, we have the following theorem. \begin{theorem}\label{theo3} Let $e_0$ and $J$ be as in Lemma \ref{lem7} or Lemma \ref{lem8}, $I$ satisfying (H2) and let $E \in ( e_0 + \mathop{\rm sing\,upp}_{\rm a}(\mu)) \cap I$ be such that $ ( E - \mathop{\rm supp}(\mu)) \subset J$. Then for all $h$-independent complex neighborhood $\Omega$ of $E$, there exist $h_0=h(\Omega)>0$ sufficiently small and $C = C( \Omega) >0$ such that for $ h \in ]0,h_0[$, $$ \#\{z\in \Omega; z\in {\rm Res}P(h)\} \ge C_\Omega h^{-n}. $$ \end{theorem} \section{Lower bound of the number of resonances near a conic singularity of the density of states} In this section we study resonances near a singularity of $\rho(\lambda)$ generated by a bands crossing. We assume that $\lambda_j$ is a double eigenvalues $$ \lambda_{j-1}(k_0)<\lambda_{j}(k_0)=e_0=\lambda_{j+1}(k_0) <\lambda_{j+2}(k_0) $$ and that for all $k\not =k_0$ such that $\lambda_i(k)=e_0$, $\lambda_i(k)$ is simple and $\nabla \lambda_i(k)\not=0$. Since $P_k$ is analytic in $k$, this implies that for $\vert k-k_0\vert \leq \delta $ (with $\delta$ small enough), the span $V(k)$, of the eigenvectors of $P_k$ corresponding to eigenvalues in the set $\{e : \vert e-e_0 \vert \leq \delta \}$ has a basis $\psi_j(x,k), \psi_{j+1}(x,k)$, which is orthonormal and real analytic in $k$. The restriction of $P_k$ to $V(k)$ has the matrix $$ \begin{pmatrix} \alpha(k) & \overline{b(k)} \cr b(k) & \beta(k) \end{pmatrix}, $$ which can be written as $$ \begin{pmatrix} a(k)+c(k) & {b_1(k)-ib_2(k)} \cr b_1(k)+ib_2(k) & a(k)-c(k)\end{pmatrix}, $$ where $a(k)=({\alpha(k)+\beta(k)) / 2}$, $ c(k)=({\alpha(k)-\beta(k))/ 2}$, $b_1(k)$ and $ b_2(k)$ are real valued. Next the periodic potential is assumed to have the symmetry $V(x)=V(-x)$. This symmetry is typical of metals. This symmetry forces $b(k)$ to be real valued (i.e., $b_2(k)=0$). Consequently, near $k_0$ we have $$ E_j(k)=a(k)-\sqrt{c^2(k)+b^2(k)},\quad E_{j+1}(k)=a(k)+\sqrt{c^2(k)+ b^2(k)}. $$ The case $n=2$ is treated in \cite{d1}. We consider here the case $n =3$. We assume that $\nabla b(k_0),\nabla c(k_0)$ are independent and \begin{equation}\label{eq10} \Vert \nabla_{b,c} a (k_0)\Vert<1 \end{equation} Nedelec in \cite{d2} section 6 studied singularity of volumes of matrix problem in some equivalent situations. She gets $C^\infty$ singularities. Following the same method we get a more precise result. \begin{lemma}\label{lem9} We assume that $ a/_{\{b=c=0\}}$ is non-degenerate at $e_0$. Then, there exist $f$ and $g$, analytic near $e_0 $, such that \begin{equation}\label{eq11} \rho(e) = f(e-e_0) + H(e-e_0) g(\sqrt {e-e_0}), \end{equation} with $g(.) \neq 0$. \end{lemma} \begin{proof} Without loss of generality we may assume that $e_0=0$ and $k_0 =0$. Let $S=\{k \in \mathbb{R}^3 ; b(k)=c(k) = 0 \}$. Since $\nabla b(k_0)$, $\nabla c(k_0)$ are independent then the system $(\nabla b(k_0), \nabla c(k_0),v)$ is a basis of $\mathbb{R}^3$ for all $v \neq 0$ in $T_{k_0}S$, (where $T_{k_0}S $ denotes the tangent space of $S$ at $k_0$). Therefore, we can choose as coordinates $$ y_1 = b(k),\quad y_2 = c(k),\quad z = v.k $$ With this change of variables we get \begin{align*} \rho (e) &= \int_{\{ G(y,z) - \vert y \vert \leq e,\, (y,z) \in W \}} J(y,z) dy \,dz \\ &\quad + \int_{\{ G (y,z) + \vert y \vert \leq e ,\, (y,z) \in W \}} J(y,z) dy\, dz + h(e) \end{align*} where $J$ is analytic in $W$ a complex neighborhood of $(0,0)$, $G(y,z) = a (k)$ and $h$ is analytic near $0$. By polar coordinates $ y \to r ( \cos(\theta), \sin(\theta)) := r \omega$, $W$ moves into $W_1$ and we obtain \begin{align*} \rho (e) &= \int_{\{ G(r \omega,z) - r \leq e ,\,(r,\omega,z) \in W_1 \}} J(r\omega ,z) rdrd\omega dz \\ &\quad + \int_{\{ G(r \omega,z) + r \leq e ,\,(r, \omega,z) \in W_1 \}} J(r\omega ,z) r\,dr\,d\omega\, dz + h(e)\\ & = -\int_{\{ G(r \omega,z) + r \leq e ,\,(-r,-\omega,z) \in W_1 \}} J(r\omega ,z) r\,dr\,d\omega \,dz\\ &\quad + \int_{\{ G(r \omega,z) + r \leq e ,\,(r, \omega,z) \in W_1 \}} J(r\omega ,z) r\,dr\,d\omega \,dz + h(e) \end{align*} In the first integral of the last equation we have use the change $(r,\omega) \to (-r,-\omega)$. The assumption that $ a/_{\{b=c=0\}}$ is non-degenerate implies $G(0,0)=0$, $\partial_zG(0,0) = 0$ and $\nabla_z^2G(0,0) \neq 0$. We may assume that $\nabla_z^2G(0,0) >0$. Applying Taylor's formula to the function $ y \to a(y,z)$, we get $$ G(r \omega,z) = G(0,z) + r G_1(r, \omega, z), $$ The condition ($\ref{eq10}$) yields $\vert G_1 \vert <1$. $$ G(r \omega,z) +r = G(0,z) + r (G_1(r, \omega, z)+1). $$ The change of variable $\tilde r = r (G_1(r, \omega, z)+1)$ leads to \begin{align*} \rho(e) &= -\int_{\{ G(0,z) + \tilde r \leq e ,\, \tilde r <0, W_1 \}} J_1(\tilde r,\omega ,z) d\tilde r d\omega dz \\ &\quad + \int_{\{ G(0,z) + \tilde r \leq e ,\, \tilde r >0, W_1 \}} J_1(\tilde r,\omega ,z) d\tilde r d\omega dz + h(e). \end{align*} Since $G(0,0) = 0$, $\partial_z G(0,0) =0 $ and $\nabla_z^2 G(0,0) > 0$, there exists $ \alpha (z)$ such that $G(0,z)= \alpha (z) z^2$, with $\alpha(0) >0$. Hence, \begin{align*} \rho(e) &= -\int_{\{ z^2 + \tilde r \leq e ,\, \tilde r <0, W_2 \}} J_2(\tilde r,\omega ,z) d\tilde r d\omega dz \\ &\quad +\int_{\{ z^2 + \tilde r \leq e ,\, \tilde r >0, W_2 \}} J_2(\tilde r,\omega ,z) d\tilde r d\omega dz +h(e) \\ &= -\int_{\{ z^2 + \tilde r \leq e ,\, W_2 \}} J_2(\tilde r,\omega ,z) d\tilde r d\omega dz \\ &\quad + 2\int_{\{ z^2 + \tilde r \leq e ,\, \tilde r >0, W_2 \}} J_2(\tilde r,\omega ,z) d\tilde r d\omega dz +h(e) \end{align*} The first integral in the last equation is an analytic function in $e$ near 0. If $e<0$ the set $\{ z^2 +\tilde r \leq e :\tilde r >0,W_2\}$ is empty, then $\rho (e)$ is reduced to the first integral. If $e>0$ the second integral is a non vanishing function near 0. Moreover this function is analytic in term of $\sqrt e$. This yields analytic singularity for $\rho$. \end{proof} This lemma and \cite[Theorem 1.6]{d2} lead to the following theorem. \begin{theorem}\label{theo4} Let $J$ be an open interval in which \eqref{eq11} is valid. Let $I$ satisfying (H2) and let $E \in I \cap ( e_0 + \mathop{\rm sing\,supp}_{\rm a}(\mu)) $ be such that $ ( E - \mathop{\rm supp}(\mu)) \subset J$. Then for all $h$-independent complex neighborhood $\Omega$ of $E$, there exist $h_0=h(\Omega)>0$ sufficiently small and $C = C( \Omega) >0$ such that for $ h \in ]0,h_0[$, $$ \#\{z\in \Omega; z\in {\rm Res}P(h)\} \ge C_\Omega h^{-n}. $$ \end{theorem} \begin{thebibliography}{00} \bibitem{d1} M. Dimassi, M. Mnif, \emph{Lower bounds for the counting function of resonances for a perturbation of a periodic Schr\"odinger operator by decreasing potential}. C. R. Acad. Sci. Paris, Ser. I335, 1013-1016 (2002). Zbl1032.35063 \bibitem{d2} M. Dimassi, M. Zerzeri, \emph{A local trace formula for resonances of perturbed periodic Schr\"odinger operators}. Journal of Functional Analysis 198, 142-159 (2003). Zbl pre01901766 \bibitem{n1} L. Nedelec, \emph{Localization of resonances for matrix Schr\"odinger operators}. Duke Math. J. 106, no. 2, 209--236 (2001). \bibitem{r1} M. Reed, B. Simon \emph{Methods of Modern Mathematical Physics, analysis operators}. \rm Academic Press, New York-London, (1978). \bibitem{s1} J. Sj\"ostrand, \emph{A trace formula for resonances and application to semi-classical Schr\"odinger operators}. S\'eminaire \'equations aux d\'eriv\'ees partielles, expos\'e no 11 (1996-97). \end{thebibliography} \end{document}