0,$ are defined as $n_\pm(\l)=n_\pm(\l;\LF)=$ $\#\{n:\;\pm \l_n^{\pm}(\LF)>\l\}$. The singular numbers $s_n(\LF)$ of the operator $\TF^\LF$ are just the absolute values of $\l_n^{\pm}$ ordered non-increasingly, and their distribution function equals $n(\l;\LF)=n_+(\l;\LF)+n_-(\l,\LF)$. We denote by $\X(\l)$ the function $\frac12 \frac{|\ln \l|}{\ln|\ln \l|}$. \begin{proposition}\label{5:EstimateAboveL} For an operator $\LF$ with bounded compactly supported coefficients, \begin{equation}\label{5:EstimateAbove.0} \limsup_{\l\to0} n_\pm(\l;\LF)\X(\l)^{-1}\le1. \end{equation} \end{proposition} The estimate \eqref{5:EstimateAbove.0} means that the eigenvalues of $\TF^{\LF}$ converge to zero extremely rapidly, super-exponentially: \begin{equation}\label{5:EstimateAbove.01} \limsup_{n\to\infty}(n!\l_n^\pm(\LF) )^{\frac1{n}}<\infty. \end{equation} The estimates \eqref{5:EstimateAbove.0}, \eqref{5:EstimateAbove.01} do not exclude the possibility that the operator $\TF^{\LF}$ has only a finite number of positive and/or negative eigenvalues, counting multiplicity. \begin{proof} We start by proving a special version of the proposition, dealing with the case when the operator $\LF$ is just the multiplication by a bounded compactly supported function $W$. The corresponding Toeplitz operator will be denoted by $\TF^W$ We apply the classical variational approach. Consider the quadratic form $\tF^W[v]$ of the operator $\TF^W$: \begin{equation}%\label{5:EstimateAbove.1} \nonumber \tF^W[v]=\int W(x)|v(x)|^2 dx,\; v\in \Hc_0. \end{equation} The variational description of eigenvalues (Glazman lemma) implies \begin{equation}%\label{5:EstimateAbove.2} \nonumber n_+(\l;W)= \min\codim\{\Ls\subset \Hc_0: \tF^W[v]\le \l \|v\|^2, \, v\in\Ls\}. \end{equation} We recall the description of the space $\Hc_0$ to get \begin{equation}\label{5:EstimateAbove.3} n_+(\l;W)= \min\codim\{\Ms\subset\Fc^\bb:\tF^W[e^{-\P}f]\le \l \|e^{-\P}f\|^2, f\in\Ms\}, \end{equation} where $\Fc^\bb$ is the space of entire analytical functions, defined in Sect.2. Now we are going to estimate the quantity in \eqref{5:EstimateAbove.3} from above by means of changing the forms and the spaces. First, if we replace $\|e^{-\P}f\|^2$ in \eqref{5:EstimateAbove.3} by something smaller, then there will be fewer subspaces $\Ms$ where the inequality holds and therefore $\min \codim$ may become only larger. So we can replace $\|e^{-\P}f\|^2$ by $C\|e^{-(1+\e)\P^\circ}f\|^2$ where $\P^\circ$ is the potential defined in \eqref{2:Creation.00} and $C$ is some constant, since, due to our estimates of $\Psi$, discussed in Sect.3, $e^{-\P}\ge C e^{-(1+\e)\P^\circ}$. After this, if we replace the enveloping space $\Fc^\bb$ in \eqref{5:EstimateAbove.3} by a larger one, $\Fc_{(1+\e)\Bn}$, then the codimension of subspaces may only grow. Therefore, we arrive at \begin{gather}\label{5:EstimateAbove.4} n_+(\l,W)\le\\ \notag \min\codim\{\Ms\subset\Fc_{(1+\e)\Bn}:\tF^W[e^{-\P}f]\le \l \|C e^{-(1+\e)\P^\circ}f\|^2, f\in\Ms\}. \end{gather} Now we notice that $\tF^W[e^{-\P}f]=\int We^{-2\P}|f|^2 dx$, and $We^{-2\P}$ is again a bounded function with compact support. Therefore we can apply estimates for the right-hand side in \eqref{5:EstimateAbove.4} established in \cite{RaiWar}, Proposition~3.2 or \cite{MelRoz}, Lemma~6.1, which lead to the inequality \eqref{5:EstimateAbove.0}. The estimate we have just established can even be improved, if we use the results of \cite{FilPush} to majorize the right-hand side in \eqref{5:EstimateAbove.4}: \begin{equation}%\label{5:EstimateAbove.5} \nonumber \limsup_{n\to\infty}(n!\l_n^\pm(W) )^{\frac1{n}}\le \frac{\Bn}2 \Cp(\supp W), \end{equation} where $\Cp(\supp W)$ is the logarithmical capacity of the support of $W$. Now we pass to the general case. Let all coefficients $p_{\a\b}$ of the operator $\LF$ have support in the disk with radius $\Rb$ centered at $0$ and $|p_{\a\b}|\le \Mb$. Denote by $\chi$ the characteristic function of the disk $\Db$ of radius $2\Rb$. By Cauchy integral formula, applied as in Lemma~\ref{3:LemComp}, each term of the form $p_{\a\b}\partial_1^{\a}\partial_2^{\b}u$, $u\in \Hc_0$ can be expressed as $T_{\a\b}( \chi u)$, with a bounded operator $T_{\a\b}$. The upper bound for the norm of this operator depends only on the order of derivatives $\a+\b$, on the size of the coefficient $p_{\a\b}$ and on the radius $\Rb$. Since the quadratic form $\tF^\LF$ of the operator $\TF^\LF$ has the form $\tF^\LF[u]=\|\sum p_{\a\b}\partial_1^{\a}\partial_1^{\b}u\|^2$ $=\|\sum p_{\a\b}\partial_1^{\a}\partial_1^{\b}\chi u\|^2$, we have \begin{equation}\label{5:Above.4} |\tF^\LF[u]|\le \varkappa^{-1} \|\chi u\|^2=\varkappa^{-1}\tF^\chi[u], \; u\in \Hc_0 \end{equation} for a certain $\varkappa$. Thus the singular numbers of the operator $\TF^\LF$ are majorated by the ones of the operator $\TF^W$ with $W=\chi,$ and for such operators the required estimate is already proved. \end{proof} It is more convenient for our needs to formulate Proposition~\ref{5:EstimateAboveL}, directly in the variational form, declaring the existence of subspaces with prescribed properties. \begin{proposition}\label{5:PropAbove} There exists a constant $\varkappa$ depending only on $l, \Rb, \Mb$, such that for any $\e>0$ and for any $\l>0,$ sufficiently small, there exists a subspace $\Ms(\l)\in \Hc^0$ such that the codimension of $\Ms(\l)$ is no greater than $(1+\e)\X(\l)$, and for any order $l$ differential operator $\LF$ with coefficients $p_{\a\b}$ supported in the disk with radius $\Rb$ and satisfying $|p_{\a\b}|\le \Mb$ \begin{equation}\label{5:Above.3} \tF^\LF[u]\le \varkappa\l\|u\|^2, u\in \Ms(\l). \end{equation} \end{proposition} The important feature in this formulation is that the subspace $\Ms(\l)$ of controlled codimension can be chosen in such way that it services simultaneously all Toeplitz-like operators generated by differential operators $\LF$ of fixed order and with coefficients subject to the above restrictions. \begin{proof} By \eqref{5:Above.4}, as soon as the chosen subspace $\Ms(\l)$ services the Toeplitz operator $\TF^\chi$, the inequality \eqref{5:Above.3} holds for all operators $\LF$ subject to our conditions.\end{proof} We formulate the eigenvalue estimate from below for operators $\TF^W$ only, under the condition that the function $W$ is non-negative. This latter condition can be somewhat relaxed, however some kind of positivity requirements still remain. In fact, starting from \cite{RaiWar}, \cite{MelRoz}, the positivity condition for lower spectral estimates remains a serious obstacle in this field. \begin{proposition}\label{5:EstimateBelow} Let the function $W$ be non-negative and be greater than some $c>0$ on an open set. Then \begin{equation}%\label{5:EstimateBelow.0} \nonumber \liminf_{\l\to0} n_+(\l;W)\X(\l)^{-1}\ge 1. \end{equation} \end{proposition} \begin{proof} We use the variational description of $ n_\pm(\l;W)$: \begin{equation}\label{5:EstimateBelow.1} n_+(\l;W)= \max\dim\{\Ls\subset \Hc_0: \tF_0[v]< \l \|v\|^2, \, v\in\Ls, v\ne0\}. \end{equation} Again, using the description of the space $\Hc_0$ of zero modes, we can rewrite \eqref{5:EstimateBelow.1} as \begin{equation}\label{5:EstimateBelow.2} n_+(\l;W)= \max\dim\{\Ms\subset\Fc^\bb:\tF_0[e^{-\P}f]> \l \|e^{-\P}f\|^2, f\in\Ms, f\ne 0\} \end{equation} By replacing $\|e^{-\P}f\|^2$ by some larger quantity, we narrow the choice of subspaces $\Ms$ where the inequality in \eqref{5:EstimateBelow.2} holds. So, we replace $\|e^{-\P}f\|^2$ by $\|Ce^{-(1-\e)\P^\circ}f\|^2$ with an arbitrary $\e\in(0,1)$ and some $C$. This leads to decreasing of the right-hand side of \eqref{5:EstimateBelow.2}. A further decreasing is obtained by narrowing the enveloping space: we replace $\Fc^\bb$ by $\Fc_{(1-\e)\Bn}$. This transforms the right-hand side in \eqref{5:EstimateBelow.2} to the form, where the estimate from below is already proved, again in \cite{RaiWar}, Proposition~3.2 or \cite{MelRoz}, Lemma~6.1. \end{proof} We will need the uniform variational version of the latter estimate. \begin{proposition}\label{5:ToeplitzSubspaces} Let $\Omega$ be an open set, $c>0$. Then for any $\e>0$ and for any $\l>0$, small enough, depending on $\e$ there exists a subspace $\Ls=\Ls(\l)\subset \Hc_0$ such that $(Wv,v)> \l \|v\|^2, \, v\in\Ls\setminus\{0\}$ for all functions $W\ge0$ satisfying $W\ge c$ on $\Omega$ and $ \dim \Ls(\l)\ge (1-\e)\X(\l)$.\end{proposition} The result follows from the estimate $(Wv,v)\ge c(\chi_\Omega v,v)$. \section{Perturbed eigenvalues} The passage from the spectral estimates for Toeplitz-type operators to the ones for the Pauli operator has been performed in a different way in \cite{Raikov1}, \cite{RaiWar} and in \cite{MelRoz}. A short and neat reasoning in \cite{Raikov1}, \cite{RaiWar} based upon the proper version of the Birman-Schwinger principle requires positivity of the perturbing operator which fails to be positive in our case. Therefore we follow the approach of \cite{MelRoz}, Sect.~9, using variational considerations, however certain new features appear. The particular form of the variational principle we use, derived immediately from the spectral theorem, is the following. For a self-adjoint operator $\AF$ and an interval $(s,r)$ in the real axis we denote by $N(s,r ; \AF)$ the total multiplicity of the spectrum of $\AF$ in $(s,r)$; when this quantity is finite, it is the number of eigenvalues of $\AF$ in $(s,r)$, counting multiplicities. \begin{proposition}\label{6:VarPrinc} Let $\AF$ be a self-adjoint operator with domain $\dom{\AF}$. Then, for $s < r$, \begin{eqnarray} \lefteqn{ \hspace*{-0.7cm} N(s,r ; \AF) } \nonumber \\ & & \hspace*{-1.55cm} = \max {\rm dim} \left\{ {\Ls} \subset \dom{\AF} : \| (\AF-\m)u \|^{2} < \t^{2} \| u \|^{2}, u \in {\Ls} \backslash \{ 0 \} \right\} \label{6:VarPrinc.01} \\ & & \hspace*{-1.55cm} = \min {\rm codim} \! \left\{ {\Ls} \! \subset \! \dom{\AF} : \| (\AF\!-\!\m)u\|^{2} \! \geq \!\t^{2} \| u \|^{2}, u \! \in \! {\Ls} \right\}, \label{6:VarPrinc.02} \end{eqnarray} where $\m=(r+s)/2$ and $\t=(r-s)/2$. \end{proposition} When using Proposition~\ref{6:VarPrinc}, one looks for a subspace $\Ls$ where the inequality in \eqref{6:VarPrinc.01} is fulfilled. The dimension of this subspace gives an estimate from below for $N(s,r ; \AF)$. On the other hand, having found a subspace $\Ls$ where the inequality in \eqref{6:VarPrinc.02} is satisfied, we can be sure that the codimension of this subspace estimates $N(s,r ; \AF)$ from above. Considering perturbations of the Landau Hamiltonian by general electric and magnetic fields, we should accept the possibility that the eigenvalues may split away from the Landau level both up and down. Therefore we need rather advanced notations. We fix a Landau level $\L=\L_q=2q\Bn$ and set $s_\pm=\L\pm \Bn$. For given $\l\in(0,\Bn/4)$ we define $\m_\pm=(\L\pm\l +s_\pm)/2$, $\t= \frac12\Bn-\frac{\l}{2}$. Thus, $\m_\pm$ is the midpoint of the interval between $\L\pm\l$ and $s_\pm$, $\t$ is the half-length of this interval. For the perturbation given by an operator $\VF$ we denote by $N_+(\l;q,\bb,\VF)$ the eigenvalue counting function $N(\L_q+\l,s_+;\PF_-+\VF)$ where $\PF_-$ is the Pauli operator with magnetic field $\Bb=\Bn+\bb$; similarly, $N_-(\l;q,\bb,\VF)$ is defined as $N(s_-,\L_q-\l;\PF_-+\VF)$. Thus, $N_+$, $N_-$ count the eigenvalues converging to $\L$ from above, resp., from below. \subsection{Upper estimate}\label{Above} We start with the upper estimate. It declares that the eigenvalues split away from the Landau level at least superexponentially rapidly. This estimate is rather robust: is does not require much smoothness of the perturbing magnetic field or of the electric potential. It is convenient to keep all the previous notations in the paper for a smooth case, and use the superscript $\sharp$ for the non-smooth versions. So, let $\bs\in C^2_0(\R^2)$ be a magnetic field with compact support, $\ps\in C^4$ be its scalar potential, $\D\ps=\bs$, and $\as\in C^3$ its vector potential. The Pauli operator associated to the field $\Bn+\bs$ will be denoted by $\PF_-^\sharp$. We consider the smoothened magnetic field, $ \bb=\omega*\bs $, where $\omega\in C_0^\infty$ is a smooth function with compact support, $\int \omega dx=1$, and $\omega(x)$ depends only on $|x|$. Thus the field $\bb$ is smooth and compactly supported. Of course, $\p=\omega*\ps$ is a potential for $\bb$, $\D\p=\bb$. Moreover, since $\ps$ is a harmonic function outside the support of $\bs$, by the mean value property, $\p$ coincides with $\ps$ outside some compact. Correspondingly the vector potential $\ab$ of the field $\bb$ coincides with $\as$ outside a compact. As a result, the Pauli operator $\PF_-^\sharp$ can be expressed as \begin{equation}\label{6:Upper.Pert1}\nonumber \PF_-^\sharp=\PF_-+\MF, \end{equation} where $\PF_-$ is the Pauli operator with smooth magnetic field $\Bb=\Bn+\bb$ and $\MF$ is a first order differential operator with $C^2$-coefficients having compact support. Further on, we consider the operator $\PF_-^\sharp+V$ as a perturbation of $\PF_-$, \begin{equation}\label{6:Upper.Pert2}\nonumber \PF_-^\sharp+V=\PF_-+\VF,\; \VF=\MF+V, \; V\in C_0^2(\R^2). \end{equation} Now we can formulate the main upper estimate. \begin{theorem}\label{6:Upper} If the perturbations $\bs$ and $V$ belong to $C^2_0(\R^2)$ then \begin{equation}\label{6:Upper.0} \limsup_{\l\to 0+}N_\pm(\l,\PF_-^\sharp+V)\X(\l)^{-1}\le1. \end{equation} \end{theorem} \begin{proof} We will prove the 'minus' version of the Theorem; the 'plus' version differs just by non-essential details. Keeping in mind Proposition~\ref{6:VarPrinc}, we are going to construct a subspace $\Ls$ in the domain of the operator $\PF_-$ such that the inequality \begin{equation}\label{6:Upper.1}\nonumber \|(\PF_-+\VF-\m)u\|^2 \ge \t^2\|u\|^2\end{equation} holds for all $u\in\Ls$, and estimate its codimension. We represent an arbitrary function $u$ as $u=u_1+u_2$ where $u_1\in \Gc=\Gc_q$ belongs to the approximate spectral subspace constructed in Section~\ref{Subspaces3} and $u_2$ belongs to the subspace $\hat{\Gc} $ orthogonal to $\Gc$, $u_1=Q_qu$, $u_2=\hat{Q}_qu$, $\hat{Q}_q=1-Q_q$. Then the inequality we aim for takes the form \begin{equation}\label{6:Upper.2} \|(\PF_-+\VF-\m)(u_1+u_2)\|^2-\t^2\|u_1\|^2-\t^2\|u_2\|^2\ge 0. \end{equation} The left-hand side in \eqref{6:Upper.2} can be written as \begin{gather}\notag \bl\|(\PF_-+\VF-\m)u_1\|^2-\t^2\|u_1\|^2\br + \bl\|(\PF_-+\VF-\m)u_2\|^2-\t^2\|u_2\|^2\br\\ \label{6:Upper.3} +2\Re((\PF_-+\VF-\m)u_1,(\PF_-+\VF-\m)u_2). \end{gather} Following the pattern of the proof of Proposition 9.2 in \cite{MelRoz}, we are going to find subspaces of controlled codimension in $\Gc$ and in $\hat{\Gc}$, where the first and the second terms in \eqref{6:Upper.3} are positive, with some margin, while the third term in \eqref{6:Upper.3} is majorized by this margin. \textbf{The first term.} We start with the first term. Since $u_1\in \Gc$, by our construction, $u_1$ can be written in a unique way as $u_1=\Qa^q v_1$, $v_1\in \Hc_0$, and thus the term takes the form \begin{gather}\label{6:Upper.4} \|(\PF_-+\VF-\m)u_1\|^2-\t^2\|u_1\|^2=\|(\PF_-+\VF-\m)\Qa^q v_1\|^2 -\t^2\|\Qa^q v_1\|^2\\ \notag =\|(\PF_--\m)\Qa^qv_1\|^2-\t^2\|\Qa^q v_1\|^2 +2\Re((\PF_--\m)\Qa^q v_1, \VF\Qa^q v_1)\\ \notag+(\VF\Qa^q v_1,\VF\Qa^q v_1). \end{gather} The first two terms in \eqref{6:Upper.4} can be transformed by means of Proposition~\ref{3:basicPropositionBP}; they produce $C_q((\L-\m)^2 -\t^2)\|v_1\|^2$ as well as $C_q(W_1v_1,v_1)$ with some compactly supported function $W_1$. The last term is non-negative and can be ignored when estimating the whole expression from below. As for the remaining, next to last term in \eqref{6:Upper.4}, we have \begin{equation}\label{6:Upper.4.1}\nonumber 2\Re((\PF_--\m)\Qa^q v_1, \VF\Qa^q v_1)\ge -(\|\chi\LF_1v_1 \|^2 +\|\LF_2v_1 \|^2), \end{equation} where $\chi\LF_1,\LF_2$ are order $q+2$ and order $q+1$ differential operators with bounded compactly supported coefficients, $\LF_1=(\PF_--\m)\Qa^q$, $\LF_2=\VF\Qa^q$, and $\chi$ is a characteristic function of an open set containing the support of coefficients of $\VF$, so that $\chi\VF=\VF$. As a result, we obtain \begin{gather}\label{6:Upper.5} \|(\PF_-+\VF-\m)u_1\|^2-\t^2\|u_1\|^2\ge C_q((\L-\m)^2 -\t^2)\|v_1\|^2\\ \nonumber+(Wv_1,v_1)-\|\chi\LF_1v_1 \|^2 -\|\LF_2v_1 \|^2. \end{gather} The expression $((\L-\m)^2 -\t^2)$ equals $(\L-\m-\t)(\L-\m+\t)=\l (\L-s)$. Now we are going to chose our first subspace. We fix $\e\in(0,\frac13)$. By Proposition~\ref{5:PropAbove}, there exists a subspace $\Ms_1$ in $\Hc_0$ such that $|(Wv_1,v_1)|, \|\chi\LF_1v_1 \|^2, \|\LF_2v_1 \|^2$ are all no greater than $\frac1{10}\l (\L-s)\|v_1\|^2$ and the codimension of $\Ms_1$ in $\Hc_0$ is less than $(1+\e)\X(\frac1{10}(\L-s)\l)$ for $\l$ small enough. So, for $v_1\in \Ms_1$ the right-hand side of \eqref{6:Upper.5} is no less than $\frac34C_q(\L-s)\l\|v_1\|^2$. This means that on the subspace $\Ls_1=\Qa^q\Ms_1$, which has the same codimension in $\Gc$ as $\Ms_1$ has in $\Hc_0$, the left hand side of \eqref{6:Upper.5} is no less than $\frac34C_q(\L-s)\l\|v_1\|^2$. The operator $\Qa^q:\Hc_0\to\Gc$ is boundedly invertible, so we can majorize $\|v_1\|$ by $||u_1||$, as in Corollary~\ref{3:Corollary}, therefore, for some constant $c_1$, not depending on $\e,\l$, \begin{equation}%\label{6:Upper.6} \nonumber \|(\PF_-+\VF-\m)u_1\|^2-\t^2\|u_1\|\ge c_1 \l\|u_1\|^2, \; u_1\in\Ls_1. \end{equation} The subspace $\Ls_1$ has codimension no greater than $(1+\e)\X(\frac1{10}(\L-s)\l)$ in $\Gc$. \textbf{The second term}. In the next term in \eqref{6:Upper.3}, $\|(\PF_--\m+\VF)u_2\|^2$ $-\t^2\|u_2\|^2$, we can write $u_2=(1-Q_q)u_2=(1-P_q)u_2+(P_q-Q_q)u_2$ where $P_q$ is the spectral projection of $\PF_-$ corresponding to the neighborhood $\d_q=(\L-\g,\L+\g)$ of $\L=\L_q$, defined in Section 3. We denote by $\tilde{P}$ the spectral projection of $\PF_-$ corresponding to the set $(s-\Bn/2,\L-\g]\cup[\L+\g,\L+\Bn)$. Since the spectrum of $\PF_-$ is discrete in $(\L-2\Bn, \L)\cap(\L,\L+2\Bn)$, the projection $\tilde{P}$ has finite rank, and thus $u_2$ can be expressed as \begin{equation}\label{6:Upper2.1} u_2=(1-P_q-\tilde{P})u_2+(P_q-Q_q+\tilde{P})u_2 =g+h. \end{equation} Constructed as above, the function $g$ is lying in the spectral subspace of $\PF_-$ corresponding to the exterior of the interval $(\L-3\Bn/2,\L+\Bn)$. So, by means of the spectral theorem, we can estimate $\|(\PF_--\mu)g\|$ from below as $\|(\PF_--\mu)g\|\ge (\t+\Bn/2)\|g\|$. Therefore \begin{equation}\label{6:Upper2.2} \|(\PF_--\mu)g\|^2-\t^2\|g\|^2\ge c_2\|g\|^2, \end{equation} as well as \begin{equation}\label{6:Upper2.3} \|(\PF_--\mu)g\|^2-\t^2\|g\|^2\ge c_3\|(\PF_--\mu)g\|^2 \end{equation} with some constants $c_2,c_3$ not depending on $\l$. Now let us consider $h=(P_q-Q_q+\tilde{P})u_2$. As we have proved in Section~\ref{Projections4}, the operator $P_q-Q_q$ is compact and remains compact after multiplication by any combination of creation and annihilation operators. This property remains intact after adding the finite rank operator $\tilde{P}$. So, we can write \eqref{6:Upper2.1} as \begin{equation}\label{6:Upper2.5} u_2=g+\KF u_2 \end{equation} with a compact operator $\KF$. Therefore, on a subspace of finite codimension, \eqref{6:Upper2.5} can be rewritten as \begin{equation}%\label{6:Upper2.6} \nonumber u_2=g+\KF'g \end{equation} with a compact operator $\KF'$ which, again, remains compact after multiplication by $\Q,\Qa$. As a result, for $\e$ fixed, we can find a subspace of finite codimension in $\Gc'\cap \Dom(\PF_-)$ such that \begin{equation}%\label{6:Upper2.7} \nonumber \frac{ \|u_2\|}{\|g\|}\in(1-\e,1+\e),\; \frac{\|(\PF_--\m)u_2\|}{\|(\PF_--\m)g\|}\in(1-\e,1+\e). \end{equation} Therefore the inequalities of the form \eqref{6:Upper2.2}, \eqref{6:Upper2.3} can be written for $u_2$ as well, just with a slight worsening of constants, the value of which is not important: \begin{equation}%\label{6:Upper2.8} \nonumber \|(\PF_--\mu)u_2\|^2-\t^2\|u_2\|^2\ge c_4\|u_2\|^2, \end{equation} as well as \begin{equation}\label{6:Upper2.9} \|(\PF_--\mu)u_2\|^2-\t^2\|u_2\|^2\ge c_5\|(\PF_--\mu)u_2\|^2. \end{equation} Now we can consider the complete second term in \eqref{6:Upper.3}. This expression can be written as \begin{gather}\label{6:Upper2.10} \|(\PF_-+\VF-\m)u_2\|^2-\t^2\|u_2\|^2= \|(\PF_--\m)u_2\|^2-\t^2\|u_2\|^2\\ \notag +\|\VF u_2\|^2+2\Re((\PF_--\m)u_2,\VF u_2), \; u_2\in\hat{\Gc}. \end{gather} The form $\|\VF u_2\|^2$ is compact with respect to $\|(\PF_--\m)u_2\|^2$, therefore on a subspace of finite codimension we have $\|\VF u_2\|^2<\frac{c_5}4 \|(\PF_--\m)u_2\|^2$. Further on, by Cauchy inequality,\begin{equation}\label{6:Upper2.11} |2\Re((\PF_--\m)u_2,\VF u_2)|\le \frac{c_5}{4}\|(\PF_--\m)u_2\|^2 +\frac{4}{c_5}\|\VF u_2\|^2.\end{equation} Again, using the same relative compactness, we can restrict ourselves to a subspace of finite codimension so that the last term in \eqref{6:Upper2.11} is no greater than $\frac{c_5}4\|(\PF_--\mu)u_2\|^2$. After substituting these estimates into \eqref{6:Upper2.10} and recalling \eqref{6:Upper2.9}, we obtain \begin{equation}\label{6:Upper2.12} \|(\PF_-+\VF-\m)u_2\|^2-\t^2\|u_2\|^2\ge \frac{3c_5}{4}\|(\PF_--\mu)u_2\|^2\ge c_6 \|u_2\|^2 \end{equation} on a subspace $\Ls_2$ of finite codimension in $\Gc'\cap\Dom(\PF_-)$. Note that the subspaces here do not depend on the value of $\l$. \textbf{The third term.} We are going to show now that the third term in \eqref{6:Upper.3} is majorized by the sum of two first ones. We can write it in the form \begin{equation}\label{6:Upper3.1} 2\Re ((\PF_-+\VF-\m)u_1,(\PF_-+\VF-\m)u_2)= 2\Re((\PF_-+\VF-\m)^2u_1,u_2). \end{equation} The function $u_1\in\Gc_q$ can be written as $u_1=\Qa^q v_1$ for some $v_1\in \Hc_0$. So we have \begin{gather}\label{6:Upper3.2} (\PF_-+\VF-\m)^2u_1=(\Qa\Q -\mu)^2 \Qa^q v_1 +(\PF_--\mu)\VF u_1+\\ \notag \VF(\PF_--\mu) u_1 +V^2 u_1 = (\Qa\Q -\mu)^2 \Qa^q v_1 +G. \end{gather} We start with the first term in \eqref{6:Upper3.2}. In the expression $\Qa\Q\Qa^q v_1$ we perform our usual commuting procedure moving $\Q$ to the right all the time. As soon as $\Q$ reaches the utmost right position, the corresponding term vanishes. Thus we arrive at the representation \begin{gather}\label{6:Upper3.3} \Qa\Q\Qa^q v=2\sum\Qa^j(\Bn+\bb)\Qa^{q-j}v_1\\ \notag=2q\Bn \Qa^qv_1 +\sum_{j=1}^q\Qa^j\bb\Qa^{q-j}v=\L u_1 + F_1. \end{gather} In a similar way, we transform the expression $\Qa\Q\Qa\Q u_1= \Qa\Q\Qa\Q \Qa^q v_1$: we move both copies of the operator $\Q$ to the utmost right position, where they vanish, being applied to $v_1\in \Hc_0$. What is left has the form \begin{equation}\label{6:Upper3.4} \Qa\Q\Qa\Q \Qa^q v_1 =\L^2 u_1 +F_2. \end{equation} The functions $F_1,F_2$ in \eqref{6:Upper3.3}, \eqref{6:Upper3.4} are compositions of several, no more than $q$, operators $\Qa$ and at least one function with compact support, applied to $v_1$. The term $G=((\PF_--\mu)\VF + \VF(\PF_--\mu) +\VF^2) \Qa^q v_1$ in \eqref{6:Upper3.2} involves the differential operator of order $q+2$ with bounded compactly supported coefficients, applied to $v_1$. So we have \begin{equation}%\label{6:Upper3.5} \nonumber (\PF_-+V-\m)^2u_1 =(\L-\m)^2u_1-2\m F_1+F_2+G. \end{equation} We substitute this expression into \eqref{6:Upper3.1} and use the fact that $u_1$ and $u_2$ are orthogonal. This leads us to \begin{gather}\label{6:Upper3.6}\nonumber 2|\Re((\PF_-+V-\m)^2u_1,u_2)|=2|\Re((-2\m F_1+F_2+G),u_2)|\\ \notag \le M\|-2\m F_1+F_2+G\|^2+M^{-1}\|u_2\|^2, \end{gather} where $M>0$ can be chosen arbitrarily. We fix $M$ larger than $16 c_5^{-1}$, where $c_5$ is the constant in \eqref{6:Upper2.12}. After this, we take $\e$ smaller than $\frac{c_5}{16M}$. By Proposition~\ref{5:PropAbove} we can choose the subspace $\Ms_1\subset\Hc_0$, constructed when considering the first term in \eqref{6:Upper.2}, in such way that the codimension of $\Ms_1$ in $\Hc_0$ is no greater than $(1+\e)\X(\varkappa\l)$ for some $\kappa$ and that $\|-2\m F_1+F_2+G\|^2$ is no greater than $ \frac{c_1}{2}\l\|v_1\|^2$. With all parameters chosen in this way, for $u_1\in\Ls_1=\Qa^q\Ms_1$ and $u_2\in\Ls_2$ the third term in \eqref{6:Upper.3} is majorated by the sum of the first two terms, and thus the whole expression in \eqref{6:Upper.3} is non-negative for $u=u_1+u_2\in \Ls_1\oplus\Ls_2$. The subspace $\Ls=\Ls_1\oplus\Ls_2$ has codimension no greater than $(1+\e)\X(\kappa\l)+N(\e)$. As a result, due to $\e$ being arbitrary and to the fact that $\X(\kappa\l)\equiv \X(\l), \l\to0$, we have \begin{equation}%\label{6:UpperFinal} \nonumber \limsup_{\l\to0}N_-(\l;q,\bb,\VF)\X(\l)^{-1}= \limsup_{\l\to0}N_-(\l;q,\bb,\VF)\X(\varkappa\l)^{-1}\le(1+\e) \end{equation} Finally, we let $\e\to0$, and obtain \eqref{6:Upper.0}. \end{proof} \subsection{Lower estimate}\label{below} Now we establish an estimate for $N_\pm(\l)$ from below. The conditions for this estimate to hold are expressed in the terms of an effective potential constructed for each LL from the magnetic field $\bb$ and the electric potential $V$. The complicated form of this effective potential reflects the complicated character of the influence of the magnetic perturbation onto the behavior of eigenvalues: we remind again that the latter dependence is not monotone. Our construction requires considerable smoothness of $\bb$ and $V$; in order to avoid extra technicalities, we suppose here that both are infinitely smooth. For a fixed LL $\L=\L_q$, with the same $s_\pm,\m_\pm,\t$ as before, we define the effective potentials as \begin{gather}\label{6:Lower.00} \!\!\! W_\pm=W_\pm[\bb,V,\l]=\!-\!(\L\pm\l\!+\!2\Bn)(s+2\Bn)\Zc_q[\bb] -\Zc_{q+2}[\bb]\\\nonumber+2(\m_\pm+3\Bn)\Zc_{q+1}[\bb] -\Xc_q[\bb, 4(2\Bb-\bb+\m_\pm)\bb-(4\Bb-2\m_\pm+V)V]\\\nonumber-2\Xc_{q+1}[\bb,V-3\bb]-4\Im\Yc_q[\bb,\pd V-2\pd b], \end{gather} where the expressions $\Zc, \Xc, \Yc$ are described in Proposition~\ref{3:basicPropositionBP}. \begin{theorem}\label{6:Lower} Suppose that for $\l$ close to zero the potential $W_\pm$ is non-negative and, moreover, there exists an open set $\Omega$ where $W_\pm\ge c>0$ for $c$ independent on $\l$. Then \begin{equation}\label{6:Lower.0} \liminf_{\l\to0}N_{\pm}(\l,q,\bb,V)\X(\l)^{-1}\ge 1 \end{equation} \end{theorem} \begin{proof} We consider the eigenvalues below $\L$ and prove the estimate \eqref{6:Lower.0} with 'minus' sign. The other case is proved in an almost identical way. We are going to construct the subspace $\Ls$ in $\Gc=\Gc_q$, having dimension asymptotically greater than $(1-\e)\X(\l)$ such that the inequality \begin{equation}\label{6:Lower.1} \|(\PF_-+V-\m)u\|^2 < \t^2\|u\|^2 \end{equation} is satisfied for all $u\in \Ls\setminus \{0\}.$ To do this, we set $u=\Qa^q v$, $v\in \Hc_0$, and thus reduce both parts of \eqref{6:Lower.1} to quadratic forms on the lowest Landau level using Proposition~\ref{3:basicPropositionBP} and then apply Proposition~\ref{5:EstimateBelow}. So, we have \begin{gather}\label{6:Lower.2} \|(\PF_-+V-\m)u\|^2 - \t^2\|u\|^2=(\PF_-u,\PF_-u)+2\Re(Vu,\PF_-u)+\\ \nonumber (\m^2-\t^2)\|u\|^2+(Vu,Vu)-2\m(Vu,u)-2\mu(\PF_-u,u). \end{gather} We substitute $u=\Qa^q v,\; v\in \Hc_0$ into \eqref{6:Lower.2} and consider each term. First, by \eqref{3:BP:equation}, we have \begin{equation}\label{6:Lower.3} (\m^2-\t^2)\|u\|^2=(\m^2-\t^2)C_q\|v\|^2+(\m^2-\t^2)(\Zc_q[\bb]v,v). \end{equation} By \eqref{3:BP:equation.V} applied for $U=V$ and then for $U=V^2$, we obtain \begin{equation}\label{6:Lower.4} -2\m(Vu,u)=-2\m(\Xc_q[\bb,V]v,v);\; (Vu,Vu)=(\Xc_q[\bb,V^2]v,v). \end{equation} For the last term in \eqref{6:Lower.2}, we have \begin{gather}\label{6:Lower.5} -2\mu(\PF_-u,u)=-2\mu(\Qa\Q u,u)=-2\mu(\Q\Qa u,u)+4\mu\Bn(u,u)\\ \nonumber +4\mu(\bb u,u)= -2\mu C_{q+1}\|v\|^2\\ \nonumber-2\mu (\Zc_{q+1}[\bb]v,v)+4\mu\Bn C_q\|v\|^2+4\mu\Bn(\Zc_{q}[\bb]v,v)+4\m (\Xc_q[\bb,\bb]v,v). \end{gather} Further on, commuting $\Q,\Qa$, we get \begin{gather}\label{6:Lower.6} 2\Re(Vu,\PF_-u)=2\Re(Vu, \Qa\Q u)=2\Re(\Qa Vu,\Qa u )-4(\Bb V u,u)\\ \nonumber =2( V\Qa u,\Qa u)+2\Re([\Qa,V]u,\Qa u)-4(\Bb V u,u) . \end{gather} To the first and the third terms in \eqref{6:Lower.6}, we apply \eqref{3:BP:equation.V}, and to the second term we apply \eqref{3:BP:equation.VQ} (with $U=-2i\pd V=[\Qa,V]$), obtaining \begin{gather}\label{6:Lower.7} 2\Re(Vu,\PF_-u)\\ \nonumber =2(\Xc_{q+1}[\bb,V]v,v)-4(\Xc_{q}[\bb,\Bb V ]v,v)+4\Im(\Yc_q[\bb, \pd V]v,v). \end{gather} Finally, for the term $(\PF_-u,\PF_-u)$ we have \begin{gather}\label{6:Lower.8} \|\PF_-u\|^2=((\Q\Qa -2\Bb)u,(\Q\Qa -2\Bb)u)=(\PF_-\Qa u,\Qa u) \\ \nonumber +4\Bn^2(u,u) +4((2\Bn+\bb)\bb u,u) - 4\Bn \|\Qa u\|^2 -4\Re(\Qa\bb u, \Qa u). \end{gather} All these terms are of the form already considered in \eqref{6:Lower.5}, \eqref{6:Lower.6},\eqref{6:Lower.7}. Thus we obtain \begin{gather}\label{6:Lower.9} (\PF_-u,\PF_-u)=C_{q+2}\|v\|^2+ (\Zc_{q+2}[\bb]v,v)-2\Bn C_{q+1}\|v\|^2\\ \nonumber -2\Bn(\Zc_{q+1}[\bb]v,v) -2(\Xc_{q+1}[\bb,\bb]v,v)+4\Bn^2(C_q\|v\|^2+(\Zc_q[\bb]v,v))\\ \nonumber- 4\Bn (C_{q+1}\|v\|^2+(\Zc_{q+1}[\bb]v,v)) + 4(\Xc_q[\bb,(2\Bn+\bb)\bb ]v,v)\\ \nonumber-4(\Xc_{q+1}[\bb,\bb]v,v)-8\Im(\Yc_q[\bb,\pd \bb]v,v). \end{gather} Now we collect the expressions \eqref{6:Lower.3}, \eqref{6:Lower.4}, \eqref{6:Lower.5}, \eqref{6:Lower.7} \eqref{6:Lower.9} to get \begin{gather}\label{6:Lower.10} \!\!\!\!\|(\PF_-\!+V-\!\m)u\|^2\! -\! \t^2\|u\|^2\!=\!(\m^2-\t^2)(C_q\|v\|^2+(\Zc_q[\bb]v,v))\\ \nonumber -2\m(\Xc_q[\bb,V]v,v)+(\Xc_q[\bb,V^2]v,v)-2\mu C_{q+1}\|v\|^2-2\mu (\Zc_{q+1}[\bb]v,v)\\ \nonumber+4\mu\Bn C_q\|v\|^2+4\mu\Bn(\Zc_{q}[\bb]v,v)+4\m (\Xc_q[\bb,\bb]v,v)+2(\Xc_{q+1}[\bb,V]v,v)\\ \nonumber-4(\Xc_{q}[\bb,\bb V ]v,v)+4\Im(\Yc_q[\bb, \pd V]v,v)+C_{q+2}\|v\|^2 + (\Zc_{q+2}[\bb]v,v)\\ \nonumber-2\Bn C_{q+1}\|v\|^2 -2\Bn(\Zc_{q+1}[\bb]v,v) -2(\Xc_{q+1}[\bb,\bb]v,v)+4\Bn^2C_q\|v\|^2\\ \nonumber+4\Bn^2(\Zc_q[\bb]v,v)- 4\Bn C_{q+1}\|v\|^2 -4\Bn(\Zc_{q+1}[\bb]v,v)\\\nonumber + 4(\Xc_q[\bb,(2\Bn+\bb)\bb ]v,v)-4(\Xc_{q+1}[\bb,\Bb]v,v)-8\Im(\Yc_q[\bb,\pd \bb]v,v). \end{gather} The terms with $\|v\|^2$ have the coefficient \begin{gather}\nonumber (\m^2-\t^2)C_q -2\mu C_{q+1} +4\mu\Bn C_q +C_{q+2}-2\Bn C_{q+1} +4\Bn^2C_q \\ \nonumber - 4\Bn C_{q+1}=((\m^2-\t^2)-2\m\L+\L^2)C_q=\l(\L-s)C_q. \end{gather} The sum of all remaining terms in \eqref{6:Lower.10} equals exactly $(-W_-v,v)$, where $W_-$ is given by \eqref{6:Lower.00}. Now, by Proposition~\ref{5:ToeplitzSubspaces} it is possible to find a subspace $\Ms(\l)\subset\Hc_0$ such that $(Wv,v)>\l(L-s)C_q\|v\|^2$ for $v\in \Hc_0$ and dimension of $\Ms(\l)$ is greater than $\X(\l(L-s)C_q)(1-o(\l))$. This is the subspace we need. \end{proof} If the conditions of Theorem~\ref{6:Lower} are satisfied then the conditions of Theorem~\ref{6:Upper} are satisfied as well and we arrive at the asymptotic formula \begin{equation}\label{6:asymptotics} \lim_{\l\to0}N_{\pm}(\l,q,\bb,V)\X(\l)^{-1}= 1 \end{equation} \section{Proofs of technical lemmas}\label{technical} \begin{proof}[Proof of Lemma~\ref{2:LemmaBound}] As usual, we may prove the inequality for functions $u$ in the Schwartz space first, and then extend it to all functions for which the right-hand side of \eqref{2:LemmaBound.1} is finite, by continuity. We will prove Lemma by a double induction. For $N=1$ the statement is obvious: $||\Q u||^2=(\PF_-u,u),$ $||\Qa u||^2=(\PF_+u,u)=(\PF_-u,u)+2(\Bb u,u).$ Now, suppose that for some $N_0$ we have established \eqref{2:LemmaBound.1} for all $N