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this paper is a follow - up of @xcite and establishes wegner - type ( more precisely , wegmer  stollmann - type ) bounds for random continuous schrdinger operators . we focus here on a two - particle interactive anderson model in a continuous space , subject to a random external field with a continuous argument . the infinite - volume hamiltonian of the model is a schrdinger operator @xmath2 acting on functions @xmath3 : @xmath4 here @xmath5 is the kinetic energy operator : @xmath6 where @xmath7 is the laplacian in variable @xmath8 corresponding to the @xmath9th particle : @xmath10    the randomness in hamiltonian @xmath11 is concentrated in the potential energy function @xmath12 which is written in the form @xmath13 the term @xmath14 represents the ( nonrandom ) interaction potential while the sum @xmath15 $ ] describes the action of an external potential field on the two - particle system . more precisely , the external potential field is represented by a family of real random variables ( rvs ) @xmath16 , or , in probabilistic terminology , by a real - valued random field with a continuous argument ( or simply ) . a sample of such a random field is a ( measurable ) function @xmath17 , and its its graph is a ` hypersurface ' in @xmath18 , with a unique point of intersection with any straight line orthogonal to the ` argument space ' @xmath0 . the argument @xmath19 in the notation @xmath20 stresses randomness of the external potential . the distribution of the @xmath21 is denoted by @xmath22 : it is a probability measure on a ( suitably chosen ) sigma - algebra @xmath23 in the space of sample functions @xmath24 . formal definitions are given in section [ sectextpot ] ; cf . a classical example is a gaussian distribution on @xmath23 ; see , e.g. , @xcite . the randomness introduced in hamiltonian @xmath25 perplexes its spectral properties . a physically - motivated result expected here is that the spectrum of @xmath25 near its ` lower edge ' is pure point with probability one , and the corresponding eigenfunctions decay exponentially in space . ( this is called lifshits - tail - based exponential localisation . ) similar results have been established for a variety of single - particle models where the hamiltonian acts on a function @xmath26 as @xmath27 , @xmath28 . see @xcite , @xcite and references therein . as far as localisation is concerned , the main difference between a single - particle model and its two - particle counterpart is that replacing term @xmath20 by the sum @xmath29 leads to ` strong ' dependencies between rvs @xmath12 and @xmath30 , no matter how far points @xmath31 are positioned from each other . the ensuing difficulty can not be relieved no matter how quickly correlations between rvs @xmath20 and @xmath32 decay when @xmath33 are far from each other . @xcite where this discussion has been conducted for the so - called tight - binding two - particle anderson model on a lattice .    an important ingredient of the existing localisation proofs for single - particle anderson models or their two - particle counterparts is the so - called wegner - type ( or wegner  stollmann type ) estimates for the corresponding hamiltonian . [ exceptions are one - dimensional models where the whole physics of localisation is rather special . ] this explains the title of this paper . the wegner - type estimates are produced for the eigen - values of a finite - volume approximation of the hamiltonian ; cf . @xcite . in our situation , these are operators @xmath34 acting on functions @xmath35 : @xmath36 with @xmath37 and @xmath12 defined , as before , by equation ( 1.2.2 ) . here @xmath38 is a bounded domain in @xmath39 which we choose to be the cartesian product of two @xmath40-dimensional cubes : for @xmath41 and @xmath42 , @xmath43 where , for @xmath44 and @xmath45 , @xmath46 . \eqno ( 1.8)\ ] ] the shorthand notation @xmath38 for @xmath47 ( and @xmath48 for @xmath49 ) will be systematically used in this paper . further , @xmath50 and @xmath51 will stand for cubes @xmath52 and @xmath53 forming the projections of ` two - particle ' parallelepiped @xmath54 onto single - particle configurational spaces ( and similarly with @xmath55 and @xmath56 ) . next , @xmath57 is the laplacian in variable @xmath58 confined to cube @xmath59 , @xmath60 , with a specified boundary condition which we choose to be dirichlet s : @xmath61 in fact , a possible choice of boundary conditions is rather broad and includes periodic and ` elastic ' , in particular , neumann s . ( what we need for our method to work is that the self - adjoint extension specified by a boundary condition is half - bounded and has a compact resolvent @xmath62 for @xmath63 away from the real line . ) see @xcite , @xcite for details . thus , operator @xmath64 in eqn ( 1.6 ) is understood in terms of the sesquilinear form @xmath65 in @xmath66 , where @xmath67 stand for the standard inner product . a similar meaning is attributed to the multiplication operator by @xmath12 in @xmath66 ; consequently , we treat the function @xmath12 , @xmath68 , as an element of @xmath66 defined for @xmath22-a.a . additional conditions imposed below mean that this function is bounded on @xmath38 , and the supremum @xmath69 of its absolute value has certain moments ; see below . moreover , under the assumptions introduced in this paper , the ( self - adjoint ) operator @xmath70 is bounded from below and has , with probability one , a discrete spectrum of a finite multiplicity . it is convenient to write its eigenvalues @xmath71 in an increasing order : @xmath72 in the ` one - volume ' wegner estimate one assesses the probability that at least one eigenvalue @xmath73 of operator @xmath70 falls in a ( narrow ) interval around a given point @xmath74 on the spectral axis : @xmath75    next , the ` two - volume ' wegner estimate addresses the probability that the eigenvalues @xmath73 and @xmath76 of operators @xmath77 and @xmath78 come near to each other in a given interval @xmath79 , for two ( distant ) parallelepipeds @xmath80 and @xmath81 . that is , @xmath82 here and below , @xmath22 stands for the corresponding probability measure on the underlying probability space ( see section [ sectextpot ] ) .    from the probabilistic point of view , the estimates for probabilities ( 1.11 ) and ( 1.12 ) are examples of concentration inequalities , albeit for rather implicit rvs @xmath73 carrying a considerable amount of dependence . for single - particle anderson models , assuming natural properties of the random term @xmath20 , the wegner estimates are rather straightforward . for the two - particle models , the study of these estimates , with a view of localisation , started in @xcite , @xcite , @xcite and is continued in this paper .    in the main body of this work , the interaction potential @xmath83 satisfies the following property :    \(i ) _ @xmath83 is a ( measurable ) bounded real function @xmath84 obeying _ @xmath85 here @xmath86 is the interaction radius , @xmath87 stands for the permutation of the ( vector ) variables ( @xmath88 for @xmath89 ) and @xmath90 denotes the max - norm in @xmath0 : for @xmath91 , @xmath60 : @xmath92 \eqno ( 1.14)\ ] ]    we can also allow the case where @xmath83 has a hard core , i.e. , @xmath93 with @xmath94 being the diameter of the hard core . however , we have to assume that @xmath95 remains uniformly bounded for @xmath96 with @xmath97 .    in section [ sectextpot ] , we give formal conditions upon the structure of the potential energy terms in hamiltonian ( 1.5 ) . a useful notion is the ` shadow ' @xmath98 of a parallelepiped @xmath99 : @xmath100 it can be a cube or a union of two cubes in @xmath0 , possibly disjoint . in what follows we call sets of this kind ` cellular ' .    throughout the paper , @xmath101 stands for the ( euclidean ) volume of parallelepiped @xmath102 and @xmath103 for that of a projection cube @xmath104 . in addition , we use a similar notation @xmath105 for a cellular set @xmath106 .    finally , note that an extension of the results of the present paper to the general case with @xmath107 particles is also possible . to this end , one needs to apply the technique proposed recently in @xcite . we plan to publish such an extension in a forthcoming paper . a common model of a is a gaussian random field on @xmath0 , with its characteristic ` linearity structure ' ; it also serves as a ` base ' for producing wider families of . our presentation will take this fact into account : comments on properties of the external potential field will be made from a ` gaussian perspective ' . however , the class of the external fields under consideration is much larger and includes various ` perturbations ' of and ` operations ' with gaussian . we will not venture in this direction in the current paper but plan to address this issue elsewhere . a convenient way to describe our assumptions on @xmath21 is as follows . fix a ( measurable ) function @xmath108 which is ( strictly ) positive - definite : @xmath109 ( measurable ) function @xmath110 , the lebesgue integral @xmath111 and @xmath112 unless @xmath113 a.e .. in these inequalities we allow the integral in the lhs of ( 2.1 ) to equal @xmath114 . however , given a cellular set @xmath115 , we denote by @xmath116 the set of functions @xmath117 with support in @xmath118 and finite @xmath119 : @xmath120 equipped with the inner product @xmath121 and the norm @xmath122 . we will often suppose that an orthonormal basis in @xmath116 has been given , @xmath123 , where @xmath124 here and below , @xmath125 stands for the indicator function of set @xmath126 and the normalising constant is given by @xmath127 in the case where @xmath128 admits a bound @xmath129 where @xmath130 , we have that @xmath131 . our first condition on @xmath21 is :    \(s ) _ summability _ : @xmath109 cellular set @xmath106 and @xmath132 : @xmath133 consequently , we set @xmath134(\om ) = \int_{{\ra } } \zeta ( y)v(y;\om ) { \rm d}y , \eqno ( 2.7)\ ] ] which yields a correctly defined ( and a.s . finite ) rv @xmath135 $ ] . in particular , given a basis @xmath136 in @xmath116 , we obtain a sequence of rvs @xmath137(\om ) $ ] .    in view of condition ( s ) , we can represent a random realisation @xmath138 , @xmath139 , of external potential @xmath21 in @xmath126 as the sum @xmath140(\om ) \eta^{{\ra}}_i(x),\eqno ( 2.8.1)\ ] ] where the series is considered in space @xmath116 . for that reason , we will employ an alternative notation @xmath141(\om ) $ ] and the decomposition @xmath142 and call rvs @xmath143 coefficient random variables in @xmath116 . the whole sequence @xmath144 of the coefficient rvs in @xmath116 is denoted by @xmath145 ( it is , of course , basis - dependent ) . furthermore , if @xmath146 is a member of the basis @xmath136 in @xmath116 , then we denote by @xmath147}$ ] the sequence of the remaining rvs @xmath148 .    to illustrate the role of space @xmath116 , take the example of a gaussian @xmath21 in @xmath0 with the covariance kernel @xmath128 . in this case , rvs @xmath149 $ ] and @xmath150 $ ] defined in ( 2.7 ) for functions @xmath151 with @xmath152 are independent normal @xmath153 . when @xmath154 , with @xmath54 , the series ( 2.8.1 ) and ( 2.8.2 ) also determine the map @xmath155 as a ( random ) multiplication operator in @xmath66 . this in turn allows us to define the ( random ) hamiltonian @xmath70 in @xmath66 ; cf . eqn ( 1.4 ) . given a basis in @xmath156 , the randomness in @xmath77 is represented by a sequence of coefficient rvs @xmath157 . we denote by @xmath158 the sigma - algebra ( in the space of locally square - integrable functions @xmath159 ) generated by the rvs @xmath135 $ ] , @xmath160 . then the sigma - algebra @xmath23 is the smallest one containing @xmath158 , @xmath109 cellular sets @xmath126 . the underlying probability distribution @xmath22 is defined on @xmath23 ; the restriction of @xmath22 to @xmath161 is denoted by @xmath162 and the expectation relative to @xmath163 by @xmath164 . our second condition on @xmath165 is :    \(t ) _ temperedness _ : @xmath109 cellular set @xmath106 , the rv @xmath166 is @xmath163-a.s . finite , and has a finite moment @xmath167 in this condition , we employ the representation @xmath168 following eqn ( 2.8.2 ) . although the choice of coefficient rvs @xmath169 depends on the basis in @xmath116 , condition t is basis - independent : when it holds for a particular choice of the orthonotrmal basis in @xmath116 , it also holds for all bases .    * remark 2.1.*[rem21 ] condition ( 2.10 ) holds for a gaussian @xmath170under a mild assumption that its samples are ` regular ' . ( in fact , a much stronger property takes place , guaranteeing ( 2.10 ) . ) see , e.g. , @xcite and references therein .    given a pair of disjoint cellular sets @xmath171 , we will be working with the conditional distribution functions @xmath172}\right)$ ] defined as follows . let us fix a basis @xmath173 in @xmath174 with @xmath175 . then set : @xmath176}\right ) : = \p\left(\left[{\wh{\mathbf 1}}_{\ra}\right]<y\big| \left\{\left[\eta^{{\ra}\cup{\ra}'}_j\right]:\;j\geq 1\right\ } \right),\end{array}\eqno ( 2.12)\ ] ] for the coefficient rv @xmath177 = \left[\eta^{\ra}_0\right]$ ] , given @xmath178}_{{\ra}\cup{\ra}'}= \left\{\left[\eta^{\ra}_i\right]:\;i\geq 1\right\}$ ] , a collection of other basis - related coefficient rvs in @xmath174 . in fact , in eqn ( 2.12 ) we are interested in sets of the form @xmath179 and @xmath180 where parallelepipeds @xmath181 and @xmath182 are distant apart , namely with @xmath183 . \eqno ( 2.13)\ ] ]    more precisely , we exploit a simple geometric fact stated in lemma [ geom ] :    [ geom ] _ consider two parallelepipeds @xmath181 and @xmath184 and suppose that condition _ ( 2.13 ) _ holds true . then there are two possibilities ( which in general do not exclude each other ) : _ \(i ) @xmath38 _ and @xmath48 are ` completely separated ' , when _ @xmath185 > 0 . \eqno ( 2.14)\ ] ]    \(ii ) @xmath38 _ and @xmath48 are ` partially separated ' . in this case one ( or more ) of the four possibilities can occur : _ @xmath186 ] & > 0,\\ { \rm{(b ) } } & \dist [ \pi_2\blam ,   [ \pi_1\blam \cup \pi\blam ' ] ] & > 0,\\ { \rm{(c ) } } & \dist [ \pi_1\blam ' , [ \pi\blam \cup \pi_2\blam ' ] ] & > 0,\\ { \rm{(d ) } } & \dist [ \pi_2\blam ' , [ \pi\blam \cup \pi_1\blam ' ] ] & > 0 . \end{array } \eqno ( 2.15)\ ] ]    the proof of lemma [ geom ] is straightforward ; it has been given in our earlier paper @xcite and applied to the lattice case , but geometrical arguments in @xcite actually refer to parallelepipeds in euclidean space @xmath0 . pictorially , case ( ii ) is where one of the cubes @xmath187 , @xmath188 , @xmath60 , is disjoint from the union of the rest of the projections of @xmath189 and @xmath48 . we note that the use of the max - norm @xmath190 is convenient here as it leads to the constant @xmath191 ( equal to @xmath192 times @xmath193 , the number of projections @xmath59 and @xmath188 , @xmath60 ) which does not depend on dimension @xmath40 . we make use of lemma [ geom ] as follows . first , for a given pair of bounded cellular sets @xmath171 , with @xmath194 , we set : @xmath195}}}\;\ ; \big[\;f_{{\ra}}\left(\,{\ry}+{\rb};\;\bgam _ { { \ra}\cup{\ra}'}^{\bot\left[{\wh{\mathbf 1}_{{\ra}}}\right]}\right ) -f_{{\ra}}\left(\,{\ry};\;\bgam_{{\ra}\cup{\ra}'}^{\bot\left [ { \wh{\mathbf 1}_{{\ra}}}\right]}\right)\big]\ , . \end{array } \eqno ( 2.16)\ ] ] here , it is assumed that we have been given a basis @xmath173 in @xmath174 , with @xmath196 .    in particular , with @xmath197 , we have : @xmath198}\right ) -f_{{\ra}}\left(\,{\ry};\;\bgam_{{\ra}}^{\bot { \wh{\mathbf 1}_{{\ra}}}}\right)\big]\,.\end{array } \eqno ( 2.17)\ ] ] we then set : @xmath199    next , given @xmath200 , we denote : @xmath201\;\hbox { and } \ ; \pi\blam\cap\pi\blam'=\emptyset\big\}\ , , \end{array } \eqno ( 2.19)\ ] ] @xmath202\qquad{}\\ \hbox { and } \pi_1\blam\cap\big(\pi_2\blam\cup \pi\blam'\big)=\emptyset\big\ } \end{array } \eqno ( 2.20)\ ] ] and @xmath203\qquad{}\\ \text { and } \pi_2\blam\cap\big(\pi_1\blam\cup \pi\blam'\big)=\emptyset\big\ } \,.\end{array } \eqno ( 2.21)\ ] ] ( we suppose here that we have bases @xmath204 in @xmath205 , with @xmath206 in eqn ( 2.19 ) , @xmath207 in eqn ( 2.20 ) and @xmath208 in eqn ( 2.21 ) . )    finally , @xmath209 . \eqno ( 2.22)\ ] ]    * remark 2.2.*[rem22 ] quantity @xmath210 in eqn ( 2.16 ) describes a ` conditioned continuity modulus ' of ( the distribution function of ) rv @xmath211 , given a sample @xmath212 . it is important to have a grasp of the magnitude of this rv and therefore of quantity @xmath210 . as an example , we can think of variable @xmath213 $ ] as normal @xmath153 : this is the case where @xmath214 is gaussian with covariance kernel @xmath128 . in this case , @xmath215 and consequently , quantities @xmath216 and @xmath217 , @xmath218 , from eqns ( 2.18 ) and ( 2.19)(2.21 ) satisfy a similar bound .    concluding this section , we would like to comment on our approach to analysis of random fields with a continuous argument , particularly , on the meaning and use of quantities introduced in eqns ( 2.16)(2.22 ) . our main intention is to provide a benefit to a readers with a physical background . we start by referring back to eqn ( 1.3 ) specifying the structure of a two - particle potential energy under consideration . in a single - particle model , the hamiltonian @xmath219 is simply @xmath220 . in this case , wegner - type bounds were proved in @xcite , for a certain class of external potential ( including gaussian fields ) . the proof given in @xcite is based on a special kind of decomposition of an in a cube @xmath221 ( see ( 1.8 ) ) , of the form @xmath222 where the ( scalar ) random variable @xmath223 is independent of the ` fluctuation field ' @xmath224 . this suggests conditioning upon values of @xmath224 , so that the finite - volume hamiltonian @xmath225 ( again with dirichlet boundary conditions ) becomes a function of the scalar random parameter @xmath223 . naturally , its eigenvalues are also viewed as functions of @xmath223 ; their analysis , based on the so - called dirichlet  neumann bracketing and other techniques , leads to wegner - type bounds . these bounds are helpful for proving the existence of the limiting density of states for single - particle model in question . conditions on the random field @xmath138 in @xcite cover some non - gaussian , but they are somewhat restrictive even in the gaussian case .    in the present paper , we make use of a similar decomposition ( 2.8.1)(2.8.2 ) . for @xmath226 where @xmath227 , @xmath60 , it looks like @xmath228{\wh\one}_\lam(x ) + \xi_\lam ( x;\om ) , \ ; x\in\lam,\ ; \hbox { where } \;\xi_\lam ( x;\om)=\sum_{\eta_j^\lam\in \bgam_{{\lam}}^{\bot { \wh{\one}_{{\lam } } } } } \eqno ( 2.24)\ ] ] with a ( normalised ) constant ` ground level ' @xmath229 and a coefficient @xmath230 $ ] ` moderately dependent ' on the residual ` fluctuation field ' @xmath231 . such an idea is well - known in probability theory : in elementary courses of statistics , it is usually proved that `` the empiric mean of a gaussian sample is independent of the empiric variance '' . see , e.g. , @xcite .    in the context of this paper , decomposition ( 2.24 ) implies a similar decomposition for two - particle finite - volume eigenvalues @xmath232 ( see ( 1.10 ) ) . in particular , conditional on a value of @xmath233 , the eigenvalues @xmath232 are behaving , as functions of @xmath234 , in a controllable fashion , which leads to our wegner - type bounds . in the special case when random field @xmath235 is gaussian , the random variables @xmath232 are ( conditionally ) normal , with the means and variances determined by the conditions but behaving in a controllable fashion . a one - volume wegner - type bound for finite - volume hamiltonians is given in theorem [ thm1 ] below . let @xmath236 denote the ( random ) spectrum of operator @xmath77 from eqn ( 1.4 ) ( i.e. , the collection of its eigenvalues @xmath237 , without their multiplicities ) . next , set @xmath238\\ \;&\leq \ou_{\blam}\ ; + \;\ov_{\pi_1\blam}(\om ) \;+\;\ov_{\pi_2\blam}(\om ) \\ \;&\leq \ou_{\blam } + 2\ov_{\pi\blam}(\om ) . \end{array}\eqno ( 3.1)\ ] ] where @xmath239 _ assume the above conditions _ ( i ) , ( e0 ) _ and _ ( e1 ) _ on function @xmath83 and external potential @xmath21 . then , @xmath109 @xmath240 , @xmath241 , @xmath242 and @xmath243 , with @xmath54 : @xmath244\cap \sigma\left(h_{\blam}\right)\neq\emptyset\big)\\ \qquad\qquad\leq c_1\,\cdot\,\left|\blam\right|\,\cdot\ , \e\left(e+2+\ow_{\blam}\right)^{d } \;\cdot\mu_{\blam } \left(4z_{\pi\blam}\cdot\eps\right ) . \end{array}\eqno ( 3.2)\ ] ] here @xmath245 is a constant independent of @xmath74 , @xmath246 , @xmath247 and @xmath248 . _ the expression in the rhs of ( 3.2 ) includes a ` volume ' factor @xmath249 and a factor @xmath250 reflecting the growth of the ( value of the ) external potential @xmath21 . next , we have a factor @xmath251 controlling ` singularity ' of the conditional distribution functions @xmath252}\right)\;\hbox{and}\ ; f_{\pi_j\blam}\left(\,{\ry};\;\bgam_{\pi_{\oj}\blam\cup\pi\blam'}^{\bot\left [ { \wh{\mathbf 1}_{\pi_j\blam}}\right]}\right),\;j=1,2,\;{\ry}\in\r , \ ] ] with a controlled distance between projections @xmath98 and @xmath253 . here @xmath254 . in a ` smooth ' situation where this distribution function has a density that is bounded uniformly in @xmath255 and in @xmath256 , @xmath247 , @xmath257 satisfying ( 2.13 ) , ( see remark [ rem21 ] above ) , we have that @xmath258 where @xmath259 is a constant . again , such a form of the bound is important for the proof of anderson s localisation , the main area of application of wegner - type inequalities . our conditions on the random external potential field are straightforward , as well as the core argument of the proof given below . however , they require from the reader sufficient familiarity with standard constructions used in the analysis of random fields with a continuous argument . it may be of some help to the reader to know that these conditions are written down by postulating , in general terms , well - known qualitative properties of gaussian random fields . on the other hand , the obtained eigenvalue concentration bounds are not optimal , and they are not supposed to be , since our main motivation here is to lay ground for the multi - particle multi - scale analysis in a euclidean space @xmath0 , using ideas and techniques similar to those used in @xcite , @xcite . given a two - particle parallelepiped @xmath181 , the randomness in operator @xmath260 is represented by the sample of the @xmath261 . more conveniently , we can refer to a sequence @xmath262 of random variables @xmath263 representing sample @xmath264 in @xmath265 : @xmath266 it is convenient to think of @xmath260 as a family of operators @xmath267 in @xmath66 parametrised by vectors @xmath268 ( i.e. , sample vectors of @xmath269 ) : @xmath270(\ux ) = \left[h^{(0)}_{\blam}\bphi\right](\ux)+w_\blam(\ux;\bgam ) \bphi ( \ux ) \\ = -\,\diy{\frac{1}{2}}\sum\limits_{j=1,2 } \delta^{(\lam_{l_j}(u_j))}_j\bphi(\ux ) +   \left(u(\ux)+\sum\limits_{j=1,2}\sum\limits_{i=0,1,\ldots } { \bgam}_i\eta^{\pi\blam}_i(x_j)\right ) \bphi(\ux),\\ \qquad\qquad\ux=(x_1,x_2)\in\blam , \;\;\bphi\in\l_2(\blam ) . \end{array}\eqno(3.3)\ ] ] vectors @xmath271 are then made random , subject to the distribution @xmath272 of @xmath273 . consequently , the eigenvalues @xmath73 of @xmath267 ( written in the non - decreasing order ) are parametrised by @xmath271 , i.e. , are random variables on @xmath274 . we can denote them by @xmath275 but in fact , to stress the dependence on function @xmath276 , we will use an alternative notation @xmath277 , @xmath278 .    in this setting , it is convenient to choose a basis @xmath279 in @xmath280 with @xmath281 and work with @xmath282 , the probability distribution for rv @xmath283\;$ ] conditional on a given vector of values @xmath284 of rvs @xmath285}_{\pi\blam}= \left\{\gam^{\pi\blam}_i , i\geq 1\right\}$ ] . our first remark is that the additive change @xmath286 in the sample of the external potential field @xmath287 generates the change @xmath288 in the value of the potential energy function @xmath289 , @xmath68 . correspondingly , the eigenvalues @xmath290 and @xmath291 of the operator @xmath77 before and after the change are related with @xmath292 recalling that @xmath293 , we obtain that @xmath109 @xmath294 : @xmath295 \big|\bgam^{\geq 1)}\right)\leq \mu_{\pi\blam } \left(4z_{\pi\blam_l}\eps\right).\eqno ( 3.4)\ ] ]    the second remark is that , when we pass from ( 3.4 ) to ( 3.2 ) , we only have to examine a finite number of eigenvalues @xmath296 of hamiltonian @xmath77 . more precisely , denote by @xmath297 the eigenvalues of the kinetic energy operator @xmath298 . then the bound @xmath299 implies that , @xmath109 @xmath300 , the conditional probability @xmath244\cap \sigma\left(h_{\blam}\right)\neq\emptyset\big|\ov_{\pi\blam } \in [ l-1,l)\big)\\ \leq \#\left\{\hbox{eigenvalues of $ h^0_{\blam}$ in $ [ e-\eps -l , e+\eps + l]$}\right\}\;\cdot\mu_{\pi\blam } \left(4z_{\pi\blam}\eps\right).\end{array}\eqno ( 3.6)\ ] ]    the final remark is that @xmath301$}\right\}\\ \leq \#\left(\z^d\times\z^d\cap\left\{\ux\in\r^d \times\r^d:\;\|\ux\|_{\rm{euclid}}\leq e+\eps + l\right\ } \right)\\ \leq c_1\left|\blam\right|(e+\eps + l)^{d}\end{array}\eqno ( 3.7)\ ] ] where @xmath302 is the the constant from weyl s formula . hence , by taking expectation , we obtain from ( 3.6 ) that the lhs in ( 3.2 ) is @xmath303 i.e. , the bound ( 3.2 ) holds true . this completes the proof of theorem [ thm1 ] . we now turn to a two - volume wegner - type bound , dealing with a pair of parallelepipeds @xmath181 and @xmath184 ( more precisely , with the corresponding hamiltonians @xmath77 and @xmath78 ) , under an assumption that the distance between @xmath38 and @xmath48 is of the same order of magnitude as the size of these parallelepipeds . given an interval @xmath304 , set : @xmath305\\ \qquad\qquad= \inf\;\left[\ ; \left|e^{(\blam ) } _ k - e^{(\blam')}_{k'}\right|:\;\ ; e^{(\blam ) } _ k , e^{(\blam')}_{k'}\in j\ ; \right],\end{array}\eqno ( 3.8)\ ] ] where @xmath73 are the eigenvalues of @xmath77 and @xmath306 those of @xmath78 , + @xmath307 . because the potential @xmath308\eqno ( 3.9)\ ] ] is a symmetric function of the pair @xmath309 , with @xmath310 consequently , the spectra @xmath311 and @xmath312 are identical , and the same is true for @xmath313 and @xmath314 . @xmath109 @xmath315 , @xmath316 with @xmath317   \eqno(3.10)\ ] ] and @xmath109 @xmath318 and interval @xmath319\subset\r$ ] , where @xmath320 and @xmath321 : _ @xmath322 \leq \eps \big)\qquad{}\qquad{}\\ \qquad\leq\;c_2\,\cdot\ , \left|\blam\right|\;\left|\blam'\right|\ ; \e\big[({\rb}+\delta + 1+\ow_{\blam})^d({\rb}+\delta + 1+\ow_{\blam'})^d\big]\\ \qquad\qquad\qquad\qquad\times\ ; { \operatornamewithlimits{\max}\limits_{j=1,2}}\ ; \left[\omu_{\pi\blam}\left(4\eps\cdot z_{\pi\blam}\right),\ ; \omu_{\pi\blam'}\left(4\eps\cdot z_{\pi\blam'}\right)\right]\ ,   .\end{array}\eqno ( 3.11)\ ] ]    * remark 3.1.*[rem31 ] as in the case of the one - volume wegner - type estimate ( 3.2 ) , the rhs in ( 3.11 ) is composed by ` volume ' factors @xmath323 and @xmath324 and factors @xmath325\eqno ( 3.12)\ ] ] and @xmath326\,.\eqno ( 3.13)\ ] ] these factors assess various aspects of randomness introduced in hamiltonians @xmath77 and @xmath78 . for us , an immediate use of theorems [ thm1 ] and [ thm2 ] is in proving anderson s localisation in a two - particle model ( with interval @xmath319 $ ] at the  edge of the spectrum , specified by further assumptions on rf @xmath165 ) . owing to lemma [ geom ] , parallelepipeds @xmath189 and @xmath48 obeying ( 3.10 ) satisfy either ( i ) or ( ii ) , i.e. they are either completely or partially separated . consider first case ( i ) ( complete separation ) . write @xmath322\leq \eps \big)\\ \qquad= \esm { \p\big(\dist\left[\sigma\left(h_{\blam}\right ) \cap j , \sigma\left(h_{\blam'}\right)\cap j\right ] \leq \eps \ , \big| \,\bfv_{\pi\blam ' } \big)}. \end{array}\eqno ( 3.14)\ ] ] note first that , under conditioning in eqn ( 3.14 ) , the eigen - values @xmath306 , @xmath327 , forming the set @xmath328 are non - random . therefore , it makes sense to use the following inequality : @xmath329 \leq \eps \ , \big| \,\bfv_{\gamma ' } \big)\\ \qquad{}\leq   n_{\blam'}\left(j;\ow_{\blam'}\right ) { \operatornamewithlimits \sup\limits_{e\in j}}\ ; \p\big(\dist\left[\sigma\left(h_{\blam}\right)\cap j , e\right ] \leq \eps\big|\bfv_{\pi\blam'}\big ) , \end{array}\eqno ( 3.15)\ ] ] where @xmath330 is the number of the eigenvalues @xmath306 that can ` eventuallly ' fall in @xmath331 . as in ( 3.11 ) , for @xmath332 $ ] and with @xmath333 standing for the eigenvalues of @xmath334 , we have : @xmath335$}\right\}\\ \quad\leq b_{2d}(l'_1l'_2)^{d}(b+\ow_{\blam'})^{d}. \end{array}\eqno ( 3.16)\ ] ]    next , as in theorem [ thm1 ] , @xmath336 \leq \eps\big|\bgam_{\pi\blam'}\big)\\ \qquad\leq c_1\,\cdot\,\left|\blam\right| \e\big[\left(e+2+\ow_{\blam}\right)^{d}\big| \bgam_{\pi\blam'}\big]\quad{}\\ \;\\ \qquad\quad\times \omu_{\pi\blam}\left(4\eps z_{\pi\blam}\right ) . \end{array}\eqno ( 3.17)\ ] ] this yields @xmath329 \leq \eps \ , \big| \,\bgam_{\pi\blam'}\big)\qquad\qquad\qquad{}\\ \qquad\leq c^{(1)}_2\;|\blam|\;|\blam'|\ ; ( b+2+\ow_{\blam'})^d\;\e\big[(b+2+\ow_{\blam})^d \big|\bgam_{\pi\blam'}\big]\qquad{}\\ \qquad\quad\times \omu_{\pi\blam}\left(4\eps z_{\pi\blam}\right ) , \end{array}\eqno ( 3.18)\ ] ] implying that @xmath337\leq \eps \big)\\ \qquad\leq c^{(1)}_2\;|\blam|\;|\blam'|\;\e\big[(b+2 + \ow_{\blam})^d(b+2+\ow_{\blam'})^d\big]\qquad{}\\ \qquad\quad\times \omu_{\pi\blam}\left(4\eps z_{\pi\blam}\right ) , \end{array}\eqno ( 3.19)\ ] ] where @xmath338 is a constant . \(ii ) now consider the case of partial separation . for example , assume case ( a ) from lemma [ geom ] ( see eqn ( 3.11 ) ) , when projection @xmath50 is disjoint from the union of the rest of the projections of @xmath38 and @xmath48 : @xmath339 = \emptyset.\eqno ( 3.20)\ ] ] we then estimate the probability in the lhs of ( 3.16 ) with the help of the conditional expectation @xmath337\leq \eps \big)\\ \qquad= \esm { \p\big(\dist\left[\sigma\left(h_{\blam}\right ) , \sigma\left(h_{\blam'}\right)\right ] \leq \eps \ , \big| \,\bgam_{\pi_2\blam\cup\pi\blam ' } \big)\big|\bgam_{\pi\blam'}}. \end{array } \eqno(3.21)\ ] ]    note that , owing to ( 3.20 ) , the sigma - algebra generated by @xmath340 does not include any of the rvs @xmath341 forming @xmath342 . thus , the argument used in the proof of theorem [ thm1 ] is still applicable if , instead of @xmath22 , we work with the probability distribution @xmath343 , conditional on @xmath344 and restricted to the the sigma - algebra generated by @xmath345 . this allows us to write @xmath346 \leq \eps \ , \big| \,\bgam^{\geq 1}_{\pi_1\blam},\bgam_{\pi_2\blam\cup\pi\blam ' } \big)\\ \qquad\leq c^{(2)}_2\;|\blam|\;|\blam'|\ ; ( b+2+\ow_{\blam'})^d\\ \qquad\quad\times\e\big[(b+2+\ow_{\blam})^d \big|\bgam^{\geq 1}_{\pi_1\blam } , \bgam_{\pi_2\blam\cup\pi\blam'}\big]\ ; \omu_{\pi\blam}\left(4\eps z_{\pi\blam}\right ) \end{array}\eqno ( 3.22)\ ] ] and @xmath346 \leq \eps \,\big)\\ \qquad\leq c^{(2)}_2\;|\blam|\;|\blam'|\ ; \e\big[(b+2+\ow_{\blam})^d(b+2+\ow_{\blam'})^d\big ] \qquad{}\\ \qquad\quad\times \omu_{\pi\blam}\left(4\eps z_{\pi\blam}\right ) . \end{array}\eqno ( 3.23)\ ] ] we are then in position to deduce the required bound for the the conditional probability in the lhs of ( 3.21 ) . if , instead of ( 3.20 ) , we have one of the other disjointedness relations ( b)-(d ) in eqn ( 3.11 ) , then the argument is conducted in a similar fashion . specifically , in case ( b ) we swap projections @xmath50 and @xmath347 in the above argument . furthermore , in cases ( c ) and ( d ) , we should exchange @xmath247 and @xmath257 as compared to arguments in cases ( a ) and ( b ) .    we now briefly comment on the case where the interaction potential has a hard core ( cf . ( 1.15 ) ) . in this case hamiltonians @xmath25 and @xmath70 in eqns ( 1.1 ) and ( 1.5 ) act in subspaces @xmath348 and @xmath349 where @xmath350 more precisely , @xmath25 and @xmath70 carry additional dirihclet s boundary conditions on @xmath351 . however , the scheme of the proof of theorems [ thm1 ] and [ thm2 ] remains unchanged , and all arguments carry through . [ bcss1 ] a. boutet de monvel , v. chulaevsky , p. stollmann and y. suhov , _ wegner - type bounds for a two - particle continuous anderson model with an alloy - type external potential . _ - arxiv : math - ph/0821:2621 , 2008 .            [ cs2 ] v. chulaevsky , y. suhov , _ eigenfunctions in a two - particle anderson tight binding model_. commun . phys , doi @xmath353 ( 2009 ) . [ cs3 ] v. chulaevsky , y. suhov , _ multi - particle anderson localisation : induction on the number of particles . _ math - ph / arxiv:0811.2530 . [ fhlm ] w. fischer , t. hupfer , h. leschke , p. mller , _ existence of the density of states for multi - dimensional continuum schrdinger operators with gaussian random potentials . phys . , * 190 * ( 1997 ) , no . 1 , 133141 .                we thank the isaac newton institute , university of cambridge , for hospitality during the programme `` mathematics and physics of the anderson localization : 50 years after '' ( july  december , 2008 ) . ys thanks ihes , bures - sur - yvette , for hospitality during visit in 2008 .
we analyse a two - particle quantum system in @xmath0 with interaction and in presence of a random external potential field with a continuous argument ( an anderson model in a continuous space ) . 
 our aim is to establish the so - called wegner - type estimates for such a model , assessing the probability that random spectra of hamiltonians in finite volumes intersect with a given set . for the lattice version of the two - particle model , 
 a similar result was obtained in @xcite    h l p    00 2 ^ 2 2 ^ 2 _ # 1#1 # 1#1 # 1\{#1 } # 1 # 1#2_#1 # 1| # 1 | # 1\{#1 } # 1 \ { # 1 } # 1#2 _#1\ { # 2 }    # 1| # 1 | # 1 # 1    # 1@xmath1-s # 1@xmath1-s # 1    # 1#2#3 # 1#2 # 1