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the single - impurity anderson model ( siam ) has been a paradigm of a strongly correlated electron system . @xcite this seemingly simple model gives rise to dynamical screening of the local spin by the electrons in the fermi sea , leading to a crossover from a weak coupling system at high temperatures ( @xmath0 ) to a strongly coupled one at low @xmath0 , with the relevant temperature scale given by the kondo temperature , @xmath1 . the recent observation of the kondo effect in quantum dots ( qd s ) , whose parameters can be tuned continuously , and which allow probing of various properties , @xcite has yielded strong theoretical efforts in this direction . @xcite recent experiments on a qd embedded on one branch of the aharonov - bohm interferometer @xcite created additional interest in kondo effects in _ complex networks_.    while the high-@xmath0 behavior of the siam can be adequately described by perturbation theory or poor - man scaling , @xcite and the low-@xmath0 behavior is described by fermi liquid theory , @xcite there is no _ simple _ theory that describes the model s dynamical properties correctly in the whole temperature range , including _ both _ the high- and the low-@xmath0 limits . this crossover has been described by the computationally demanding numerical renormalization group ( nrg ) approach . @xcite other methods to describe the crossover are the  conserving t - matrix approximation " @xcite ( which overestimates the unitarity sum - rule ) , or the more limited  non - crossing approximation " , @xcite and to some extent , quantum monte carlo calculations . @xcite some of these methods have also been applied to the qd in the aharonov - bohm interferometer . @xcite however , none of these methods has the flexibility to follow _ analytically _ the effects of various network parameters ( e.g. a magnetic flux ) on experimentally measurable quantities , in particular dynamical ones .    in this paper we discuss the siam for a qd which is embedded in a _ general complex network_. many of the interesting physical properties of the system can be expressed explicitly in terms of the single electron ( retarded ) green function ( gf ) on the dot , @xmath2 , for electrons with spin component @xmath3 and energy @xmath4 ( measured relative to the fermi energy ) . here , @xmath5_+\rangle.\end{aligned}\ ] ] ( for the dot gf , @xmath6 , where @xmath7 destroys an electron with spin @xmath3 on the dot ) . one example concerns the local density of states ( ldos ) on the dot , @xmath8/\pi.\label{rho}\end{aligned}\ ] ] another example is the conductance @xmath9 between two leads which are connected to sites on the network . for a general mesoscopic structure , this ( linear response ) conductance has the form @xcite @xmath10 where @xmath11 is the fermi distribution function ( we set the fermi energy at @xmath12 ) , @xmath13 and @xmath14 is often given in terms of the retarded @xmath15 , the advanced @xmath16 and non - interacting parameters characterizing the network . for the simple siam ( dot and leads ) , @xmath14 is proportional to @xmath17 . @xcite    here we use the equations of motion method to derive a simple analytic approximate expression for @xmath15 , for a qd on a general network . our approximate formula exhibits the correct behavior both at high and at very low temperatures . for simplicity , we consider a qd with a single level ( of energy @xmath18 ) . electron - electron interactions are assumed to exist only on the qd , and we take their energy @xmath19 to be infinite . in our scheme , one first solves for the gf on the dot _ in the absence of the electron - electron interactions _ , i.e. @xmath20 . this ( spin - independent ) gf can be written in the form @xmath21 with the energy dependent ( complex ) self - energy ( se ) @xmath22 this is easily done exactly , for any finite network ; it only involves the solution of a finite set of linear equations . our approximate explicit expression for @xmath2 is then given in a single equation [ see eq . ( [ ggg ] ) below ] . somewhat surprisingly , this equation depends on the network parameters _ only _ via @xmath23 . although approximate , this expression allows for detailed systematic investigations of the kondo effect as function of the system parameters , on a variety of complex networks . even if sometimes only qualitatively correct , such systematic studies help to investigate new physical phenomena on a broad variety of mesoscopic systems furthermore , the dynamical mean field theory @xcite ( which was developed to address the physics of the periodic anderson and hubbard models ) iterates the local gf of the siam , which is calculated at each stage in terms of the effective se created by the rest of the lattice . since our gf is easily calculated in terms of @xmath24 , it is an ideal candidate for such calculations . we are not aware of alternative simple analytic expressions which obey the necessary requirements at both high and low @xmath0 .    in sec . [ ii ] we follow refs . , and derive the equation - of - motion ( eom ) for the dot gf . extending these references , this eom is derived here for a _ generalized case _ , in which the dot  sits " within an _ arbitrary complex network_. the eom for the dot gf involves higher - order gf s ( consisting of more operators ) , whose eom s introduce in turn more gf s . one then terminates this hierarchy by decoupling out averages of operators . the latter are then found using the fluctuation - dissipation relationship , from the relevant gf s . in this manner , the treatment becomes self - consistent . the successful decoupling must keep as much of the electronic correlations as possible . @xcite for example , a widely used earlier approximation @xcite neglected some dot - lead correlations and therefore gave reasonable results _ only _ at @xmath25 . for this reason , gerland _ et al . _ @xcite had to combine the eom method at high @xmath0 with the nrg at low @xmath0 . after we correct these earlier calculations , and include all the necessary correlations , we obtain good results for _ all _ @xmath0 . an earlier application of the eom method solved the integral equations numerically , on the simplest interferometer geometry . @xcite    the above decoupling scheme produces an integral equation for @xmath2 , which generalizes that found in refs . and . a new approximate analytical solution for this equation is found in sec . the low temperature limit , the fermi liquid conditions , and the kondo behavior are discussed in sec . [ fermi ] . unlike earlier papers ( spanning over four decades ) , our solution gives good qualitative results for the _ whole parameter range _ : it has the correct high-@xmath0 behavior and fulfills several low-@xmath0 fermi liquid relations .    for the simplest siam , the  network " is represented by a band with a density of states @xmath26 . in the broad band limit , the self - energy on the qd can be approximated by its value at the fermi energy , @xmath27 , and @xmath28 represents the width of the impurity state in the absence of interactions ( @xmath29 represents the approximately energy - independent dot - band coupling ) . in this case , our equations reduce to those discussed in the earlier literature . we emphasize again that even the solution for this simple case is new and non - trivial , since earlier work either did not have a close analytical formula or missed some correlations , leading to wrong results at low @xmath0 . as stated , our result for @xmath2 depends on the details of the general network only via the non - interacting self - energy , @xmath30 . to demonstrate the power of our result , we present here a few simple examples . section [ iv ] contains explicit results for the simple case of the siam , when the qd is coupled to two one - dimensional leads . for this case , we easily calculate various features of the kondo behavior , including the peak in @xmath31 at the fermi energy , the plateau in the conductance in the so - called unitary limit and the plateau in the  transmission phase " at @xmath32 .    a second example , of the so called  t " network , is solved in sec . [ t ] . in this case , one has interference between the wave function on the dot and that on the intersection point . our simple approximate formula exhibits the fano vanishing of the transmission and the associated so - called anti - kondo effect , as seen experimentally . @xcite here we also extend earlier theoretical work , which used complicated techniques . @xcite finally , sec . [ v ] contains our summary . our hamiltonian , @xmath33 contains the dot part @xmath34 with a single energy level @xmath35 and an electron - electron interaction @xmath19 ( @xmath36 ) . all the other parts of @xmath37 can be arbitrary , but non - interacting : @xmath38 describes the network without the dot , and @xmath39 describes the coupling of the dot to the network . here , @xmath40 creates an electron with spin @xmath3 and energy @xmath41 on the site @xmath42 of the network . the coefficients @xmath43 and @xmath44 represent tight - binding hopping matrix elements . in the absence of a magnetic flux , these coefficients can be chosen to be real . the flux then turns them complex , with phases which relate to the aharonov - bohm effect and with @xmath45 . @xcite    the main part of this paper concerns the above general hamiltonian , which assumes no specific details on the structure of the network . however , for some specific applications , it is convenient to attach a finite mesoscopic network to several semi - infinite leads , which connect to electron reservoirs . we thus divide @xmath46 into two parts , @xmath47 where @xmath48 has the same form as @xmath49 , except that the sum is restricted to sites which belong to the finite mesoscopic part of the net ( excluding the dot and the leads ) , and @xmath50,\nonumber\\   { \cal h}_l&=&-\sum_{rs } j_{rs } c^{(l)\dagger}_{r\sigma } c^{(l)}_{s\sigma}+h.c .. \label{leads}\end{aligned}\ ] ] here @xmath51 are the indices of the points on the mesoscopic structure which are connected to lead number @xmath52 , while @xmath53 destroys an electron of spin @xmath3 on the site @xmath54 of the lead . usually , the lead @xmath52 has a continuous spectrum , with eigenstates @xmath55 and eigenenergies @xmath56 , in a band of width @xmath57 . equation ( [ leads ] ) can then be written as @xmath58,\nonumber\\   { \cal h}_l&=&\sum_{k}\epsilon_k c^{(l)\dagger}_{k\sigma } c^{(l)}_{k\sigma},\label{leads1}\end{aligned}\ ] ] and also @xmath59 with obvious definitions . when the network consists only of the dot and the leads ( i.e. @xmath60 ) then this has exactly the form of the standard siam model , discussed broadly in the literature . if all the leads have similar bands , then the electrons in the whole system also have a continuous spectrum in the range @xmath61 . all integrals over @xmath4 below will be thus over this energy band , and it is due to resonances in such integrals that we need to calculate the retarded gf at @xmath62 . from now on we return to the general hamiltonian , eqs . ( [ h1]-[h4 ] ) . the eom for the retarded gf on the qd reads @xmath63 where @xmath64 from now on , we shall use the shorthand @xmath4 for @xmath62 , except where it matters . specifically , @xmath65 we next define the matrix @xmath66 which represents the system without the dot . in the following , we shall need the inverse matrix , @xmath67^{-1 } , \end{aligned}\ ] ] which is the green function for the system without the dot . with this definition one finds @xmath68 where @xmath69 we can now calculate the second term on the rhs of eq . ( [ eomg0 ] ) , and show that @xmath70 with @xmath71 at @xmath20 , eq . ( [ eomg0 ] ) thus reduces to eq . ( [ g0 ] ) . clearly , @xmath24 can be calculated for the _ non - interacting _ case . all the other gf s can similarly be expressed in terms of @xmath15 . for example , @xmath72 where @xmath73 . another way to represent @xmath74 is to start from @xmath75 , and then take the complex conjugate only of the hopping coefficients @xmath76 ( namely reverse all their phases ) , without touching @xmath62 . similarly , @xmath77    since @xmath78 is an infinite matrix , its inversion may not be trivial . formally , we can denote the eigenenergies and eigenstates of @xmath46 by @xmath79 and @xmath80 , and then write @xmath81 however , in the examples it is useful first to eliminate the parts which involve the sites on the leads . examples for this procedure are presented in appendix [ a ] . the eom for @xmath82 ( with only on - site hubbard interaction on the dot ) does not involve gf s of more arguments : @xmath83 with three new gf s , @xmath84 the inhomogeneous term in eq . ( [ eomgamma ] ) , @xmath85 , represents the average number of electrons with spin @xmath3 on the dot . this number needs to be determined self - consistently , via the fluctuation - dissipation theorem : @xmath86 we shall return to this condition below . we now write the eom s for the new gf s , @xmath87 , and use several approximations for their solution . firstly , it is easy to see from the eom for @xmath88 that this gf is of order @xmath89 . therefore , it contributes to @xmath90 only at order @xmath91 , and can be ignored for @xmath92 [ we need @xmath90 only to order @xmath89 , see eq . ( [ eomg0 ] ) ] . secondly , we introduce a decoupling scheme for the new gf s that appear in the eom s of the @xmath87 s , @xmath93 we omit thermal averages of the form @xmath94 , which include two destruction ( or creation ) operators , and @xmath95 with @xmath96 , relevant only for states with a net magnetic moment . the latter assumption means that we restrict the discussion only to symmetric states , with @xmath97 . using these approximations , the eom for @xmath98 becomes @xmath99 the derivation of this equation also required an additional term , @xmath100 $ ] . however , this term vanishes . here and below , we calculate equilibrium thermal averages by using the fluctuation - dissipation theorem , @xmath101    equation ( [ g1 ] ) is practically the same as eq . ( [ gnd ] ) , which was used to derive @xmath24 in terms of the @xmath43 s . applying the same algebra to eq . ( [ g1 ] ) then yields the analog to eq . ( [ sigext ] ) , @xmath102    using the same approximations , the eom of @xmath103 is @xmath104 in which @xmath105 although the first two terms on the rhs of eq . ( [ gamma2 ] ) are similar to those in eq . ( [ g1 ] ) , we now need to calculate the other two terms . since these terms require thermal averages , which we express using the fluctuation - dissipation eq . ( [ fd ] ) , they end up with integrals which involve @xmath106 , leading finally to our integral equation for the dot gf . the details of these calculations are presented in appendix [ app2 ] .    inserting the results for the necessary thermal averages into the rhs of eq . ( [ gamma2 ] ) , the results are used in eq . ( [ xxx ] ) for the combination @xmath107 which is needed in the rhs of eq . ( [ eomgamma ] ) . adding also the corresponding combination for @xmath108 [ eq . ( [ g1sof ] ) ] , yields our final result for @xmath90 . it remains to insert it back into the eom ( [ eomg0 ] ) for the dot green function @xmath15 . this yields an integral equation for this function ,    @xmath109\nonumber\\ & + & ( 1+\sigma_0 g_{dd\sigma})\int\bigl ( -\frac{d\omega ' } { 2\pi i}\bigr ) \frac{f(\omega ' ) } { \omega -\omega ' + i\eta}\bigl [ g'_{dd-\sigma}(\sigma_0-\sigma ' _ 0)-(\omega ' \rightarrow\omega ' ^{\ast})\bigr ] , \ \ \ \eta\rightarrow 0^{+}.\label{gsof1}\end{aligned}\ ] ]    here and below , @xmath15 and @xmath24 are understood to be functions of @xmath110 , and the primes denote a dependence on @xmath111 , with @xmath112 . two comments are in place here . first , for the simplest siam , when @xmath113 is independent of @xmath4 , we have @xmath114 , so that only the parts with @xmath115 survive . furthermore , in this case one can factorize @xmath28 out of the integrals . this reproduces the integral equation of lacroix , @xcite @xmath116^\ast)\nonumber\\   & -&(1-i\delta_b g_{dd\sigma})\int\bigl ( -\frac{d\omega ' } { 2\pi i}\bigr ) \frac{f(\omega ' ) } { \omega -\omega ' + i\eta}[g'_{dd-\sigma}]^\ast\bigr ] .\end{aligned}\ ] ]    2 however , even for this simple case there has not been an analytic solution that covers the whole parameter range . second , discarding @xcite correlations between the dot and other sites on the net , e.g. @xmath117 , amounts to neglecting the terms containing @xmath118 in eq . ( [ gsof1 ] ) . this ends up with a breakdown of the fermi liquid conditions at @xmath119 , and with a bad approximation for @xmath120 . we now restrict the discussion to  non - magnetic " states , and replace @xmath121 . equation ( [ gsof1 ] ) can then be written as @xmath122 where @xmath123 and @xmath124 and @xmath125 are functions of @xmath4 , given by @xmath126,\nonumber\\ & & { \cal b}(\omega ) = \int\bigl ( -\frac{d\omega ' } { 2\pi i}\bigr ) \frac{f(\omega ' ) } { \omega -\omega ' + i\eta}\nonumber\\ & & \times\bigl [ g'_{dd}(\sigma_{0}'-\sigma_{0})-(\omega ' + i\eta \rightarrow\omega'-i\eta)\bigr ] .\label{ab}\end{aligned}\ ] ]    for reasons that will be explained later , it is now convenient to replace @xmath127 inside the integrals by @xmath128 + 1/2 $ ] . the part related to the constant 1/2 can then be calculated using the kramers - kronig relations , @xmath129 and thus we find @xmath130,\nonumber\\ & & { \cal b}(\omega ) = \int\bigl ( -\frac{d\omega ' } { 2\pi i}\bigr ) \frac{f(\omega ' ) -1/2}{\omega -\omega ' + i\eta}\nonumber\\ & & \times\bigl [ g'_{dd}(\sigma_{0}'-\sigma_{0})-(\omega ' + i\eta \rightarrow\omega'-i\eta)\bigr ] .\label{abnew}\end{aligned}\ ] ] we next follow lacroix , @xcite and assume that the integrals are dominated by the region @xmath131 , namely that @xmath132 and thus @xmath133 where @xmath134 and @xcite @xmath135,\end{aligned}\ ] ] ( @xmath136 is the digamma function ) . a similar procedure yields @xmath137 . \label{aapprox}\end{aligned}\ ] ]    the reason for using the above transformation is that when one analyzes the equivalent equations for finite @xmath19 , then one requires particle hole symmetry , namely the relation @xmath138 where the ` tilde ' denotes particle - hole transformed quantities . in that case , this relation is equivalent to the relations @xmath139 these relations should hold also for any approximate solution . @xcite it is easy to check that these relations are obeyed by our approximate expressions ( [ bapprox ] ) and ( [ aapprox ] ) , but would not hold if we made the approximation ( [ lacapp ] ) on the original equations , before shifting @xmath127 by 1/2 . @xcite    taking @xmath140 as a parameter , and defining the scaled variable @xmath141 the solution of eq . ( [ gsof ] ) can be written as @xmath142 with @xmath143^{1/2},\label{qq}\\   { \rm s}&=&z^2+\frac{9}{16}-z\re x+(\delta n-\frac{3}{4})\im x , % \frac{1}{4\delta^{2}|g|^{2}}-\frac{\re [ { \rm % x}^{\ast}/g]}{2\delta } + \delta n\im{\rm x}. \label{ss}\end{aligned}\ ] ] and we have omitted the explicit dependence on @xmath4 for brevity . for some purposes it is more convenient to write eq . ( [ apprg ] ) as @xmath144 where the real functions @xmath145 and @xmath146 , which result from the strong interactions on the dot , depend on @xmath0 and on @xmath23 and are given by @xmath147 equation ( [ ggg ] ) forms our general result . we emphasize again that all we need to know is the non - interacting self - energy , @xmath30 .    as seen explicitly from the above equations , our analytic expression for @xmath148 depends on the parameter @xmath149 , which needs to be determined self - consistently , via eq . ( [ selfcon ] ) . we discuss this condition in a specific example below . at @xmath119 , one easily sees that @xmath150 diverging at the fermi energy @xmath151 . at @xmath152 we thus have @xmath153 and @xmath154 , with @xmath155}{[z+\sqrt{z^2+\delta n(1.5-\delta n)}]^2+\delta n^2},\end{aligned}\ ] ] and @xmath156/[2\delta(0)]$ ] . therefore , @xmath157+i\bigr ) } .\label{gt0}\end{aligned}\ ] ] for a very deep level on the dot , @xmath158 , @xmath159 approaches @xmath160 , and @xmath148 approaches @xmath161 , ending up with high plateaus in the ldos and in the phase of the complex @xmath162 sticking to @xmath32 . all of these characteristics , which are hallmarks of the kondo behavior , @xcite are observed in the examples below . as @xmath35 moves up from large negative values , eq . ( [ gt0 ] ) shows a gradual crossover away from this  unitary " limit , and @xmath148 becomes mainly real and small as @xmath163 . ( [ gt0 ] ) also ensures that the gf obeys the fermi - liquid relations @xcite at @xmath119 : ( i ) the imaginary part of the se on the fermi level is the same as in the absence of interactions , i.e. @xmath164 , confirming the unitarity limit ; @xcite ( ii ) for @xmath165 , the ldos there [ i.e. , @xmath166 approaches the finite value @xmath167 $ ] . we emphasize that these results apply for _ any _ complex network ; the only input is @xmath30 . the kondo effect concerns the behavior of the density of states near the fermi energy at low temperature . in that regime , @xmath148 is dominated by its imaginary part , and thus @xmath168.\end{aligned}\ ] ] for @xmath169 , rne has @xmath170 , @xmath171 and the leading @xmath4-dependence of @xmath172 is @xmath173 we thus end up with a logarithmic cusp in @xmath174 at @xmath12 . the related narrow peak reaches one half of its peak value ( equal to @xmath167 $ ] ) when @xmath175 . one definition of the kondo energy @xmath1 is to identify @xmath1 with the half - width of this peak . solving @xmath176 and using eq . ( [ asym ] ) thus yields @xmath177 , i.e. @xmath178 . this result , which agrees with that of lacroix , @xcite is qualitatively similar but quantitatively different from the presumably exact @xmath1 as given by haldane , @xcite which has @xmath179 . however , @xmath1 only represents some crossover energy scale , and we expect the above solution to follow the qualitative variations of @xmath1 with the system parameters , which are contained in @xmath180 and in @xmath181 . it should be emphasized that @xmath1 depends on _ both _ @xmath181 _ and _ @xmath180 ( some authors ignore the part coming from the real part of the se , @xmath180 ) . for example , for the aharonov - bohm interferometer @xmath1 oscillates with the flux . @xcite    at the fermi energy , @xmath12 , and for fixed @xmath182 , one expects @xmath183 to decrease with increasing @xmath0 . in this limit , @xmath184 , with @xmath185 . for @xmath186 we then find the asymptotic relation @xmath187 this density of states decreases to one half of its maximum @xmath119 value at a temperature @xmath188 , with @xmath1 given in eq . ( [ tkondo ] ) . although the result ( [ rhot ] ) is qualitatively correct , and is also consistent with our @xmath1 , its explicit low - temperature dependence disagrees with exact expectations : we obtain a logarithmic behavior , whereas fermi liquid theory predicts a @xmath189 dependence at low @xmath0 . @xcite this is an artifact of our approximation . however , at high @xmath0 we do recover the usual logarithmic variation .    in the opposite limit , of @xmath190 , one finds that @xmath191 , and eq . ( [ apprg ] ) implies that @xmath192 approaches @xmath193 . the appearance of @xmath140 in the numerator , in place of 1 , results from the fact that some weight of the spectral function is pushed to infinite @xmath19 . @xcite another interesting point concerns the factor @xmath194 in eqs . ( [ gg ] ) and ( [ ggg ] ) . this factor may be explained heuristically @xcite by the fact that while both spin directions are accessible for tunnelling into the dot , only a single one can tunnel out of it . in the following two examples , we assume simple semi - infinite one - dimensional leads , with identical nearest - neighbor hopping matrix elements , @xmath195 , and with a lattice constant @xmath196 . the eigenenergies of each @xmath197 are therefore @xmath198 , with eigenfunctions @xmath199 ( @xmath200 is the length of the lead ) , and @xmath201 . we also assume that the leads @xmath52 and @xmath202 are attached to the mesoscopic network only at one site , with hopping matrix elements @xmath203 and @xmath204 . for the simple siam , when the two leads are directly connected to the dot , we show in appendix [ a ] that the leads generate a self - energy @xmath205 on the dot , with @xmath206 determined by @xmath207 and with @xmath208 here we have used the notation @xmath209 with @xmath210 being the density of the band states . the width of the non - interacting resonance at @xmath35 is thus equal to @xmath211 . in the figures presented below we use the symmetric case , @xmath212 . substituting @xmath213 into eqs . ( [ apprg])-([ss ] ) then yields @xmath148 . we start by discussing the self - consistency condition for @xmath214 . figure [ nd ] shows an example of our self - consistent solutions for @xmath214 , at @xmath119 : we start from an initial guess for @xmath140 , then calculate @xmath214 from eq . ( [ selfcon ] ) , and iterate ; the procedure converges after a few iterations . also shown is @xmath215 , which represents the total change in the electron occupation in the system due to the dot , as determined from the friedel phase , defined via @xcite @xmath216 \equiv \re[g_{dd}(0)]/\im[g_{dd}(0 ) ] . \label{friedel}\end{aligned}\ ] ] for @xmath217 , @xmath215 approaches 1 , while @xmath214 remains slightly smaller . this small difference could reflect an additional occupation of other sites in the leads . it could also result from the inaccuracy of our approximation for @xmath218 , which becomes worse as @xmath4 moves away from the fermi energy ; the integral in eq . ( [ selfcon ] ) contains contributions from all @xmath4 . however , this small difference has only a small effect on the other calculations presented below . = 6.truecm    although it is easy to solve for @xmath214 for each set of parameters and use the resulting @xmath140 for other calculations , the qualitative results are only weakly affected if one uses an arbitrary smooth variation of @xmath214 from 0 to 1 as @xmath35 varies from @xmath219 to @xmath220 . in the calculations below , we used such a smooth variation for @xmath140 , with a width of order @xmath28 . an alternative approximate expression for @xmath140 follows from eq . ( [ apprg ] ) . it turns out that for most of the integration range in eq . ( [ selfcon ] ) , @xmath162 is dominated by the first term there . neglecting the second term , one has @xmath221 , with @xmath222/\pi$ ] depending only on the non - interacting parameters . thus , @xmath223 . for @xmath28 between 0.1 and 0.01 and for @xmath224 , we find this estimate to be within @xmath225 from the full self - consistent value . figure [ dost ] presents the ldos [ eq . ( [ rho ] ) ] versus @xmath4 , for the simple siam , with the parameters as indicated . plots at lower @xmath0 are indistinguishable from the one shown at @xmath226 . our approximation reproduces the kondo peak at low @xmath0 , in addition to the much broader peak at @xmath35 . figure [ dosed ] shows @xmath227 at the fermi energy ( @xmath12 ) versus @xmath35 . note the very slow ( logarithmic ) increase of @xmath228 with decreasing @xmath0 , in agreement with eq . ( [ rhot ] ) . = 6.truecm    = 6.truecm    we now turn to the conductance , @xmath9 . for the simple siam , one has @xcite @xmath229 equation ( [ ggg ] ) implies that @xmath230 for @xmath119 , when @xmath231 , we also have @xmath232 . at finite temperatures , these two expressions for the conductance exhibit the same qualitative behavior as @xmath227 ( fig . [ dosed ] ) . the quantitative difference between the two expressions , which increases with @xmath0 and with large negative @xmath35 , represents a breakdown of unitarity which may be an artifact of our approximation .    at low @xmath0 , @xmath233 is practically a delta function , and eq . ( [ cond ] ) yields @xmath234 , exhibiting the qualitative behavior shown in fig . [ dosed ] . at higher @xmath0 , the peak in @xmath9 slightly below @xmath235 becomes lower and broader than that of @xmath236 . at fixed @xmath237 , increasing @xmath0 results with a decreasing @xmath9 , but with an interesting superimposed peak at @xmath238 ) ( when the peak at @xmath35 starts to contribute ) . some experiments @xcite place the qd on one branch of an open aharonov - bohm interferometer , and attempt to extract the _ transmission phase _ for scattering through the dot . the transmission phase is usually related to the phase of @xmath148.@xcite without getting into the question of what is really measured in the interferometer , @xcite it is still of interest to study the latter phase . figure [ phase ] thus shows the  transmission " , represented by @xmath239 , and the  transmission phase " , represented by [ see eq . ( [ friedel ] ) ] @xmath240 as function of @xmath35 for several temperatures . interestingly , at high @xmath0 this phase simply grows smoothly from zero to @xmath241 through the resonance , similarly to the non - interacting case or to the coulomb blockade case . however , as @xmath0 decreases , the peak in the  transmission " broadens towards negative @xmath35 , eventually reaching a plateau for @xmath119 ( see also fig . [ dosed ] ) . at the same time , the phase develops an intermediate plateau at @xmath32 . @xcite this plateau begins at an energy @xmath35 which is roughly given by @xmath242 , as defined in eq . ( [ tkondo ] ) . studying the energy where this phase grows from zero to @xmath32 thus suggests another way to define the crossover temperature @xmath1 . our equations also become very simple for the  t " network , when the dot sits on a side branch . such a network has recently attracted both theoretical @xcite and experimental @xcite interest , as the simplest realization of the fano - kondo effect . the mesoscopic network now consists of a single site  0 " . this site has one bond connected to the dot , with hopping @xmath243 , and two bonds connected to two leads , with hopping @xmath203 and @xmath204 . in this case , the matrix @xmath244 in appendix [ a ] is of order @xmath245 , and eqs . ( [ dd ] ) and ( [ aaa ] ) yield @xmath246 $ ] . using @xmath28 from eq . ( [ delta ] ) , we have @xmath247 in this case , @xmath248 of eq . ( [ cond ] ) is given by @xcite @xmath249 the calculation of @xmath250 follows directly from eqs . ( [ gdm ] ) and ( [ glm ] ) , @xmath251 therefore , @xmath252 is a linear combination of @xmath253 and of @xmath254 , with coefficients which depend only on the non - interacting parameters . these equations reproduce those found e.g. in refs . and . however , at this point those authors use complicated numerical schemes to obtain @xmath148 , which do not allow for systematic studies of the dependence on the various parameters . in contrast , we can easily obtain the approximate @xmath148 analytically , with the same ease as for the previous example . all we need to do is to substitute eq . ( [ sig0 t ] ) into eqs . ( [ apprg])-([ss ] ) .    for the non - interacting case , @xmath148 is simply given by eq . ( [ g0 ] ) . figure [ tcond ] then compares the results for the conductance ( in units of @xmath255 ) , with and without interactions . without interactions ( thin lines ) , for @xmath256 the upper panel in fig . [ tcond ] shows a symmetric fano vanishing of the conductance at @xmath235 . as @xmath257 becomes more negative ( lower panel there ) , the conductance assumes a typical non - symmetric fano shape , with smaller conductances at large @xmath258 . the interactions ( thick lines ) have a negligible effect at @xmath259 , when the dot is not occupied . however , at @xmath182 and @xmath119 ( thickest lines ) the interactions cause strong changes . for @xmath256 and large negative @xmath35 , the occupation on the dot causes the so - called anti - kondo effect , @xcite where the formation of the kondo singlet causes destructive interference which yields zero conductance . for @xmath260 , the fano zero is shifted to lower @xmath35 , the peak disappears and the asymptotic conductance increases . as @xmath0 increases ( medium thick lines ) , the behavior at negative @xmath35 gradually returns to that of the non - interacting case , and the fano zero disappears : the conductance is always non - zero . again , all of these phenomena are qualitatively similar to those found in refs . , and . = 6truecm    a slightly more complex t - network is obtained by placing another non - interacting site ,  1 " , between the site 0 and the dot , with energy @xmath261 and with hopping elements @xmath243 to both site 0 and the dot . our mesoscopic system now contains two sites , 0 and 1 , and the methods of appendix [ a ] yield @xmath262 @xmath263 with @xmath264    figure [ t11 ] shows a few examples on how the interplay between @xmath257 and @xmath261 can change the dependence of the conductance on @xmath35 for the fully interacting case at @xmath119 . for @xmath265 , the graph looks exactly like that for the simple siam , discussed in sec . v. clearly , the presence of the intermediate point turned the destructive interference into a constructive one . changing @xmath257 to positive ( negative ) values then simply shifts the whole curve to the left ( right ) . changing @xmath261 to non - zero values generates either a `` fano '' zero in the conductance ( when @xmath260 ) , similar to the results in fig . [ tcond ] , or a  fano " resonance ( when @xmath266 ) . at @xmath256 one observes a change from a zero to a resonance as @xmath261 changes from positive to negative values . all of these parameters can be easily changed using the various gate voltages , e.g in the setup of ref . . without going into much further discussion , it is clear that the intermediate point on the side branch is very effective in changing the interference pattern between fano - kondo resonances and anti - resonances . data from such experiments can then be used to obtain information on @xmath148 . we have derived an approximate analytic expression for a complex mesoscopic network , which contains a quantum dot with electron - electron interactions , and which may connect to several leads . our formulae correct and generalize earlier expressions , and give a good qualitative interpolation between the fermi liquid behavior at very low temperatures and the simpler high temperature one . although our formulae reproduce many features required by the fermi liquid theory , they are still approximate , and should thus only be used for discussing the qualitative variation of various quantities on the parameters characterizing the network . however , the simplicity of our expressions allows for relatively easy comparisons with experiments and with more complicated numerical work . we have demonstrated the use of our formulae for the two simple cases of a single quantum dot attached to two leads and of the t " network , where the dot is connected to another site which couples to the two leads . indeed , we have reproduced and extended all the expected phenomena for these two cases . in an upcoming publication @xcite we shall apply this scheme to the interesting case of the aharonov - bohm interferometer . * acknowledgements *    we acknowledge helpful discussions with a. schiller , and support to ym from the israel science foundation ( isf ) . this project was carried out in a center of excellence supported by the isf ( oe and aa ) .    2 we need to invert the matrix @xmath268 . we start by dividing our hilbert space into two parts , called  1 " and  2 " . then , we write @xmath269 and the inverse matrix @xmath270 as @xmath271 it is now easy to obtain the dyson - like equation , @xmath272^{-1}m_{21}\bigr ) ^{-1}.\label{dd}\end{aligned}\ ] ]    in our calculations we only need @xmath273 for @xmath274 within the finite mesoscopic part of the network ( possibly including the dot ) . identifying this part with the subspace 1 " above , we thus only need the matrix elements of the finite matrix @xmath244 . to obtain these via eq . ( [ dd ] ) , we need the green function for the leads , @xmath275^{-1}$ ] . for our eq . ( [ leads ] ) , the matrix @xmath276 separates for the different leads , and we end up with @xmath277^{-1},\label{minv1}\end{aligned}\ ] ] with the self - energy matrices @xmath278 here , the matrices @xmath276 and @xmath279 are all of order @xmath280 , where @xmath281 is the number of sites in the mesoscopic part of the network . also , @xmath282 is the green function for the disconnected lead @xmath52 .    for the one - dimensional leads , we use the identity @xmath283 with @xmath284 , to obtain the self - energy due to the lead @xmath52 , which is attached at the single point @xmath51 , @xmath285 the same result also applies when the mesoscopic part is empty , and the leads are connected directly to the dot . following the same logic that led to eqs . ( [ sigext ] ) and ( [ g1sof ] ) , the formal solution to eq . ( [ gamma2 ] ) gives @xmath287.\label{gg2}\end{aligned}\ ] ] calculating the thermal average @xmath288 via the fluctuation - dissipation eq . ( [ fd ] ) , and using eqs . ( [ gmd ] ) and ( [ gdm ] ) , we end up with an integral which requires the sum @xmath289 for this and for similar sums , it is helpful to use the identity @xmath290 which follows from eq . ( [ mm ] ) , together with a careful use of @xmath291 . using eq . ( [ sig0 ] ) , the sum in eq . 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we study the kondo effect of a quantum dot placed in a complex mesoscopic structure . assuming that electronic interactions are taking place solely on the dot , and focusing on the infinite hubbard interaction limit , we use a decoupling scheme to obtain an explicit analytic approximate expression for the dot green function , which fulfills certain fermi - liquid relations at zero temperature . 
 the details of the complex structure enter into this expression only via the self - energy for the non - interacting case . 
 the effectiveness of the expression is demonstrated for the single impurity anderson model and for the t - shaped network .    2