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in spectral analysis , one of the most powerful tools is the method of the conjugate operator , also called mourre s commutator method after the seminal work of mourre in the early eighties . this approach has reached a very high degree of precision and abstraction in @xcite ; see also @xcite for further developments . in order to study the nature of the spectrum of a selfadjoint operator @xmath0 , the main idea of the standard method is to find an auxiliary selfadjoint operator @xmath1 such that the commutator @xmath2 $ ] is strictly positive when localized in some interval of the spectrum of @xmath0 . more precisely , one looks for intervals @xmath3 of @xmath4 such that @xmath5e(j)\geq a e(j)\ ] ] for some strictly positive constant @xmath6 that depends on @xmath3 , where @xmath7 denotes the spectral projection of @xmath0 on the interval @xmath3 . an additional compact contribution to ( [ mou ] ) is allowed , greatly enlarging the range of applications . when strict positivity is not available , one can instead look for an @xmath1 such that the commutator is positive and injective , _ i.e. _  @xmath8>0.\ ] ] this requirement is close to the one of the kato - putnam theorem , _ cf . _ xiii.28 ) . a new commutator method based on such an inequality was proposed in @xcite . by analogy to the method of the conjugate operator , it has been called the method of the weakly conjugate operator ( mwco ) . under some technical assumptions , both approaches lead to a limiting absorption principle , that is , a control of the resolvent of @xmath0 near the real axis . in the case of the usual method of the conjugate operator , this result is obtained locally in @xmath3 , and away from thresholds . the mwco establishes the existence of the boundary value of resolvent also at thresholds , but originally applies only to situations where the operator @xmath0 has a purely absolutely continuous spectrum . this drawback limits drastically the range of applications . however , in some recent works @xcite the mwco has also been applied successfully to examples with point spectrum . this review intend to present and illustrate some of these extensions through applications to the spectral theory of operators acting on groups and graphs . compared to the huge number of applications based on an inequality of the form , the number of papers that contain applications of the mwco is very small . let us cite for example the works @xcite that deal with the original form of the theory , and the papers @xcite that contain very close arguments . the derivation of the limiting absorption principle of @xcite has been abstracted in @xcite . the framework of @xcite is still the one of the mwco , but since its result applies a certain class of two - body schrdinger operators which have bound states below zero , it can also be considered as the first extension of the mwco dealing with operators that are not purely absolutely continuous . the main idea of @xcite is that @xmath0 itself can add some positivity to . the new requirement is the existence of a constant @xmath9 such that @xmath10>0.\ ] ] this inequality , together with some technical assumptions , lead to a limiting absorption principle which is either uniform on @xmath4 if @xmath11 or uniform on @xmath12 if @xmath13 . the extensions of the mwco developed in @xcite are of a different nature . in these papers , the operators @xmath0 under consideration admit a natural conjugate operator @xmath1 that fulfills the inequality @xmath2\geq0 $ ] , namely , the commutator is positive but injectivity may fail . in that situation , the authors considered a decomposition @xmath14 , and the restrictions of @xmath0 and @xmath2 $ ] to these subspaces . in favorable circumstances the injectivity can be restored in one of the subspace , and a comprehensible description of the vectors of the second subspace can be given . this decomposition leads again to statements that are close to the ones of the mwco , but which apply to operators with arbitrary spectrum . this extension is described in section [ ext ] . we would also like to mention the references @xcite . they pereform what can be considered as a unitary version of the mwco and extend the kato - putnam analysis of unitary operators to the case of unbounded conjugate operators . the main applications concern time - depending propagators . the content of this review paper is the following . in section [ mofc ] we recall the original method of the weakly conjugate operator , and then present an abstract version of the approach used in @xcite . in section [ secad ] we present applications of this approach to the study of adjacency operators on graphs . a similar analysis for operators of convolution on locally compact groups is performed in section [ seclcg ] . let us finally fix some notations . given a selfadjoint operator @xmath0 in a hilbert space @xmath15 , we write @xmath16 , @xmath17 , @xmath18 , @xmath19 and @xmath20 respectively for the continuous , absolutely continuous , singularly continuous , singular and pure point subspaces of @xmath15 with respect to @xmath0 . the corresponding parts of the spectrum of @xmath0 are denoted by @xmath21 , @xmath22 , @xmath23 , @xmath24 and @xmath25 .    * acknowledgements . * m. m. was partially supported by the chilean science foundation _ fondecyt _ under the grant 1085162 . s. r. and r. t. d a. thank the swiss national science foundation for financial support . in this section we recall the basic characteristics of the method of the weakly conjugate operator , as originally introduced and applied to partial differential operators in @xcite . we then present the abstract form of the extension developed in @xcite . the method works for unbounded operators , but for our purposes it is enough to assume @xmath0 bounded . we start by introducing some notations . the symbol @xmath15 stands for a hilbert space with scalar product @xmath26 and norm @xmath27 . given two hilbert spaces @xmath28 and @xmath29 , we denote by @xmath30 the set of bounded operators from @xmath28 to @xmath29 , and put @xmath31 . we assume that @xmath15 is endowed with a strongly continuous unitary group @xmath32 . its selfadjoint generator is denoted by @xmath1 and has domain @xmath33 . in most of the applications @xmath1 is unbounded . [ defc1 ] a bounded selfadjoint operator @xmath0 in @xmath15 belongs to @xmath34 if one of the following equivalent condition is satisfied :    1 .   the map @xmath35 is strongly differentiable , 2 . the sesquilinear form @xmath36 is continuous when @xmath33 is endowed with the topology of @xmath15 . we denote by @xmath37 the strong derivative in ( i ) , or equivalently the bounded selfadjoint operator associated with the extension of the form in ( ii ) . the operator @xmath37 provides a rigorous meaning to the commutator @xmath2 $ ] . we write @xmath38 if @xmath37 is positive and injective , namely if @xmath39 for all @xmath40 . the operator @xmath1 is _ weakly conjugate to _ the bounded selfadjoint operator @xmath0 if @xmath41 and @xmath42>0 $ ] .    for @xmath38 let us consider the completion @xmath43 of @xmath15 with respect to the norm @xmath44 . the adjoint space @xmath45 of @xmath43 can be identified with the completion of @xmath46 with respect to the norm @xmath47 . one has then the continuous dense embeddings @xmath48 , and @xmath37 extends to an isometric operator from @xmath43 to @xmath45 . due to these embeddings it makes sense to assume that @xmath49 restricts to a @xmath50-group in @xmath45 , or equivalently that it extends to a @xmath50-group in @xmath43 . under this assumption ( tacitly assumed in the sequel ) we keep the same notation for these @xmath50-groups . the domain of the generator of the @xmath50-group in @xmath43 ( resp . @xmath45 ) endowed with the graph norm is denoted by @xmath51 ( resp . @xmath52 ) . in analogy with definition [ defc1 ] the requirement @xmath53 means that the map @xmath54 is strongly differentiable , or equivalently that the sesquilinear form @xmath55 is continuous when @xmath51 is endowed with the topology of @xmath43 . here , @xmath26 denotes the duality between @xmath43 and @xmath45 . finally let @xmath56 be the banach space @xmath57 defined by real interpolation , see for example ( * ? ? ? * proposition 2.7.3 ) . one has then the natural continuous embeddings @xmath58 and the following results ( * ? ? ? * theorem 2.1 ) :    [ thmabstract ] assume that @xmath1 is weakly conjugate to @xmath0 and that @xmath42 $ ] belongs to @xmath59 . then there exists a constant @xmath60 such that @xmath61 for all @xmath62 , @xmath63 and @xmath64 . in particular the spectrum of @xmath0 is purely absolutely continuous . the global limiting absorption principle plays an important role for partial differential operators , _ cf . _  @xcite . however , for the examples included sections 3 and 4 , the space @xmath56 involved has a rather obscure meaning and can not be greatly simplified , so we shall only state the spectral results . the extension proposed in @xcite relies on the following observation . assume that @xmath0 is a bounded selfadjoint operator in a hilbert space @xmath15 . assume also that there exists a selfadjoint operator @xmath1 such that @xmath65 , and that @xmath66=k^2\ ] ] for some selfadjoint operator @xmath67 , which is bounded by hypothesis . it follows that @xmath2\geq0 $ ] but injectivity may not be satisfied . so let us introduce the decomposition of the hilbert space @xmath68 with @xmath69 . the operator @xmath37 is reduced by this decomposition and its restriction @xmath70 to @xmath71 satisfies @xmath72 . formally , the positivity and injectivity of @xmath70 are rather promising , but are obviously not sufficient for a direct application of the mwco to @xmath70 . however , let us already observe that some informations on @xmath73 can be inferred from . indeed , it follows from the virial theorem ( * ? ? ? 7.2.10 ) that any eigenvector @xmath74 of @xmath0 satisfies @xmath75f\>=0 $ ] . so @xmath76 , _ i.e. _  @xmath77 , and one has proved :    if @xmath78 and @xmath2=k^2 $ ] , then @xmath79 . let us now come back to the analysis of @xmath70 . the space @xmath43 can still be defined in analogy with what has been presented in the previous section : @xmath43 is the completion of @xmath71 with respect to the norm @xmath44 . then , the adjoint space @xmath45 of @xmath43 can be identified with the completion of @xmath80 with respect to the norm @xmath47 . but in order to go further on , some compatibility has be imposed between the decomposition of the hilbert space and the operators @xmath0 and @xmath1 . let us assume that both operators @xmath0 and @xmath1 are reduced by the decomposition @xmath81 of @xmath15 , and let @xmath82 and @xmath83 denote their respective restriction to @xmath71 . it clearly follows from the above lemma that @xmath82 has no point spectrum . we are now in a suitable position to rephrase theorem [ thmabstract ] in our present framework . we freely use the notations introduced above . [ mo ] assume that @xmath78 and that @xmath84=k^2 $ ] . assume furthermore that both operators @xmath0 and @xmath1 are reduced by the decomposition @xmath85 of @xmath15 and that @xmath70 belongs to @xmath86 . then a limiting absorption principle holds for @xmath82 uniformly on @xmath4 , and in particular the spectrum of @xmath82 is purely absolutely continuous . a straightforward consequence of this statement is that @xmath87    we will see in the applications below that theorem [ mo ] applies to various situations , and that it really is a useful extension of the original method of the weakly conjugate operator . we present in this section the results of @xcite on the spectral analysis for adjacency operators on graphs . we follow the notations and conventions of this paper regarding graph theory . a _ graph _ is a couple @xmath88 formed of a non - void countable set @xmath89 and a symmetric relation @xmath90 on @xmath89 such that @xmath91 implies @xmath92 . the points @xmath93 are called vertices and couples @xmath94 such that @xmath91 are called edges . so , for simplicity , multiple edges and loops are forbidden in our definition of a graph . occasionally @xmath88 is said to be a simple graph . for any @xmath93 we denote by @xmath95 the set of neighbours of @xmath96 . we write @xmath97 for the degree or valence of the vertex @xmath96 and @xmath98 for the degree of the graph . we also suppose that @xmath88 is uniformly locally finite , _ i.e. _  that @xmath99 . a path from @xmath96 to @xmath100 is a sequence @xmath101 of elements of @xmath89 , usually denoted by @xmath102 , such that @xmath103 , @xmath104 and @xmath105 for each @xmath106 .    throughout this section we restrict ourselves to graphs @xmath88 which are simple , infinite countable and uniformly locally finite . given such a graph we consider the adjacency operator @xmath0 acting in the hilbert space @xmath107 as @xmath108 due to ( * ? ? ? * theorem 3.1 ) , @xmath0 is a bounded selfadjoint operator with @xmath109 and spectrum @xmath110 $ ] . results on the nature of the spectrum of adjacency operators on graphs are quite sparse . some absolutely continuous examples are given in @xcite , including the lattice @xmath111 and homogeneous trees . for cases in which singular components are present we refer to @xcite , @xcite , @xcite and @xcite . we now introduce the key concept of @xcite . sums over the empty set are zero by convention . [ adaptat ] a function @xmath112 is semi - adapted to the graph @xmath88 if    1 .   there exists @xmath113 such that @xmath114 for all @xmath115 with @xmath91 , 2 .   for any @xmath115 one has @xmath116=0.\ ] ]    if in addition for any @xmath115 one has @xmath117\[\phi(z)-\phi(y)\ ] \[2\phi(z)-\phi(x)-\phi(y)\]=0,\ ] ] then @xmath118 is adapted to the graph @xmath88 .    for a function @xmath118 semi - adapted to the graph @xmath88 we consider in @xmath15 the operator @xmath67 given by @xmath119f(y),\quad f\in\h,~x\in x.\ ] ] the operator @xmath67 is selfadjoint and bounded due to the condition ( i ) of definition [ adaptat ] . it commutes with @xmath0 , as a consequence of condition ( [ semi ] ) . we also decompose the hilbert space @xmath15 into the direct sum @xmath120 , where @xmath71 is the closure of the range @xmath121 of @xmath67 , thus the orthogonal complement of the closed subspace @xmath122 it is shown in ( * ? ? ? 4 ) that @xmath0 and @xmath67 are reduced by this decomposition , and that their restrictions @xmath82 and @xmath123 to the hilbert space @xmath71 are bounded selfadjoint operators . a rather straightforward application of the general theory presented in [ ext ] gives    [ unu ] assume that @xmath118 is a function semi - adapted to the graph @xmath88 . then @xmath82 has no point spectrum . [ doi ] let @xmath118 be a function adapted to the graph @xmath88 . then the operator @xmath82 has a purely absolutely continuous spectrum . the role of the weakly conjugate operator is played by @xmath124 and the assumptions imposed on @xmath118 make the general theory work .    for a certain class of admissible graphs , the result of theorem [ doi ] on the restriction @xmath82 can be turned into a statement on the original adjacency operator @xmath0 . the notion of admissibility requires ( among other things ) the graph to be directed . thus the family of neighbors @xmath125 is divided into two disjoint sets @xmath126 ( fathers ) and @xmath127 ( sons ) , @xmath128 . we write @xmath129 if @xmath130 and @xmath131 if @xmath132 . on drawings , we set an arrow from @xmath100 to @xmath96 ( @xmath133 ) if @xmath131 , and say that the edge from @xmath100 to @xmath96 is positively oriented . we assume that the directed graph subjacent to @xmath89 , from now on denoted by @xmath134 , is _ admissible _ with respect to these decompositions , _ i.e. _  ( i ) it admits a position function and ( ii ) it is uniform . _ a position function _ is a function @xmath135 such that @xmath136 whenever @xmath129 . it is easy to see that it exists if and only if all paths between two points have the same index ( which is the difference between the number of positively and negatively oriented edges ) . the directed graph @xmath134 is called _ uniform _ if for any @xmath115 the number @xmath137 $ ] of common fathers of @xmath96 and @xmath100 equals the number @xmath138 $ ] of common sons of @xmath96 and @xmath100 . thus the admissibility of a directed graph is an explicit property that can be checked directly , without making any choice . the graph @xmath88 is admissible if there exists an admissible directed graph subjacent to it . [ first ] the adjacency operator of an admissible graph @xmath88 is purely absolutely continuous , except at the origin , where it may have an eigenvalue with eigenspace @xmath139    many examples of periodic graphs , both admissible and non - admissible , are presented in ( * ? ? ? 6 ) . in particular , it is explained that periodicity does not lead automatically to absolute continuity , especially ( but not only ) if the number of orbits is infinite . @xmath140-products of graphs , as well as the graph associated with the one - dimensional xy hamiltonian , are also treated in @xcite . we recall in figures [ graph2 ] , [ graph0 ] , and [ graph7 ] some two - dimensional @xmath141-periodic examples taken from ( * ? ? ? more involved , @xmath142-periodic situations are also available . in this section we consider locally compact groups @xmath89 , abelian or not , and convolution operators @xmath143 , acting on @xmath144 , defined by suitable measures @xmath145 belonging to @xmath146 , the banach @xmath147-algebra of complex bounded radon measures on @xmath89 . using the method of the weakly conjugate operator , we determine subspaces @xmath148 and @xmath149 of @xmath144 , explicitly defined in terms of @xmath145 and the family @xmath150 of continuous group morphisms @xmath151 , such that @xmath152 and @xmath153 . this result , obtained in @xcite , supplements other works on the spectral theory of operators on groups and graphs @xcite . let @xmath89 be a locally compact group ( lcg ) with identity @xmath154 , center @xmath155 and modular function @xmath156 . let us fix a left haar measure @xmath157 on @xmath89 , using the notation @xmath158 . on discrete groups the counting measure ( assigning mass @xmath159 to every point ) is considered . the notation _ a.e . _ stands for  almost everywhere " and refers to the haar measure @xmath157 . we consider in the sequel the convolution operator @xmath143 , @xmath160 , acting in the hilbert space @xmath161 , _ @xmath162 where @xmath163 and @xmath164 the operator @xmath143 is bounded with norm @xmath165 , and it admits an adjoint operator @xmath166 equal to @xmath167 , the convolution operator by @xmath168 defined by @xmath169 . if the measure @xmath145 is absolutely continuous w.r.t . the left haar measure @xmath157 , so that @xmath170 with @xmath171 , then @xmath172 is also absolutely continuous w.r.t . @xmath157 and @xmath173 , where @xmath174 for _ a.e . _ @xmath175 . in such a case we simply write @xmath176 for @xmath177 . we shall always assume that @xmath143 is selfadjoint , that is @xmath178 . given @xmath160 , let @xmath179 be such that the linear functional @xmath180 is bounded . then there exists a unique measure in @xmath181 associated to @xmath182 , due to the riesz - markov representation theorem . we write @xmath183 for this measure , and we simply say that @xmath184 is such that @xmath185 . we call _ real character _ any continuous group morphism @xmath151 .      1 . a real character @xmath118 is _ semi - adapted to @xmath187 _ if @xmath188 , and @xmath189 . the set of real characters that are semi - adapted to @xmath145 is denoted by @xmath190 . a real character @xmath118 is _ adapted to @xmath145 _ if @xmath118 is semi - adapted to @xmath191 , and @xmath192 . the set of real characters that are adapted to @xmath145 is denoted by @xmath193 . the cases @xmath197 or @xmath198 are interesting ; in the first case @xmath143 has no eigenvalues , and in the second case @xmath143 is purely absolutely continuous . a more precise result is obtained in a particular situation . [ precis ] let @xmath89 be a lcg and let @xmath186 . assume that there exists a real character @xmath118 adapted to @xmath145 such that @xmath199 is equal to a nonzero constant on @xmath200 . then @xmath143 has a purely absolutely continuous spectrum , with the possible exception of an eigenvalue located at the origin , with eigenspace @xmath201 . corollary [ precis ] specially applies to adjacency operators on certain classes of cayley graphs , which are hecke - type operators in the regular representation , thus convolution operators on discrete groups .    to see how the method of the weakly conjugate operator comes into play , let us sketch the proof of the second inclusion in theorem [ princ ] . in quantum mechanics in @xmath202 , the position operators @xmath203 and the momentum operators @xmath204 satisfy the relation @xmath205 $ ] with @xmath206 , and the usual conjugate operator is the generator of dilations @xmath207 . so if we regard @xmath208 as a position operator on @xmath89 , it is reasonable to think of @xmath209 $ ] as the corresponding momentum operator and to use @xmath210 as a tentative conjugate operator . in fact , simple calculations using the hypotheses of definition [ admis].(b ) show that @xmath211=k^2.\ ] ] therefore @xmath212 and the commutator @xmath213 $ ] is a positive operator . however , in order to apply the theory of section [ mofc ] , we need strict positivity . so , we consider the subspace @xmath214^\perp$ ] of @xmath15 , where @xmath213\equiv k^2 $ ] is strictly positive . the orthogonal decomposition @xmath14 with @xmath69 , reduces @xmath143 , @xmath67 , and @xmath1 , and their restrictions @xmath82 , @xmath123 , and @xmath83 to @xmath71 are selfadjoint . it turns out that all the other conditions necessary to apply theorem [ mo ] can also be verified in the hilbert space @xmath71 . so we get the inclusion @xmath215 since @xmath208 is arbitrary , this implies the second inclusion of theorem [ princ ] . the construction of weakly conjugate operators for @xmath143 relies on real characters . so a small vector space @xmath216 is an obstacle to applying the method . a real character @xmath118 maps compact subgroups of @xmath89 to the unique compact subgroup @xmath217 of @xmath218 . consequently , abundancy of compact elements ( elements @xmath93 generating compact subgroups ) prevents us from constructing weakly conjugate operators . the extreme case is when @xmath89 is itself compact , so that @xmath219 . actually in such a case all convolution operators @xmath143 have pure point spectrum . we review now briefly some of the groups for which we succeeded in @xcite to apply the mwco . if the group @xmath89 is unimodular , one can exploit commutativity in a non - commutative setting by using central measures ( _ i.e. _  elements of the center @xmath220 $ ] of the convolution banach @xmath147-algebra @xmath146 ) . for instance , in the case of central groups @xcite , we have the following result ( @xmath221 stands for the closed subgroup generated by the set of compact elements of @xmath89 ) :    [ centreaza ] let @xmath89 be a central group and @xmath222 a central measure such that @xmath223 is compact and not included in @xmath221 . let @xmath224 with @xmath225 and set @xmath226 . then @xmath227 . let @xmath228 , where @xmath229 is the symmetric group of degree @xmath230 . the group @xmath229 has a presentation @xmath231 , and its conjugacy classes are @xmath232 , @xmath233 and @xmath234 . set @xmath235 and choose @xmath236 two finite symmetric subsets of @xmath141 , each of them containing at least two elements . then @xmath237 if @xmath238 . let @xmath239 , where @xmath240 is the group ( with haar measure @xmath241 ) of @xmath242 unitary matrices of determinant @xmath243 . for each @xmath244 $ ] let @xmath245 be the conjugacy class of the matrix @xmath246 in @xmath240 . a direct calculation ( using for instance euler angles ) shows that @xmath247 for each @xmath248 $ ] with nonzero lebesgue measure . set @xmath249 , @xmath250 , @xmath251 , @xmath252 , and @xmath253 . then @xmath237 if @xmath238 .    let @xmath89 be a central group , let @xmath254 , and set @xmath255 for some @xmath224 with @xmath225 . then @xmath145 satisfies the hypotheses of proposition [ centreaza ] , and we can choose @xmath256 such that @xmath257 ( note in particular that @xmath258 iff @xmath259 and that @xmath260 ) . thus @xmath261 . but @xmath163 belongs to @xmath262 iff @xmath263 for _ a.e . _ @xmath93 . this periodicity w.r.t . the non - compact element @xmath264 implies that the @xmath265-function @xmath74 should vanish _ _ and thus that @xmath266 .    if @xmath89 is abelian all the commutation relations in definition [ admis ] are satisfied . moreover one can use the fourier transform @xmath267 to map unitarily @xmath143 on the operator @xmath268 of multiplication with @xmath269 on the dual group @xmath270 of @xmath89 . so one gets from from theorem [ princ ] a general lemma on muliplication operators . we recall some definitions before stating it . the function @xmath271 is _ differentiable at @xmath272 along the one - parameter subgroup @xmath273 _ if the function @xmath274 is differentiable at @xmath275 . in such a case we write @xmath276 for @xmath277 . higher order derivatives , when existing , are denoted by @xmath278 , @xmath279 . we say that the one - parameter subgroup @xmath280 is in @xmath281 if @xmath282 is twice differentiable w.r.t . @xmath184 and @xmath283 . if , in addition , @xmath282 is three times differentiable w.r.t . @xmath184 and @xmath284 too , we say that @xmath184 belongs to @xmath285 . we end up this section by considering a class of semidirect products . let @xmath291 be two discrete groups with @xmath292 abelian ( for which we use additive notations ) , and let @xmath293 be a group morphism . let @xmath294 be the @xmath295-semidirect produt of @xmath296 by @xmath292 . the multiplication in @xmath89 is defined by @xmath297 so that @xmath298 in this situation it is shown in @xcite that many convolution operators @xmath176 , with @xmath299 of finite support , have a non - trivial absolutely continuous component . for instance , we have the following for a type of wreath products . take @xmath292 a discrete abelian group and put @xmath300 , where @xmath301 is an arbitrary discrete group and @xmath3 is a finite set on which @xmath292 acts by @xmath302 . then @xmath303 defines an action of @xmath292 on @xmath304 , thus we can construct the semidirect product @xmath305 . if @xmath306 and @xmath307 are finite subsets with @xmath308\ne\varnothing$ ] , then @xmath309 satisfies all the conditions of ( * ? ? ? thus @xmath310 if @xmath311 .                      a.  boutet de monvel and m.  mntoiu . the method of the weakly conjugate operator . in _ inverse and algebraic quantum scattering theory ( lake balaton , 1996 ) _ , volume 488 of _ lecture notes in phys . _ , pages 204226 , springer , berlin , 1997 . j. breuer . singular continuous and dense point spectrum for sparse tree with finite dimensions . in _ probability and mathematical physics _ d. dawson , v. jaksic and b. vainberg ) , crm proc . and lecture notes 42 , pages 6583 , 2007 .
in this review we present some recent extensions of the method of the weakly conjugate operator . 
 we illustrate these developments through examples of operators on graphs and groups .