PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION ON QUANTUM GRAPHS AND THE RESONANCE ASYMPTOTICS

Transcription

1 PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION ON QUANTUM GRAPHS AND THE RESONANCE ASYMPTOTICS JIŘÍ LIPOVSKÝ Department of Physcs, Faculty of Scence, Unversty of Hradec Králové, Roktanského 6, Hradec Králové, Czecha Abstract. In ths note we explan the method how to fnd the resonance condton on quantum graphs, whch s called pseudo orbt expanson. In three examples wth standard couplng we show n detal how to obtan the resonance condton. We focus on non-weyl graphs,.e. the graphs whch have fewer resonances than expected. For these graphs we explan benefts of the method of deletng edges for smplfyng the graph. PACS: Ge, Nk, 0.0.Ox. Introducton Resonances are a phenomenon, whch occurs often n physcs and can be easly understood heurstcally. Nevertheless, studyng t mathematcally rgorously s more dffcult. There are two man defntons of resonances resolvent resonances poles of the meromorphc contnuaton of the resolvent nto the non-physcal sheet) and scatterng resonances poles of the meromorphc contnuaton of the scatterng matrx). Non-compact quantum graph, where halflnes are attached to a compact part of the graph, provdes a good background for studyng resonances. It has been proven n [EL07] that on quantum graph the above two defntons are almost equvalent; to be precse, the set of resolvent resonances s equal to the unon of the set of scatterng resonances and the set of the egenvalues supported only on the nternal part of the graph. There s a large bblography on resonances n quantum graphs; for the quantum chaos communty e.g. the papers [KS03, KS04] mght be nterestng. The pseudo orbt expanson s a powerful tool for the trace formula expanson and the secular equaton on compact quantum graphs. We refer the reader to [BHJ] and the references theren. The method has been recently adjusted to fndng the resonance condton for non-compact quantum graphs [Lp5]. The resonance asymptotcs on non-compact quantum graphs was frst studed n [DP], where t was observed that some graphs do not obey expected Weyl behavour and a crteron for dstngushng non-weyl graphs wth standard couplng has been obtaned. Ths crteron was later generalzed n [DEL0] to all possble couplngs. Asymptotcs for magnetc graphs was studed n [EL]. The paper [Lp5] shows how to fnd the constant by the leadng term of the asymptotcs for non-weyl graphs and gves bounds on ths constant. E-mal address: jr.lpovsky@uhk.cz.

2 PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION In the current note we contnue n nvestgatng ths problem and llustrate the man results of [Lp5] on several smple examples wth standard couplng. In partcular, we focus on the pseudo orbt expanson and explan to the reader n detal how to construct the resonance condton by ths method. The paper s structured as follows. In the second secton we ntroduce the model and the resonance asymptotcs. The thrd secton deals wth the pseudo orbt expanson. In the secton 4 we gve two theorems on the effectve sze of an equlateral graph. In all these three secton we gve theorems wthout proofs, whch can be found n the referred papers. The only excepton s the theorem 3., where a smpler proof than the general one from [Lp5] s gven. The last three secton are devoted the the examples of non-weyl graphs. In all of them the resonance condton s obtaned from a regular drected graph and then from a smplfed graph after deletng some of ts edges.. Prelmnares We assume a quantum graph wth attached halflnes. We consder a metrc graph Γ, whch conssts of the set of vertces V and the set of edges E. There are N fnte edges, parametrzed by the segments 0, l j ), and M nfnte edges, parametrzed by 0, ). The graph s equpped wth the quantum Hamltonan H actng as negatve second dervatve wth the doman consstng of functons n the Sobolev space W, Γ) ths space conssts of Sobolev spaces on the edges) whch fulfll the couplng condtons at the vertces U j I)Ψ j + U j + I)Ψ j = 0..) Here U j s a d d untary matrx d s the degree of the gven vertex), I s d d unt matrx, Ψ j s the vector of lmts of functonal values at the gven vertex and Ψ j s the vector of lmts of outgong dervatves. In ths paper, we wll mostly consder standard couplng sometmes also known as Krchhoff, free or Neumann) n ths case the functonal values are contnuous n the vertex and the sum of outgong dervatves s equal to zero. The correspondng couplng matrx s U j = J I, where J s a d d matrx wth all entres equal to one. d We wll study resolvent resonances. The resolvent resonance s often defned as pole of the meromorphc contnuaton of the resolvent H λd). We wll use a smpler defnton, proof that both defntons are equvalent can be done by the method of complex scalng [EL07]. Defnton.. We say that λ = k s a resolvent resonance of H f there s a generalzed egenfuncton f, whch satsfes f x) = k fx) on all edges of the graph, satsfes the couplng condtons.) at the vertces and ts restrcton to each halflne s β j expkx). Defnton.. The countng functon NR) gves the number of all resolvent resonances ncludng multplctes) n the crcle of radus R centered at the orgn n the k-plane. Now we state the surprsng result of Daves and Pushntsk [DP] on the exstence of graphs wth non-weyl asymptotcs. Theorem.3. For graphs wth standard couplng the followng bound on the countng functon holds NR) = π W R + O) as R

3 PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION 3 wth 0 W vol Γ, where vol Γ s the sum of the lengths of the nternal edges of the graph. W s strctly smaller than vol Γ ff there exsts a balanced vertex, the vertex for whch there s the same number of nternal and external edges. 3. Pseudo orbt expanson for the resonance condton In ths secton, we lay theoretcal grounds for the method of pseudo orbt expanson. The method of pseudo orbt expanson was used to fndng the spectrum and trace formula for compact quantum graphs we refer to [BHJ]). The method was recently adjusted by the author to obtan the resonance condton for graphs wth attached halflnes [Lp5]. We explan here the method, n most cases omttng the proofs, whch can be found n the above references. We defne an effectve vertex-scatterng matrx, whch gves effectve scatterng from the nternal edges to other nternal edges emanatng from a vertex wth radatng condton on the halflnes. Defnton 3.. Let us assume a vertex v of the graph wth n nternal edges, whch emanate from ths vertex, parametrzed by 0, l j ) wth x = 0 correspondng to v and m halflnes. Let the soluton of the Schrödnger equaton on these nternal edges be f j x) = αj n exp kx) + αj out exp kx) and on the external edges g s x) = β s exp kx). Then the effectve vertex-scatterng matrx σ v) s gven by the relaton α v out = σ v) α v n, where the vectors α v out and α v n have as entres the above coeffcents of the outgong and ncomng waves, respectvely. Theorem 3.. The effectve vertex-scatterng matrx for the vertex v wth n nternal and m external edges and the standard couplng s σ v) = J n+m n I n, where J n s n n matrx wth all entres equal to one and I n s n n unt matrx. In partcular, for a balanced vertex σ v) = J n n I n. Proof. The theorem s proven as corollary 4.3 n [Lp5], we wll show here a drect proof. The couplng condton yeld α out j + α n j = α out Now we fx and substtute for β s = α out n j= + α n = β s, j =,... n, s =,..., m, n m k αj out αj n ) + k β j = 0. α out From ths equaton we have from whch the result follows. + α n j= + α n and α out j = α out s= αj n ) + mα out + α n ) = 0. n ) α out = αj n α n n + m j= + α n α n j. We obtan Now we ntroduce the orented graph Γ, whch s made from the compact part of the graph Γ. Each edge s replaced by two orented edges b j, ˆb j of the same length and opposte drectons. We wll defne the followng matrces.

4 4 PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION Defnton 3.3. The N N matrx Σ s a block-dagonalzable matrx wrtten n the bass correspondng to α = αb n,..., αb n N, α n,..., α n ) T ˆb ˆbN whch s block dagonal wth blocks σ v f transformed to the bass α n b v,..., αn b v d, α n b v,..., αn b v d,... ) T, where b v j s the j-th edge endng at the vertex v. Moreover, we defne N N matrx Q = and L = dag l,..., l N, l,..., l N ). 0 IN I N 0 ), the scatterng matrx S = Q Σ Note that Σ and S may for general couplng be energy dependent. However, ths s not the case for standard couplng, snce the matrx σ s not energy dependent. The matrx S = Q Σ s constructed n the followng way. We denote ts frst N rows by b,... b N and the other N rows by the edges n the opposte drecton ˆb,... ˆb N ; smlarly we denote the columns. If b j ends n the vertex v, then we wrte nto the b j -th column and all rows correspondng to orented edges startng from v the entres of the vertex-scatterng matrx σ v). To the ˆb j -th row we wrte the dagonal term of σ v), to the other rows whch correspond to the edges emanatng from v the nondagonal terms; n the rest of the rows n ths column s zero. We contnue wth the theorem whch s proven n [Lp5], but the proof s wth the excepton of the effectve vertex-scatterng matrx the same as e.g. n [BHJ]. Theorem 3.4. The resonance condton s where I N s a N N unt matrx. det e kl Q Σ I N ) = 0, Now we defne perodc orbts, pseudo orbts and rreducble pseudo orbts. Defnton 3.5. A perodc orbt γ on the graph Γ s a closed path on Γ. A pseudo orbt γ s a collecton of perodc orbts. An rreducble pseudo orbt γ s a pseudo orbt, whch does not use any drected edge more than once. We defne length of a perodc orbt by l γ = j,b j γ l j; the length of pseudo orbt and hence rreducble pseudo orbt) s the sum of the lengths of the perodc orbts from whch t s composed. We defne product of scatterng ampltudes for a perodc orbt γ = b, b,..., b n ) t uses frst the bond b, then t contnues to b, etc., t ends n the bond b n whch s connected to b ) as A γ = S b b S b3 b... S b b n, where S b b s the entry of the matrx S n the b -th row and b -th column. For a pseudo orbt we defne A γ = Π γn γa γj. Fnally, by m γ we denote the number of perodc orbts n the pseudo orbt γ. Now we restate the prevous theorem usng rreducble pseudo orbts; the proof can be found n [BHJ]. Theorem 3.6. The resonance condton s gven by the sum over rreducble pseudo orbts ) m γ A γ e kl γ = 0. γ Note that n general A γ can be energy dependent, but ths s not the case for standard couplng.

5 PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION 5 v v v 3 ˆ ˆ Fgure. Graph Γ for two abscssas and two halflnes 4. Theorems on the effectve sze of an equlateral graph Now we focus on equlateral graphs the graphs whch have all the nternal edges of the length l. Frst, we gve a theorem how to fnd the effectve sze of an equlateral graph. Then we ntroduce a method how to reduce the number of orented edges of the graph Γ for an equlateral graph wth standard couplng and balanced vertces. Both theorems were proven n [Lp5]. Theorem 4.. Let us assume an equlateral graph nternal edges of lengths l). Then the effectve sze s l n nonzero, where n nonzero s the number of nonzero egenvalues of the matrx S = Q Σ. Theorem 4.. Let us assume an equlateral graph Γ for whch no edge starts and ends n one vertex and no two vertces are connected by more than one edge. Let us assume standard couplng and let there be a balanced vertex v n whch drected edges b, b,..., b d end. Then the followng constructon does not change the resonance condton. We delete the drected edge b of the graph Γ, whch starts n the vertex v, and replace t by ghost edges b, b,..., b d ), where the ghost edge b j) starts n the vertex v and contnues to the drected edge b j+. Contrbuton of the rreducble pseudo orbt contanng ghost edge b to the resonance condton gven by theorem 3.6 s the followng. The ghost edge does not contrbute to the length of the pseudo orbt. The scatterng ampltude from the bond b, whch ends n v, to the bond b s equal to the scatterng ampltude from b to b taken wth the opposte sgn. Every ghost edge can be n the rreducble pseudo orbt used only once. Smlarly, one can delete more edges; for each balanced vertex we delete an edge whch ends n ths vertex. 5. Example : two abscssas and two halflnes In the followng sectons we use prevous theorems n several smple examples. Graph n the frst example conssts of two nternal edges of length l connected n one vertex wth two halflnes. There s Drchlet couplng f0) = 0) at the spare ends vertces v and v 3 ) of the abscssas and standard couplng n the central vertex v ). The orented graph Γ s shown n) fgure. Snce the vertex v s balanced, we have by theorem 3. σ v) =. The vertex-scatterng matrces for the vertces v and v 3 are σ v) = σ v3) =. The matrx S = Q Σ s ˆ ˆ / 0 0 / ˆ / 0 0 / ˆ

6 6 PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION v v v 3 ˆ ˆ Fgure. Graph Γ after deletng the edge We explctly mark the edges to whch the columns and rows correspond. Egenvalues of S are, and 0 wth multplcty. From theorem 4. we obtan that the effectve sze of the graph s l. Now we fnd the resonance condton usng pseudo orbts. The contrbuton of the pseudo orbts whch do not nclude any bond s. Clearly, there are no rreducble pseudo orbts on one or three bonds. Let us look at the contrbuton of the rreducble pseudo orbts on two edges,.e. fnd the coeffcent by expkl). We have two rreducble pseudo orbts, ˆ) and, ˆ). The scatterng ampltude between ˆ and s, the scatterng ampltude between and ˆ s /. There s one perodc orbt n the pseudo orbt, ˆ), hence there s a factor of ). The contrbuton of the rreducble pseudo orbt, ˆ) s ) /) ) = /, smlarly for the pseudo orbt, ˆ). Hence the coeffcent by expkl) s. Fnally, we fnd the contrbuton the rreducble pseudo orbts on four edges. There are two rreducble pseudo orbts:,, ˆ, ˆ) and, ˆ), ˆ); the former conssts of one perodc orbt, the latter of two perodc orbts. The contrbuton of the rreducble pseudo orbt,, ˆ, ˆ) s ) /) ) = /4, the contrbuton of the rreducble pseudo orbt, ˆ), ˆ) s ) /) ) = /4. Hence the coeffcent by exp4kl) s 0, because contrbuton of the two rreducble pseudo orbts cancel. The resonance condton s expkl) = 0. Snce the vertex v s balanced, we can also delete one bond whch ends n ths vertex, say bond. We replace t by a ghost edge whch starts at v and contnues to the only other edge whch ends n v, the bond ˆ see fgure ). Now we can do pseudo orbt expanson agan. The contrbuton of pseudo orbt whch does not nclude any bond s. We have two rreducble pseudo orbts on two non-ghost edges ˆ,, ˆ) and, ˆ). The former has contrbuton / ) = / we have used the fact that the scatterng ampltude between ˆ and ˆ s +, because the scatterng ampltude between ˆ and was ), the contrbuton of the latter s ) /) ) = /. Agan we obtan that the coeffcent by expkl) s. There s no rreducble pseudo orbt, whch would use all the three remanng non-ghost edges, even f t would use the ghost edge. Clearly, there cannot be also any pseudo orbt on four edges, because we have deleted one. Agan, we obtan the same resonance condton. In ths case t was easer to fnd that the coeffcent by exp4kl) s zero. We conclude that the resolvent resonances n ths case are only egenvalues λ = k wth k = nπ, n Z. 6. Example : trangle wth attached halflnes Let us consder a graph wth three nternal edges of the lengths l n the trangle. To each vertex two halflnes are attached, so every vertex s balanced. The graph

7 PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION 7 ˆ3 ˆ ˆ3 3 ˆ Fgure 3. Graph Γ for the trangle wth attached halflnes ˆ 3 ˆ Fgure 4. Graph Γ for the trangle wth attached halflnes after deletng the edges ˆ, ˆ, ˆ3 Γ s shown) n fgure 3. The vertex scatterng matrces are n all the vertces σ v) =. The matrx S = Q Σ s 3 ˆ ˆ ˆ3 0 0 / / 0 0 / / / / ˆ / / 0 ˆ 0 / / ˆ3 0 0 / / 0 0. Its egenvalues are, / wth multplcty and 0 wth multplcty 3. Hence the effectve sze of ths graph s 3l/. Now we fnd the contrbutons of the pseudo orbts to the resonance condton. There s no rreducble pseudo orbt on edge and on 5 edges. We have the followng three rreducble pseudo orbts on two edges:, ˆ);, ˆ); 3, ˆ3). There are two rreducble pseudo orbts on three edges,, 3) and ˆ, ˆ3, ˆ), sx on four edges, ˆ), ˆ);, ˆ)3, ˆ3); 3, ˆ3), ˆ);,, ˆ, ˆ);, 3, ˆ3, ˆ); 3,, ˆ, ˆ3) and eght rreducble pseudo orbts on sx edges, ˆ), ˆ)3, ˆ3);,, ˆ, ˆ)3, ˆ3);, 3, ˆ3, ˆ), ˆ); 3,, ˆ, ˆ3), ˆ);,, 3)ˆ, ˆ3, ˆ);,, 3, ˆ3, ˆ, ˆ);, 3,, ˆ, ˆ3, ˆ); 3,,, ˆ, ˆ, ˆ3). Below, we compute ther

8 8 PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION contrbutons exp 0 :, exp kl) : ) ) 3 = 3 4, ) 3 exp 3kl) : ) = 4, exp 4kl) : 4 ) ) 3 + ) ) ) 3 = 0, exp 6kl) : ) 6 ) 3 + ) ) ) ) 3 + ) 3 ) 3 + ) + ) ) 4 ) 3 = 0. The resonance condton s 3 4 exp kl) exp 3kl) = 0. 4 The alternatve way of constructng the resonance condton s usng the method of deletng the edges. We have three balanced vertces, hence we can delete the edges ˆ, ˆ and ˆ3 and replace them by ghost edges ˆ, ˆ and ˆ3 see fgure 4). It s clear that there are no rreducble pseudo orbts on 4, 5 or 6 edges and snce no ghost edge contnues to an edge endng n a vertex from whch ths ghost edge starts, there are also no pseudo orbts on one edge. There are three rreducble pseudo orbts on two non-ghost edges,, ˆ );, 3, ˆ3 ); 3,, ˆ ) and there are two rreducble pseudo orbts on three edges,, 3) and, ˆ, 3, ˆ3,, ˆ ). Ther contrbutons are exp 0 :, ) exp kl) : ) 3 = 3 4, ) 3 exp 3kl) : ) = 4, whch gves the same resonance condton. The resonances are such λ = k wth k = nπ/l and k = π + nπ ln )/l wth multplcty two, n Z.

9 PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION 9 v v ˆ 4 ˆ4 ˆ v v ˆ3 ˆ4 4 ˆ ˆ v 4 ˆ3 3 Fgure 5. Graph Γ for the square wth attached halflnes v 3 v 4 3 Fgure 6. Graph Γ for the square wth attached halflnes after deletng the edges ˆ, ˆ, ˆ3, ˆ4 v 3 7. Example 3: square wth attached halflnes We consder a square of the edges of lengths l, n each vertex two halflnes are attached, hence every vertex s balanced. The) graph Γ s n fgure 5. The vertexscatterng matrces are agan σ v) =. The matrx S = Q Σ s 3 4 ˆ ˆ ˆ3 ˆ / / / / / / / / ˆ / / 0 0 ˆ 0 / / 0 ˆ3 0 0 / / ˆ / / 0 0 0, ts egenvalues are, and 0 wth multplcty 6; the effectve sze s l. One can see that there are no rreducble pseudo orbts on odd number of edges. The rreducble pseudo orbts on two edges are, ˆ);, ˆ); 3, ˆ3); 4, ˆ4), on four edges, ˆ), ˆ);, ˆ)3, ˆ3);, ˆ)4, ˆ4);, ˆ)3, ˆ3);, ˆ)4, ˆ4); 3, ˆ3)4, ˆ4);,, ˆ, ˆ);, 3, ˆ3, ˆ); 3, 4, ˆ4, ˆ3); 4,, ˆ, ˆ4);,, 3, 4); ˆ4, ˆ3, ˆ, ˆ), on sx edges, ˆ), ˆ)3, ˆ3);, ˆ), ˆ)4, ˆ4);, ˆ)3, ˆ3)4, ˆ4);, ˆ)3, ˆ3)4, ˆ4);,, ˆ, ˆ)3, ˆ3);,, ˆ, ˆ)4, ˆ4);, 3, ˆ3, ˆ), ˆ);, 3, ˆ3, ˆ)4, ˆ4); 3, 4, ˆ4, ˆ3), ˆ); 3, 4, ˆ4, ˆ3), ˆ); 4,, ˆ, ˆ4), ˆ); 4,, ˆ, ˆ4)3, ˆ3);,, 3, ˆ3, ˆ, ˆ);, 3, 4, ˆ4, ˆ3, ˆ); 3, 4,, ˆ, ˆ4, ˆ3); 4,,, ˆ, ˆ, ˆ4) and on eght edges, ˆ), ˆ)3, ˆ3)4, ˆ4);,, ˆ, ˆ)3, ˆ3)4, ˆ4);, 3, ˆ3, ˆ), ˆ)4, ˆ4); 3, 4, ˆ4, ˆ3), ˆ), ˆ); 4,, ˆ, ˆ4), ˆ)3, ˆ3);,, ˆ, ˆ)3, 4, ˆ4, ˆ3);, 3, ˆ3, ˆ)4,, ˆ, ˆ4);,, 3, ˆ3, ˆ, ˆ)4, ˆ4);, 3, 4, ˆ4, ˆ3, ˆ), ˆ); 3, 4,, ˆ, ˆ4, ˆ3), ˆ); 4,,, ˆ, ˆ, ˆ4)3, ˆ3);

10 0 PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION 3 4 Fgure 7. Graph Γ smplfed after deletng the edges,, 3, 4)ˆ4, ˆ3, ˆ, ˆ);,, 3, 4, ˆ4, ˆ3, ˆ, ˆ);, 3, 4,, ˆ, ˆ4, ˆ3, ˆ); 3, 4,,, ˆ, ˆ, ˆ4, ˆ3); 4,,, 3, ˆ3, ˆ, ˆ, ˆ4). Ther contrbutons to the resonance condtons are exp 0 :, exp kl) : ) ) 4 =, exp 4kl) : exp 6kl) : exp 8kl) : ) ) ) 6 + ) ) ) ) ) ) 4 ) 4 = 0, + ) ) ) ) ) ) 4 + ) ) ) 4 + ) ) ) 4 + ) ) ) 4 4 ) ) + ) 4 ) = 0 ) ) ) ) 8 + ) ) ) + ) ) ) ) ) ) 4 ) 6 ) 4 = 0. If we delete the edges ˆ, ˆ, ˆ3 and ˆ4, we obtan fgure 6. Note that the scatterng ampltude for path from to s the same as the scatterng ampltude from to 4 through ˆ ; n both cases we obtan /. Smlarly for other vertces, so the graph s equvalent to a graph n fgure 7, where all the scatterng ampltudes are /. Ths smplfes fndng the resonance condton. We have four rreducble pseudo orbts on two edges ); 4); 3); 34) and four on four edges )34); 4)3); 34); 43). Ther contrbuton s exp 0 :, ) exp kl) : ) 4 =, ) ) ) 4 exp 4kl) : ) + ) = 0.

11 PSEUDO ORBIT EXPANSION FOR THE RESONANCE CONDITION The resonance condton s exp kl) = 0, the postons of resonances are λ = k wth k = nπ/l, n Z. Acknowledgements Support of the grant 5-480Y of the Grant Agency of the Czech Republc s acknowledged. The author thanks to R. Band for a useful dscusson. References [BHJ] R. Band, J. M. Harrson, C. H. Joyner, Fnte pseudo orbt expansons for spectral quanttes of quantum graphs, J. Phys. A: Math. Theor. 45, ). DOI: 0.088/75-83/45/3/3504 [DEL0] E. B. Daves, P. Exner, J. Lpovský, Non-Weyl asymptotcs for quantum graphs wth general couplng condtons, J. Phys. A: Math. Theor. 43, ). DOI: 0.088/75-83/43/47/47403 [DP] E. B. Daves, A. Pushntsk, Non-Weyl resonance asymptotcs for quantum graphs, Analyss and PDE 4, ). DOI: 0.40/apde [EL07] P. Exner, J. Lpovský, Equvalence of resolvent and scatterng resonances on quantum graphs, n Adventures n Mathematcal Physcs Proceedngs, Cergy-Pontose 006) vol 447 Provdence, R.I.), pp ). DOI: 0.090/conm/447 [EL] P. Exner, J. Lpovský, Non-Weyl resonance asymptotcs for quantum graphs n a magnetc feld, Phys. Lett. A 375, ). DOI: 0.06/j.physleta [KS03] T. Kottos, U. Smlansky, Quantum graphs: a smple model for chaotc scatterng, J. Phys. A: Math. Gen. 36, , 003). DOI: 0.088/ /36//337 [KS04] T. Kottos, H. Schanz, Statstcal propertes of resonance wdth for open quantum systems Waves n Random Meda 4, S9 S05 004). DOI: 0.088/ /4//03 [Lp5] J. Lpovský, On the effectve sze of a non-weyl graph, arxv: [math-ph] 05).