Quantum Graphs I. Some Basic Structures

Quantum Graphs I. Som Basic Structurs Ptr Kuchmnt Dpartmnt of Mathmatics Txas A& M Univrsity Collg Station, TX, USA 1 Introduction W us th nam quantum graph for a graph considrd as a on-dimnsional singular varity and quippd with a slf-adjoint diffrntial (in som cass psudo-diffrntial) oprator ( Hamiltonian ). Thr ar manifold rasons for studying quantum graphs. Thy naturally aris as simplifid (du to rducd dimnsion) modls in mathmatics, physics, chmistry, and nginring (nanotchnology), whn on considrs propagation of wavs of diffrnt natur (lctromagntic, acoustic, tc.) through a msoscopic quasi-ondimnsional systm that looks lik a thin nighborhood of a graph. On can mntion in particular th fr-lctron thory of conjugatd molculs in chmistry, quantum chaos, quantum wirs, dynamical systms, photonic crystals, scattring thory, and a varity of othr applications. W will not discuss any dtails of ths origins of quantum graphs, rfrring th radr instad to [54] for a rcnt survy and litratur. Th problms addrssd in th quantum graph thory includ justifications of quantum graphs as approximations for mor ralistic (and complx) modls of wavs in complx structurs, analysis of various dirct and invrs spctral problms (coming from quantum chaos, optics, scattring thory, and othr aras), and many othrs. This papr dos not contain discussion of most of ths topics and th radr is rfrrd to th survy [54] and to paprs prsntd in th currnt issu of Wavs in Random Mdia for mor information and rfrncs. In this papr w addrss som basic notions and rsults concrning quantum graphs and thir spctra. Whil th spctral thory of combinatorial graphs is a rathr wll stablishd topic (.g., books [12, 21, 22, 23, 62] and 1

rfrncs thrin), th corrsponding thory of quantum graphs is just dvloping (.g., xampls of such studis in [3, 5, 6, 7, 8, 9, 15], [16] - [20], [25], [29] - [35], [39], [43] - [49], [56] - [58], [64, 65, 70, 74, 79, 80] and furthr rfrncs in [54]). Lt us dscrib th contnts of th following sctions. Sction 2 is dvotd to introducing basic notions of a mtric and quantum graph. Th largst Sction 3 dals with th dtaild dscription of slf-adjoint vrtx conditions for scond drivativ Hamiltonians on quantum graphs. Tratmnt of infinit graphs rquird som rstrictions on thir structur. Th vrtx conditions ar writtn in th form that nabls on to dscrib asily th quadratic forms of th oprators and to classify all prmutation-invariant conditions. Sction 4 is dvotd to rlations btwn quantum and combinatorial spctral problms that will b sn as spcially hlpful in th plannd nxt part [56] of this articl. Th papr nds with short sctions containing rmarks and acknowldgmnts. Th radr should not that this papr is of th survy natur and hnc most of th rsults ar not nw (although th xposition might diffr from othr sourcs). Som rfrncs ar providd throughout th txt, albit th bibliography was not intndd to b comprhnsiv, and th radr is dirctd to th survys [54, 55] for mor dtaild bibliography. It was initially plannd to addrss svral nw issus in this papr, among which on can mntion abov all a mor dtaild spctral tratmnt of infinit graphs (bounds for gnralizd ignfunctions, Schnol s thorms, priodic graphs, som gap opning ffcts, and discussion of bound stats), howvr th articl siz limitations rsultd in th ncssity of postponing thos to th nxt papr [56]. For th sam rason, th author has also rstrictd considrations to th cas of th scond drivativ Hamiltonian only, whil on can xtnd ths without much of a difficulty to mor gnral Schrödingr oprators (.g., [46]). This, as wll as som othr topics will b dalt with lswhr. So, th papr is plannd to srv as an introduction that could b usful whil rading othr articls of this issu of Wavs in Random Mdia, and also as th first part of [56]. 2 Quantum graphs As it was mntiond in th introduction, w will b daling with quantum graphs, i.. graphs considrd as on-dimnsional singular varitis rathr 2

than purly combinatorial objcts and corrspondingly quippd with diffrntial (or somtims psudo-diffrntial ) oprators (Hamiltonians) rathr than discrt Laplac oprators. 2.1 Mtric graphs A graph Γ consists of a finit or countably infinit st of vrtics V = {v i } and a st E = { j } of dgs conncting th vrtics. Each dg can b idntifid with a pair (v i, v k ) of vrtics. Although in many quantum graph considrations dirctions of th dgs ar irrlvant and could b fixd arbitrarily (w will not nd thm in this papr), it is somtims mor convnint to hav thm assignd. Loops and multipl dgs btwn vrtics ar allowd, so w avoid saying that E is a subst of V V. W also dnot by E v th st of all dgs incidnt to th vrtx v (i.., containing v). It is assumd that th dgr (valnc) d v = E v of any vrtx v is finit and positiv. W hnc xclud vrtics with no dgs coming in or going out. This is natural, sinc for th quantum graph purposs such vrtics ar irrlvant. So far all our dfinitions hav dalt with a combinatorial graph. Hr w introduc a notion that maks Γ a topological and mtric objct. Dfinition 1. A graph Γ is said to b a mtric graph (somtims th notion of a wightd graph is usd instad), if its ach dg is assignd a positiv lngth l (0, ] (notic that dgs of infinit lngth ar allowd). Having th lngth assignd, an dg will b idntifid with a finit or infinit sgmnt [0, l ] of th ral lin with th natural coordinat x along it. In most cass w will drop th subscript in th coordinat and call it x, which should not lad to any confusion. This nabls on to intrprt th graph Γ as a topological spac (simplicial complx) that is th union of all dgs whr th nds corrsponding to th sam vrtx ar idntifid. Th radr should not that w do not assum th graph to b mbddd in any way into an Euclidan spac. In som applications such a natural mbdding dos xist (.g., in modling quantum wir circuits or photonic crystals), and in such cass th coordinat along an dg is usually th arc lngth. In som othr applications (.g., in quantum chaos) th graph is not assumd to b mbddd. Graph Γ can b quippd with a natural mtric. If a squnc of dgs { j } M j=1 forms a path, its lngth is dfind as l j. For two vrtics v and w, th distanc ρ(v, w) is dfind as th minimal path lngth btwn thm. 3

Sinc along ach dg th distanc is dtrmind by th coordinat x, it is asy to dfin th distanc ρ(x, y) btwn two points x, y of th graph that ar not ncssarily vrtics. W lav this to th radr. W also impos som additional conditions: Condition A. Th infinit nds of infinit dgs ar assumd to hav dgr on. Thus, th graph can b thought of as a graph with finit lngth dgs with additional infinit lads or nds going to infinity attachd to som vrtics. This situation ariss naturally for instanc in scattring thory. Sinc ths infinit vrtics will nvr b tratd as rgular vrtics (in fact, in this papr such vrtics will not aris at all), on can just assum that ach infinit dg is a ray with a singl vrtx. Condition B. Whn studying infinit graphs, w will impos som assumptions that will imply in particular that for any positiv numbr r and any vrtx v thr is only a finit st of vrtics w at a distanc lss than r from v. In particular, th distanc btwn any two distinct vrtics is positiv, and thr ar no finit lngth paths of infinitly many dgs. This obviously mattrs only for infinit graphs (i.., graphs with infinitly many dgs) and is automatically satisfid for th class of infinit mtric graphs that will b introducd latr. So, now on can imagin th graph Γ as a on-dimnsional simplicial complx, ach 1D simplx (dg) of which is quippd with a smooth structur, with singularitis arising at junctions (vrtics) (s Fig. 1). Th radr should notic that now th points of th graph ar not only its vrtics, but all intrmdiat points x on th dgs as wll. On can dfin in th natural way th Lbsgu masur dx on th graph. Functions f(x) on Γ ar dfind along th dgs (rathr than at th vrtics as in discrt modls). Having this and th masur, on can dfin in a natural way som function spacs on th graph: Dfinition 2. 1. Th spac L 2 (Γ) on Γ consists of functions that ar masurabl and squar intgrabl on ach dg and such that f 2 L 2 (Γ) = E f 2 L 2 () <. In othr words, L 2 (Γ) is th orthogonal dirct sum of spacs L 2 (). 2. Th Sobolv spac H 1 (Γ) consists of all continuous functions on Γ 4

Figur 1: Graph Γ. that blong to H 1 () for ach dg and such that f 2 H 1 () <. E Not that continuity in th dfinition of th Sobolv spac mans that th functions on all dgs adjacnt to a vrtx v assum th sam valu at v. Thr sm to b no natural dfinition of Sobolv spacs H k (Γ) of ordr k highr than 1, sinc boundary conditions at vrtics dpnd on th Hamiltonian (s dtails latr on in this papr). Th last stp that is ndd to finish th dfinition of a quantum graph is to introduc a slf-adjoint (diffrntial or mor gnral) oprator (Hamiltonian) on Γ. This is don in th nxt sction. 2.2 Oprators Th oprators of intrst in th simplst cass ar: th ngativ scond drivativ a mor gnral Schrödingr oprator f(x) d2 f dx 2, (1) 5

f(x) d2 f + V (x)f(x), (2) dx2 or a magntic Schrödingr oprator ( ) 2 1 d f(x) i dx A(x) + V (x)f(x). (3) Hr x dnots th coordinat x along th dg. Highr ordr diffrntial and vn psudo-diffrntial oprators aris as wll (s,.g. th survy [54] and rfrncs thrin). W, howvr, will concntrat hr on scond ordr diffrntial oprators, and for simplicity of xposition spcifically on (1). In ordr for th dfinition of th oprators to b complt, on nds to dscrib thir domains. Th natural conditions rquir that f blongs to th Sobolv spac H 2 () on ach dg. On also clarly nds to impos boundary valu conditions at th vrtics. Ths will b studid in th nxt sction. 3 Boundary conditions and slf-adjointnss W will discuss now th boundary conditions on would lik to add to th diffrntial xprssion (1) in ordr to crat a slf-adjoint oprator. 3.1 Graphs with finitly many dgs In this sction w will considr finit graphs only. This mans that w assum that th numbr of dgs E is finit (and hnc th numbr of vrtics V is finit as wll, sinc w assum all vrtx dgrs to b positiv). Notic that dgs ar still allowd to hav infinit lngth. W will concntrat on local (or vrtx) boundary conditions only, i.. on thos that involv th valus at a singl vrtx only at a tim. It is possibl to dscrib all th vrtx conditions that mak (1) a slf-adjoint oprator (s [43, 41] and a partial dscription in [34]). This is don by ithr using th standard von Numann thory of xtnsions of symmtric oprators (as for instanc dscribd in [1]), or by its mor rcnt vrsion that amounts to finding Lagrangian plans with rspct to th complx symplctic boundary form that corrsponds to th maximal oprator (s for instanc 6

[3, 26, 27, 28, 41, 42, 43, 69, 71] for th accounts of this approach that gos back at last as far as [63], whr it was prsntd without us of words symplctic or Lagrangian ). On of th most standard typs of such boundary conditions is th Kirchhoff condition: f(x) is continuous on Γ and at ach vrtx v on has E v df dx (v) = 0, (4) whr th sum is takn ovr all dgs containing th vrtx v. Hr th drivativs ar takn in th dirctions away from th vrtx (w will call ths outgoing dirctions ), th agrmnt w will adhr to in all cass whn ths conditions ar involvd. Somtims (4) is calld th Numann condition. It is clar that at loos nds (vrtics of dgr 1) it turns into th actual Numann condition. Bsids, as th Numann boundary condition for Laplac oprator, it is natural. Namly, as it will b sn a littl bit latr, th domain of th quadratic form of th corrsponding oprator dos not rquir any conditions on a function bsids bing in H 1 (Γ) (and hnc continuous). It is also usful to not that undr th boundary conditions (4) on can liminat all vrtics of dgr 2, conncting th adjacnt dgs into on smooth dg. Thr ar many othr plausibl vrtx conditions (som of which will b discussd latr), and th qustion w want to addrss now is how to dscrib all of thos that lad to a slf-adjoint ralization of th scond drivativ along th dgs. Sinc w ar intrstd in local vrtx conditions only, it is clar that it is sufficint to addrss th problm of slf-adjointnss for a singl junction of d dgs at a vrtx v. Bcaus along ach dg our oprator acts as th (ngativ) scond drivativ, on nds to stablish two conditions pr an dg, and hnc at ach vrtx th numbr of conditions must coincid with th dgr d of th vrtx. For functions in H 2 on ach dg, th conditions may involv only th boundary valus of th function and its drivativ. Thn th most gnral form of such (homognous) condition is A v F + B v F = 0. (5) Hr A v and B v ar d d matrics, F is th vctor (f 1 (v),..., f d (v)) t of th vrtx valus of th function along ach dg, and F = (f 1(v),..., f d (v))t 7

is th vctor of th vrtx valus of th drivativs takn along th dgs in th outgoing dirctions at th vrtx v, as w hav agrd bfor. Th rank of th d 2d matrix (A v, B v ) must b qual to d (i.., maximal) in ordr to nsur th corrct numbr of indpndnt conditions. Whn this would not lad to confusion, w will drop th subscript v in ths matrix notations, rmmbring that th matrics dpnd on th vrtx (in fact, for non-homognous graphs thy ssntially hav no othr choic). Now on is intrstd in th ncssary and sufficint conditions on matrics A and B in (5) that would guarant slf-adjointnss of th rsulting oprator. All such conditions wr compltly dscribd in [43] (s also th arlir papr [33] for som spcial cass and [41, 48]for an altrnativ considration that rprsnts th boundary conditions in trms of vrtx scattring matrics). W will formulat th corrsponding rsult in th form takn from [43]. Thorm 3. [43] Lt Γ b a mtric graph with finitly many dgs. Considr th oprator H acting as d2 dx 2 on ach dg, with th domain consisting of functions that blong to H 2 () on ach dg and satisfy th boundary conditions (5) at ach vrtx. Hr {A v, B v v V } is a collction of matrics of sizs d v d v such that ach matrix (A v B v ) has th maximal rank. In ordr for H to b slf-adjoint, th following condition at ach vrtx is ncssary and sufficint: th matrix A v B v is slf-adjoint. (6) Th proof of this thorm can b found in [43]. W would lik now to dscrib th quadratic form of th oprator H corrsponding to th (ngativ) scond drivativ along ach dg, with slfadjoint vrtx conditions (5) (w assum in particular that (6) is satisfid). In ordr to do so, w will stablish first a coupl of simpl auxiliary statmnts. In th nxt two lmmas and a corollary w will considr matrics A and B as in (8). Sinc w will b concrnd with a singl vrtx hr, w will drop for this tim th subscripts v in A v, B v, and d v. Lt us introduc som notations. W will dnot by P and P 1 th orthogonal projctions in C d onto th krnls K = kr B and K 1 = kr B rspctivly. Th complmntary orthogonal projctors onto th rangs R = R(B ) and R 1 = R(B) ar dnotd by Q and Q 1 (hr R(M) dnots th rang of a matrix M). 8

Lmma 4. Lt d d matrics A and B b such that th d 2d matrix (A B) has maximal rank and AB is slf-adjoint. Thn 1. Oprator A maps th rang R of B into th rang R 1 of B. 2. Th mapping P 1 AP : K K 1 is invrtibl. 3. Th mapping Q 1 BQ : R R 1 is invrtibl (w dnot its invrs by B ( 1) ). 4. Th matrix B ( 1) AQ is slf-adjoint. Proof. Slf-adjointnss of AB mans AB = BA, which immdiatly implis th first statmnt of th lmma. In ordr to prov th nxt two statmnts, lt us dcompos th spac C d into th orthogonal sum R 1 K 1 and C 2d into C d C d = R K R K. Thn th matrix (A B) rprsnting an oprator from C 2d into C d can b writtn in a 2 4 block-matrix form with rspct to ths dcompositions. Taking into account th dfinitions of th subspacs R, R 1, K, and K 1 and th alrady provn first statmnt of th lmma, this lads to th block matrix (A B) = ( ) A11 A 12 B 11 0. (7) 0 A 22 0 0 For this matrix to hav maximal rank, th ntry A 22 must b invrtibl, which givs th scond statmnt of th lmma. Th third statmnt is obvious, sinc th matrix B 11 is squar and has no krnl (which has alrady bn liminatd and includd into K). Immdiat calculation shows that slf-adjointnss of AB mans that th squar matrix A 11 B11 is slf-adjoint, i.. A 11 B11 = B 11 A 11. Sinc invrtibility of B 11 has alrady bn stablishd (rcall that its invrs is dnotd by B ( 1) ), w can multiply th prvious quality by appropriat invrs matrics from both sids to gt B ( 1) A 11 = A 11B ( 1). This mans that th matrix B ( 1) A 11 is slf-adjoint and hnc th last statmnt of th lmma is provn. Corollary 5. Lt th conditions of Lmma 4 b satisfid. Thn th boundary condition (5) AF + BF = 0 is quivalnt to th pair of conditions P F = 0 9

and LQF + QF = 0, whr P, as bfor, is th orthogonal projction onto th krnl of matrix B, Q is th complmntary projctor, and L is th slfadjoint oprator B ( 1) A. Proof. W will mploy th notations usd in th prcding lmma. It is clar that (5) is quivalnt to th pair of conditions P 1 AF + P 1 BF = 0 and Q 1 AF + Q 1 BF = 0. Th lmma now shows that th first of thm can b rwrittn as A 22 P F = 0, which by th scond statmnt of th lmma is quivalnt to P F = 0. Th scond quality, again by th lmma, can b quivalntly rwrittn as AQF + B 11 QF = 0, or aftr invrting B 11 as LQF + QF = 0, which finishs th proof of th corollary. W can now r-phras gnral slf-adjoint boundary conditions in a fashion that is somtims mor convnint (for instanc, for dscribing th quadratic form of th oprator). Thorm 6. All slf-adjoint ralizations H of th ngativ scond drivativ on Γ with vrtx boundary conditions can b dscribd as follows. For vry vrtx v thr ar an orthogonal projctor P v in C dv with th complmntary projctor Q v = Id P v and a slf-adjoint oprator L v in Q v C dv. Th functions f from th domain D(H) H 2 () of H ar dscribd by th following conditions at ach (finit) vrtx v: P v F (v) = 0 Q v F (v) + L v Q v F (v) = 0. (8) In trms of th matrics A v and B v of Thorm 3, P v is th orthogonal projctor onto th krnl of B v and L v = B v ( 1) A v (whr B v ( 1) has bn dfind prviously). Proof. Adopting th dfinitions of P v and L v providd in th thorm, on can s that th thorm s statmnt is just a simpl consqunc of Thorm 3, Lmma 4, and Corollary 5 combind. Rmark 7. 1. In viw of th first condition in (8), th scond on can b quivalntly writtn as Q v F (v) + L v F (v) = 0. 2. Conditions (8) say that th P v -componnt of th vrtx valus F (v) of f must b zro (kind of a Dirichlt part), whil th P v -part of th drivativs F (v) is unrstrictd. Th Q v -part of th drivativs F (v) is dtrmind by th Q v -part of th function F (v). 10

W will nd also th following wll known trac stimat that w prov hr for compltnss. Lmma 8. Lt f H 1 [0, a], thn for any l a. f(0) 2 2 l f 2 L 2 [0,a] + l f 2 L 2 [0,a] (9) Proof. Du to H 1 -continuity of both sids of th inquality, it is sufficint to prov it for smooth functions. Start with th rprsntation f(0) = f(x) and stimat by Cauchy-Schwartz inquality This implis x 0 x 0 f (t)dt, x [0, l] (10) f (t)dt 2 f 2 L 2 [0,a] χ [0,x] 2 L 2 [0,a] = x f 2 L 2 [0,a]. x f (t)dt 2 L 2 [0,l] f 2 L 2 [0,a] l x dx = l2 2 f 2 L 2 [0,a]. 0 0 Now taking L 2 [0, l]-norms in both sids of (10) and using triangl inquality and (a + b) 2 2a 2 + 2b 2, w gt th stimat f(0) 2 l 2 f 2 L 2 [0,a] + l 2 f 2 L 2 [0,a], which implis th statmnt of th lmma. W ar rady now for th dscription of th quadratic form of th oprator H on a finit graph Γ. Lt as bfor Γ b a mtric graph with finitly many vrtics. Th slfadjoint oprator H in L 2 (Γ) acts as d2 along ach dg, dx2 with th domain consisting of all functions f(x) on Γ that blong to th 11

Sobolv spac H 2 () on ach dg and satisfy at ach vrtx v conditions (8): P v F (v) = 0 Q v F (v) + L v Q v F (v) = 0. (11) Hr, as always F (v) = (f 1 (v),...f dv (v)) t is th column vctor of th valus of th function f at v that it attains whn v is approachd from diffrnt dgs j adjacnt to v, F (v) is th column vctor of th corrsponding outgoing drivativs at v, th d v d v -matrix P v is an orthogonal projctor and L v is a slf-adjoint oprator in th krnl Q v C d v of P v. Thorm 9. Th quadratic form h of H is givn as h[f, f] = df dx 2 dx (L v ) jk f j (v)f k (v) E v V j, k E v = df dx 2 dx L v F, F, E v V (12) whr, dnots th standard hrmitian innr product in C d. Th domain of this form consists of all functions f that blong to H 1 () on ach dg and satisfy at ach vrtx v th condition P v F = 0. Corrspondingly, th ssqui-linar form of H is: h[f, g] = E df dg dx dx dx L v F, G. (13) v V Proof. Notic that Lmma 8 shows that (12) with th domain dscribd in th thorm dfins a closd quadratic form. It hnc corrsponds to a slfadjoint oprator M in L 2 (Γ). Intgration by parts in (13) against smooth functions g that vanish in a nighborhood of ach vrtx shows that on its domain M acts as th ngativ scond drivativ along ach dg. So, th rmaining task is to show that its domain D(M) consists of all functions that blong to H 2 on ach dg and satisfy th vrtx conditions (8). This would imply that M = H. So, lt us assum f D(M). In particular, f H 1 (). It is th standard conclusion thn that f H 2 () for any dg (w lav to th radr to fill in th dtails, s also th sction concrning infinit graphs). W nd now to vrify that f satisfis th vrtx conditions (8). Th condition P v F (v) = 0 dos not nd to b chckd, sinc 12

it is satisfid on th domain of th quadratic form. transforms (13) into Intgration by parts E d 2 f dx gdx F + L 2 v F, G. (14) v V Th scond trm must vanish for any g in th domain of th quadratic form. Taking into account that thn G(v) can b an arbitrary vctor such that P v G(v) = 0, this mans that for ach v th quality Q v F (v) + Q v L v F (v) = 0 (15) nds to b satisfid, whr Q v is th complmntary projction to P v. This givs us th ndd conditions (8) for th function f. It is also asy to chck in a similar fashion that as soon as a function f blongs to H 2 on ach dg and satisfis (8), it blongs to th domain of M. This provs that M in fact coincids with th prviously dscribd oprator H. Th proof is hnc compltd. Corollary 10. Th oprator H is boundd from blow. Morovr, lt S = max{ L v }, thn v whr H C Id, (16) C = 4S max{2s, max{l ( 1) }}. Proof. On can choos l := min{l } in (9) applid to any dg. Thn, du to (9) on has: v L v F (v), F (v) S ( v 2 l f 2 L 2 (Γ) + l f 2 L 2 (Γ) 2S F (v) 2 ). (17) If now 2lS 1, thn (17) and th dfinition of th quadratic form h show that th statmnt of th Corollary holds. Although on can (and oftn nds to) considr quantum graphs with mor gnral Hamiltonian oprators (.g. Schrödingr oprators with lctric and magntic potntials, oprators of highr ordr, psudo-diffrntial oprators, tc.), for th purpos of this articl only w adopt th following dfinition: 13

Dfinition 11. A quantum graph is a mtric graph quippd with th oprator H that acts as th ngativ scond ordr drivativ along dgs and is accompanid by th vrtx conditions (8). 3.2 Exampls of boundary conditions In this sction w tak a brif look from th prospctiv of th prvious sction at som xampls of vrtx conditions and corrsponding oprators. Th radr can find mor xampls in [34, 43, 59]. 3.2.1 δ-typ conditions ar dfind as follows: and at ach vrtx v, f(x) is continuous on Γ E v df dx (v) = α v f(v). (18) Hr α v ar som fixd numbrs. On can rcogniz ths conditions as an analog of conditions on obtains from a Schrödingr oprator on th lin with a δ potntial, which xplains th nam. In this cas th conditions can b obviously writtn in th form (5) with and Sinc A v = 1 1 0... 0 0 1 1... 0...... 0 1 1 α v 0... 0 0 0 0... 0 B v =............ 0 0... 0. 1 1... 1 0 0... 0 0 A v Bv =..............., 0 0... 0 α v th slf-adjointnss condition (6) is satisfid if and only if α is ral. 14

In ordr to writ th vrtx conditions in th form (8), on nds to find th orthogonal projction P v onto th krnl of B v and th slf-adjoint oprator L v = B v ( 1) A v Q v. It is a simpl xrcis to find that Q v is th ondimnsional projctor onto th spac of vctors with qual coordinats and corrspondingly th rang of P v is spannd by th vctors r k, k = 1,..., d v 1, whr r k has 1 as th k-th componnt, 1 as th nxt on, and zros othrwis. Thn a straightforward calculation shows that L v is th multiplication by th numbr α v. In particular, th dscription of th projctor P v shows d v that th quadratic form of th oprator H is dfind on functions that ar continuous throughout all vrtics (i.., F (v) = (f(v),..., f(v)) t ) and hnc blong to H 1 (Γ). Th form is computd as follows: df dx 2 dx L v F, F E v V = df dx 2 dx + α v f(v) 2 (19). E v V It is obvious from (19) that th oprator is non-ngativ if α v 0 for all vrtics v. 3.2.2 Numann (Kirchhoff) conditions Ths conditions (4) that hav alrady bn mntiond, rprsnt probably th most common cas of th δ-typ conditions (18) whn α l = 0, i.. f(x) is continuous on Γ and at ach vrtx v, E v df dx (v) = 0. (20) Th discussion abov shows that th quadratic form of H is df dx 2 dx, (21) E dfind on H 1 (Γ), and th oprator is non-ngativ 3.2.3 Conditions of δ -typ Ths conditions rmind th δ-typ ons, but with th rols of functions and th drivativs ar rvrsd at ach vrtx (s also [2]). In ordr to dscrib 15

thm, lt us introduc th notation f v for th rstriction of a function f onto th dg. Thn th conditions at ach vrtx v can b dscribd as follows: Th valu of th drivativ df dx (v) is th sam for all dgs E v and df f (v) = α v (v). dx E v (22) Hr, as bfor, df dx (v) is th drivativ in th outgoing dirction at th vrtx v. It is clar that in comparison with δ-typ cas th matrics A v and B v ar switchd: and Sinc B v = 1 1 0... 0 0 1 1... 0...... 0 1 1 α v 0... 0 0 0 0... 0 A v =............ 0 0... 0. 1 1... 1 0 0... 0 0 A v Bv =..............., 0 0... 0 α v th slf-adjointnss condition (6) is satisfid again for ral α v only. Considr first th cas whn α v = 0 for som vrtx v. Thn th krnl of B v consists of all vctor with qual coordinats, and th projctor Q v projcts orthogonally onto th subspac of vctors that hav th sum of thir coordinats qual to zro. On this subspac oprator A v is qual to zro, and hnc L v = 0. This lads to no non-intgral contribution to th quadratic form coming from th vrtx v. In particular, if α v = 0 for all vrtics, w gt th quadratic form df dx 2 dx 16

with th domain consisting of all functions from H 1 () that hav at ach vrtx th sum of th vrtx valus along all ntring dgs qual to zro. In this cas th oprator is clarly non-ngativ. Lt us look at th cas whn for a vrtx v th valu α v is non-zro. In this cas th oprator B v is invrtibl and so P v = 0, Q v = Id. It is not hard to comput that (L v ) ij = (αd) ( 1) for all indics i, j. This lads to th non-intgral trm 1 f (v) 2. α v { E v } On can think that th cas whn α v = 0 is formally a particular cas of this on, if on assums that th dnominator bing qual to zro forcs th condition that th sum in th numrator also vanishs. Th quadratic form for a gnral choic of ral numbrs α can b writtn as follows: E df dx 2 dx + {v V α v 0} 1 α v { E v} f (v) 2. Th domain consists of all functions in H 1 () that hav at ach vrtx v whr α v = 0 th sum of th vrtx valus along all ntring dgs qual to zro. Whn all numbrs α v ar non-ngativ, th oprator is clarly non-ngativ as wll. 3.2.4 Vrtx Dirichlt and Numann conditions Th vrtx Dirichlt conditions ar thos whr at ach vrtx it is rquird that th boundary valus of th function on ach dg ar qual to zro. In this cas th oprator compltly dcoupls into th dirct sum of th ngativ scond drivativs with Dirichlt conditions on ach dg. Thr is no communication btwn th dgs. Th quadratic form is clarly E df dx 2 dx on functions f H ( Γ) with th additional condition f(v) = 0 for all vrtics v. Th spctrum σ(h) is thn found as σ(h) = {n 2 π 2 /l 2 E, n Z 0}. 17

Anothr typ of conditions undr which th dgs compltly dcoupl and th spctrum can b asily found from th st of dg lngths, is th vrtx Numann conditions. Undr ths conditions, no rstrictions on th vrtx valus f (v) ar imposd, whil all drivativs f (v) ar rquird to b qual to zro. Thn on obtains th Numann boundary valu problm on ach dg sparatly. Th formula for th quadratic form is th sam as for th vrtx Dirichlt conditions, albit on a largr domain with no vrtx conditions imposd whatsovr. 3.2.5 Classification of all symmtric vrtx conditions Th radr might hav noticd that in all xampls abov th conditions wr invariant with rspct to any prmutations of dgs at a vrtx. W will now classify all such conditions (8). Th list of symmtric vrtx conditions includs svral popular classs. Howvr, quantum graph modls arising as approximations for thin structurs somtims involv non-symmtric conditions as wll, which prsrv som mmory of th gomtry of junctions (s,.g. [58, 59, 60]). As it has alrady bn mntiond, on can find discussion of othr xampls of boundary conditions in [34, 43]. Lt us rpat for th radr s convninc th boundary conditions (8) at a vrtx v, dropping for simplicity of notations all subscripts indicating th vrtx: P F (v) = 0 QF (v) + LQF (v) = 0. (23) Hr, as bfor, P is an orthogonal projctor in C d, Q = I P, and L is a slf-adjoint oprator in QC d. W ar now intrstd in th cas whn ths conditions ar invariant with rspct to th symmtric group S d acting on C d by prmutations of coordinats. Notic that this action has only two non-trivial invariant subspacs: th on-dimnsional subspac U consisting of th vctors with qual componnts, and its orthogonal complmnt U, sinc th rprsntation of S d in U is irrducibl (.g., Sction VI.4.7 in [13] or VI.3 in [78]). Hr U consists of all vctors with th sum of componnts qual to zro. Lt us dnot by φ th unit vctor φ = (d 1/2,..., d 1/2 ) C d. This is a unit basis vctor of U. Thn th orthogonal projctor onto U is φ φ (a physicist would dnot it φ φ ) acting on a vctor a as a, φ φ. Thn th complmntary 18

projctor onto U is I φ φ. In ordr for (23) to b S d -invariant, oprators P and L must b so. Du to th just mntiond xistnc of only two non-trivial S d -invariant subspacs, thr ar only four possibl orthogonal projctors that commut with S d : P = 0, P = φ φ, P = I φ φ, and P = I. Lt us study ach of ths cass: Lt first P = 0. In this cas Q = I and L acting on C d must commut with th rprsntation of th symmtric group by prmutations of coordinats. As it was discussd abov, this implis that L = αφ φ + βi. This shows that thr ar no rstrictions imposd on th vrtx valus F and th rstrictions on F ar givn as F + α F, φ + βf = 0. In othr words, on can say that th xprssion f (v) + βf (v) is dgindpndnt and α f (v) = (f (v) + βf(v)). In th particular E v cas whn α 0, β = 0, w conclud that all th valus of th outgoing drivativs f (v) ar th sam, and (f (v)) = α ( 1) f (v). On E v rcognizs this as th δ -typ conditions. If α = β = 0, on nds up with th vrtx Numann condition. Lt now P = I. Thn Q = 0 and hnc L is irrlvant. W conclud that F = 0 and no mor conditions ar imposd. This is th vrtx Dirichlt condition, undr which th dgs dcoupl. Lt P = φ φ. Thn Q = I φ φ and L is qual to a scalar α, du to irrducibility of th rprsntation in QC d. Thn E v f (v) = 0 and F (v) F (v), φ φ + αf (v) = 0. Th last quality shows that th xprssion f (v) + αf (v) is dgindpndnt and qual to f (v). This, togthr with f (v) = 0 E v E v givs all th conditions in this cas. Thr appars to b no common nam for ths conditions. Th last cas is P = I φ φ. Thn Q = φ φ and L is a scalar α again. In this cas th condition P F = 0 mans that th valus f (v) ar dg indpndnt, or in othr words f is continuous through th vrtx v. Th othr condition asily lads to f (v) = αf(v), E v which on rcognizs as th δ-typ conditions. 19

This complts our classification of symmtric vrtx conditions. Ths conditions wr found prviously in [34] by a diffrnt tchniqu. As w hav alrady mntiond bfor, non-symmtric conditions aris somtims as wll (.g., [34, 43, 58, 59, 60]). On of th natural qustions to considr is which of th conditions (6) aris in th asymptotic limits of problms in thin nighborhoods of graphs. This issu was discussd in [54], howvr it has not bn rsolvd yt. On might think that whn quantum graph modls ar dscribing th limits of thin domains, diffrnt typs of vrtx conditions could probably b obtaind by changing th gomtry of th domain nar th junctions around vrtics. This guss is basd in particular on th rsults of [59, 60]. 3.3 Infinit quantum graphs W will now allow th numbr of vrtics and dgs of a mtric graph Γ to b (countably) infinit. Our goal is to dfin a slf-adjoint oprator H on Γ in a mannr similar to th on usd for finit graphs. In othr words, H should act as th (ngativ) scond drivativ along ach dg, and th functions from its domain should satisfy (now infinitly many) vrtx conditions (5) or quivalntly (8). This would turn a mtric graph Γ into a quantum graph. Howvr, unlss additional rstrictions on th graph and vrtx conditions ar imposd, th situation can bcom mor complx than in th finit graph cas. This is tru vn for such simpl graphs as trs, whr additional boundary conditions at infinity may or may not b ndd dpnding on gomtry (s [19, 79]). On th othr hand, if on looks at th naturally arising infinit graphs, on can notic that in many cass thr is an automorphism group acting on th graph such that th orbit spac (which is a graph by itslf) is compact. This is th cas for instanc with priodic graphs and Cayly graphs of groups. W do not nd xactly th homognity, but rathr that th gomtry dos not chang drastically throughout th graph. Th assumptions that w introduc blow capturs this ida and covrs all cass mntiond abov. It also nabls on to stablish nic proprtis of th corrsponding Hamiltonians. W would also lik to notic that this class of graphs is in som sns an analog of th so calld manifolds of boundd gomtry [76]. On such manifolds studying lliptic oprators is asir than on mor gnral ons. 20

Assumption 1. Th lngths of all dgs ar uniformly boundd from blow: 0 < l 0 l. (24) Rmark 12. 1. This assumption maks sns for infinit graphs only. 2. Thr is no uppr bound assumd on th lngths of dgs. In fact, som dgs may b of infinit lngth, as thy oftn ar,.g. in scattring problms. Lt us now quip a mtric graph Γ with th ngativ scond drivativ oprator and boundary conditions (8). W will say that th Hamiltonian H is dfind, although its prcis dfinition will b providd only a littl bit latr in this sction. This maks Γ a quantum graph. Assumption 2. Th following stimat holds uniformly for all vrtics v: L v S <. (25) Th norms in (25) ar th oprator norms with rspct to th standard l 2 norms on spacs C d. Rmark 13. If th vrtx conditions ar givn in th form (5), thn th condition abov should b rplacd by th following: B ( 1) v A v Q v S <.. (26) Hr, as bfor, Q v is th orthogonal projction onto th rang of Bv and B v ( 1) is th invrs to th oprator B v acting from th rang of Bv to th rang of B v. Lt us now dfin th oprator H mor prcisly. Dfinition 14. Th (unboundd) Hamiltonian H in L 2 (Γ) acts as th ngativ scond drivativ along th dgs, dfind on th domain D(H) consisting of functions f such that: 1. f H 2 () for ach dg, 2. f 2 H 2 () <, 21

3. for ach vrtx v, conditions (8) ar satisfid: L v F (v) + Q v F (v) = 0, P v F (v) = 0. It will b shown blow that this oprator is slf-adjoint and in fact is th only rasonabl slf-adjoint ralization of our Hamiltonian (i.., its rstriction to th appropriat subspac of compactly supportd functions is ssntially slf-adjoint). W, howvr, want to dscrib th corrsponding quadratic form first. Th dfinition will b similar to th on w gav in th cas of finit graphs: Dfinition 15. Th quadratic form h is dfind as h[f, f] = df dx 2 dx L v F, F, (27) E v V whr, dnots th standard hrmitian innr product in C d v. Th domain of this form consists of all functions f that blong to H 1 () for ach dg, satisfy at ach vrtx v th condition P v F = 0, and such that <. (28) f 2 H 1 () On can asily writ th ssquilinar form for h: h[f, g] = E Som rmarks ar du concrning this dfinition. df dg dx dx dx L v F, G. (29) v V Rmark 16. 1. Analogous formulas can b writtn in trms of th matrics A v, B v of conditions (5), on just nds to rplac L v by B v ( 1) A v and rmmbr that P v is th orthogonal projction onto th krnl of B v. 2. Du to th lowr bound on th lngths of th dgs, th norms of th trac oprators that associat to ach function f H 1 () for E v its valu at th vrtx v ar boundd uniformly with rspct to v and : f(v) C f 1 H(). (30) 22

3. In ordr for th dfinition to b corrct, on nds to mak sur that both infinit sums in th formula for h convrg. Th first on (th sum of th intgrals) convrgs du to Cauchy-Schwartz inquality and (28). Morovr, (28) and (30) imply that for any f from th domain of th form on has for its tracs F (v) (rcall that F (v) is a vctor in C dv ) th inquality F (v) 2 C v f 2 H 1 () <. (31) Sinc th sam applis to th function g, this, (25), and th Cauchy- Schwartz inquality scur convrgnc of th scond sum. W ar now prpard for th discussion of th Hamiltonian. Thorm 17. Lt Γ b a quantum graph satisfying Assumptions 1 and 2. Undr th dfinitions givn abov for th quadratic form h and oprator H, th following statmnts hold: 1. Th oprator H is slf-adjoint and its quadratic form is h. 2. Lt H 0 b th rstriction of H onto th sub-domain consisting of all functions from D(H) with compact support. Thn H 0 is symmtric, ssntially slf-adjoint, and its closur is H. Bfor w procd to th proof of th thorm, lt us mntion that its scond statmnt implis that undr th prscribd vrtx conditions thr is only on rasonabl way to dfin our slf-adjoint Hamiltonian H, and it is th on w choos. On can also notic that according to [19, 79] this is not tru anymor for graphs that do not hav boundd gomtry in trms of th Assumptions 1 and 2, vn for trs, whr som boundary conditions at infinity might b ndd. Proof. First of all, it is immdiat to chck that both oprators H and H 0 ar symmtric. Nxt, th form h is closd. Indd, th stimat (31) shows that th norm M f 2 L 2 (Γ) + h[f, f] with a sufficintly larg M on th domain of h is quivalnt to th norm of th spac H = H 1 (). This implis closdnss. Now, th form h corrsponds 23

to a slf-adjoint oprator M. W will show that M coincids with H, which would prov th first statmnt of th thorm. According to th dfinition of oprator M, for any f D(M) H thr xists p L 2 (Γ) (which is thn dnotd by Mf) such that for any g D(h) on has h[f, g] = p(x)g(x)dx. Lt now g b any compactly supportd function smooth on ach dg and qual to zro in a nighborhood of ach vrtx. Thn clarly g D(h). Choosing only such functions in th prvious quality, substituting th dfinition of h for th lft hand sid, and intgrating by parts, on concluds that p(x) = Mf(x) = d2 f dx 2 on ach dg, whr th drivativs ar mant in th distributional sns. This mans that d2 f L 2 (Γ), and du to f D(h) H, w conclud that dx 2 f H 2 () and f H 2 () <. Sinc f D(h), this function satisfis th conditions P v F (v) = 0 at ach vrtx v. W nd to show that it satisfis also th rmaining vrtx conditions (thos containing drivativs). On dos this using a tst function g D(h) that is non-zro in small nighborhood of a singl vrtx v. Thn intgration by parts shows that F (v) + L v F (v), G(v) = 0. Sinc this quality must hold for any vctor G(v) such that P v G(v) = 0, this implis th complt boundary condition (8). This shows that M H. It is a straightforward calculation of xactly sam natur that shows that in fact any f D(H) blongs to D(M). Hnc, M = H and th first statmnt of th thorm is provn. Lt now f D(H). Our goal is to crat a squnc of cut-off functions φ n (x) such that f n = φ n f D(H 0 ) and f f n L2 (Γ) 0, Hf H 0 f n L2 (Γ) 0. (32) If this wr accomplishd, thn w would know that H wr th closur of H 0 and hnc H 0 wr ssntially slf-adjoint. Th ida of how functions φ n should bhav is clar: thy must b qual to 1 on an xpanding and xhausting squnc Γ n Γ of compacta, must 24

hav compact supports, must b constant in a nighborhood of ach vrtx (in ordr not to dstroy th vrtx conditions), and must fall off to zro not too fast, so that thir first and scond ordr drivativs ar uniformly boundd. Now, if a squnc of such functions is constructd, thn (32) is straightforward. Th graph natur of our varity causs som suprficial complications, so w will dscrib this construction (which could crtainly b don in many diffrnt ways). Lt us first dscrib a convnint xpanding squnc of compacta in Γ that xhaust th whol graph. Lt us fix a vrtx o Γ and considr for any natural n th st Γ n Γ that contains all (finit) dgs whos both ndpoints ar at a distanc at most n from o and all points x of infinit dgs such that ρ(x, o) n (hr ρ is th prviously dfind mtric on Γ). This is clarly an xpanding squnc of compact sts that xhausts Γ. Lt φ(x) b any smooth function on [0, l 0 /4] such that it is idntically qual to 1 in a nighborhood of 0 and idntically qual to zro clos to l 0 /4. Hr l 0 is th lowr bound for th lngths of all dgs of Γ, which is positiv du to our assumptions. W ar rady to dfin th cut-off function φ n on Γ. It is qual to 1 on Γ n and to 0 on all dgs which do not hav vrtics in Γ n. W only nd to dfin it along th dgs that hav only on vrtx in Γ n. Lt b a finit dg whos on vrtx v is containd in Γ n. Th function φ n is dfind to b qual to 1 along starting from v till th middl of th dg, thn it is continud by an appropriatly shiftd copy of φ(x) (which by construction will bcom zro at last at th distanc l /4 from th nd of th dg), and stays zro aftr that. If is an infinit dg starting at v Γ n, thn φ n is dfind to b qual to 1 along starting from v till th th distanc n from o, thn it is continud by an appropriatly shiftd copy of φ(x), and stays zro aftr that. It is clar that all our rquirmnts for th squnc of functions ar satisfid. Lt now f D(H) and f n = φ n f. Thn f n is in H 2 () for any dg and satisfis th boundary conditions. Th rason for th lattr is that φ n is constant around ach vrtx, and so multiplication by it dos not dstroy th vrtx conditions. In othr words, f n D(H 0 ). On gts th following simpl conclusion: lim f f n L2 (Γ) = lim (1 φ n )f L2 (Γ) n n C lim f L2 (Γ Γ n ) = 0. n Hr C is th maximal valu of φ(x) 1. W also hav Hf H 0 f n = (1 φ n )f φ nf φ nf. 25

Sinc f, f, and f all blong to L 2 (Γ) and th functions 1 φ n, φ n, and φ n ar uniformly boundd and supportd outsid Γ n, w also obtain th scond rquird limit lim n Hf H 0f n L2 (Γ) = 0. This finishs th proof of th thorm. 4 Rlations btwn spctra of quantum and discrt graph oprators In many (if not most) cass whn a quantum graph is involvd, on is intrstd in th spctrum of th corrsponding Hamiltonian H. This is tru for quantum chaos studis, scattring thory, photonics, tc. (s th rfrncs in [54]). W hav mphasizd throughout th txt th diffrnc btwn combinatorial graphs and corrsponding diffrnc oprators on on hand and mtric graphs quippd with diffrntial oprators on th othr. Howvr, w will show now that spctral problms for quantum graphs can somtims b transformd into th ons for diffrnc oprators on combinatorial graphs. This obsrvation gos back probably to th papr [4]. W will addrss hr th cass of finit graphs with dgs of finit lngths only. Du to th articl siz limitations, mor complx situation of infinit graphs will b tratd in [56]. Th situation of intrst for scattring thory whn svral infinit lads ar attachd to a finit graph will also b considrd lswhr. Lt us start with th following simpl rsult. Thorm 18. Lt Γ b a finit quantum graph with finit lngth dgs quippd with a Hamiltonian givn by th ngativ scond drivativ along th dgs and vrtx conditions (8). Thn its rsolvnt is of trac class, and in particular th spctrum is discrt. Proof. Th domain of H is a closd subspac of th dirct sum of th Sobolv spacs H 2 () on all dgs. Hnc, for non-ral λ th rsolvnt R(λ) = (H λ) 1 maps L 2 (Γ) continuously into this dirct sum. Now th statmnt follows form th standard mbdding thorm for th Sobolv spacs on finit intrvals. 26

According to Thorm 18, th spctrum σ(h) is discrt. W ar intrstd thrfor in solving th quation Hf = λf (33) with f L 2 (Γ). Lt v b a vrtx and b on of th outgoing dgs of lngth l and with th coordinat x countd from v. W will also dnot by w th othr nd of. Thn along this dg on can solv (33) as follows: f (x) = 1 ( sin f (v) sin λ(l x) + f (w ) sin ) λx. (34) λl This can b don as long as λ n 2 π 2 l 2 with an intgr n 0 (th formula can also b naturally intrprtd for λ = 0), i.. whn λ dos not blong to th spctrum of th ngativ scond drivativ with Dirichlt conditions on (idntifid with [0, l ]). Th last formula allows us to find th drivativ at v: f (v) = λ sin l λ ( f (w ) f (v) cos l λ ). (35) Substituting ths rlations into (23) to liminat th drivativs, on rducs (23) to a systm of discrt quations that involv only th vrtx valus: T (λ)f = 0. (36) Hr F is th vctor of dimnsion D = d v that combins all th vctor v valus F (v) of function f and T (λ) is a D D matrix. Th radr can notic that (36) is a systm of scond ordr diffrnc quations on th combinatorial vrsion of th graph Γ, whr at ach vrtx v w hav a d v -dimnsional valu F (v) of th vctor function F assignd. On asily concluds that th following statmnt holds: Thorm 19. A point λ n 2 π 2 l ( 2), n Z {0} blongs to th spctrum of H if and only if zro blongs to th spctrum of th matrix T (λ). This thorm shows that spctral problms for quantum graph Hamiltonians can b rwrittn as spctral problms for som diffrnc oprators. On can notic that whn computd, th systm (36) oftn looks rathr complx. Howvr, it simplifis significantly for som frquntly arising situations. 27

Considr for instanc a quantum graph with all dgs of sam lngth l and with δ-typ vrtx conditions. In this cas th function is continuous, and hnc th valus F (v) = f(v) do not dpnd on th dg E v. Thn (36) aftr som simpl arithmtic bcoms {w =(v,w) E v } f(w) = ( α sin l λ l λ + d v cos λ ) f(v). (37) In th particular cas whn all vrtics hav sam dgr d (i.., th graph is rgular), w conclud that λ ( n 2 π 2 /l 2 blongs to th spctrum σ(h) of th quantum graph if and only if α sin l λ l + d cos ) λ blongs to th spctrum λ σ( ) of th discrt Laplac oprator on Γ that is dfind by th lft hand sid of (37). This provids a vry usful rlation of th spctra that nabls on to pass information btwn th continuous and discrt modls. Th full advantag of using this corrspondnc will b shown in particular in [56]. On can ask what happns to th xcludd Dirichlt ignvalus n 2 π 2 /l. 2 Th following xampls show that it is hard to answr this qustion in gnral trms. Considr th vrtx Dirichlt conditions cas. Hr th whol spctrum obviously consists of th abov Dirichlt ignvalus only. Considr a ring consisting of two dgs of lngths l 1, l 2 connctd at two vrtics of dgr 2 into a loop of lngth L = l 1 + l 2 and quippd with th Kirchhoff conditions (4). As it has bn mntiond bfor, this mans that w can liminat th two vrtics of th graph and considr it as a circl of lngth L quippd with th ngativ scond drivativ Hamiltonian H. In this cas, th spctrum σ(h) is th st of numbrs {(2nπ/L) 2 }. If th numbrs l j ar rationally indpndnt, thn non of th dg Dirichlt ignvalus ar in th spctrum. Choosing in th prvious xampl th lngths l j commnsurat, on can mak sur that only a non-mpty part of th st of dg Dirichlt ignvalus blongs to σ(h). On should bwar ths dg Dirichlt ignvalus. For instanc, in th cas of an intgr lattic graph Γ = Z n, on can apply standard Floqut thory usd for priodic PDEs (.g., [51, 52, 53, 73]) to find th spctrum. 28

At th first glanc, this lads to th usual pictur of a band-gap absolutly continuous spctrum of a priodic problm. Howvr, thr is a dangr (that has matrializd in svral publications) to ovrlook th point spctrum that dos aris at th dg Dirichlt ignvalus. 5 Rmarks 1. W hav considrd for th sak of simplicity th scond drivativ Hamiltonians only. On can analogously dal with mor gnral Schrödingr oprators on graphs that involv lctric and magntic potntials [46]. Somtims matrix or highr ordr diffrntial and vn psudo-diffrntial oprators on graphs nd to b considrd [15, 36, 57, 58, 70]. W hop to addrss som of ths issus lswhr. 2. Th rsults of Sction 4 concrning rlations btwn spctra of quantum and combinatorial graph oprators in th cas of infinit graphs rquir mor analysis, sinc th spctra might not b discrt anymor. In particular, on should b abl to idntify points of th spctrum with thos whr som growing solutions (gnralizd ignfunctions) xist. This rquirs analogs of th so calld Schnol s thorms and stimats on gnralizd ignfunctions (s th PDE vrsions of ths corrspondingly in [24, 40, 75, 76] and [10, 77, 38]). Such analogs will b providd in [56]. 3. It was mntiond that in situation th discrt quations (36) that on gts whn switching from a quantum to a combinatorial graph look rathr complx. It would b nic to fit thos into som algbraic framwork that would allow a thorough analysis lik th on availabl for discrt Laplac oprators. On can intrprt (36) in trms of graph rprsntations [11, 37], although w hav not found this usful so far. A vrtx scattring matrix approach [48] might also prov to b usful hr. 4. Th rsults of this papr do not provid any dtails of th structur of th spctrum that would rflct spcific graph gomtry. This is a problm dfinitly worth studying and th on that has bn addrssd from various points of viw in svral publications (.g. [3, 5, 6, 7, 8], [16]-[20], [29]-[32], [39], [43]-[49], [54]-[61], [64, 65, 74, 79, 80], othr 29

paprs of th currnt issu of Wavs in Random Mdia, and rfrncs thrin). W plan to addrss som of th rlatd problms (.g., spctral gaps, bound stats, tc.) in [56] and othr paprs. 5. An intrsting and usful gnralization that dsrvs considration concrns oprators on multi-structurs that involv clls of diffrnt dimnsions (s,.g. [14, 50, 66, 72]). Among thos on can mntion for instanc, 2D or 3D quantum wlls joind by 1D quantum wirs, or thr-dimnsional photonic band gap mdia that somtims look as 2D surfac structurs in R 3. 6 Acknowldgmnt Th author thanks M. Aiznman, R. Carlson, P. Exnr, S. Fulling, R. Grigorchuk, V. Kostrykin, L. Kunyansky, P. Kurasov, V. Mazýa, S. Novikov, O. Pnkin, J. Rubinstin, Y. Saito, M. Schatzman, J. Schnkr, R. Schradr, H. Zng, and N. Zobin for information and discussion. Th author is also gratful to th rfrs for usful commnts. This rsarch was partly sponsord by th NSF through th DMS Grants 9971674, 0072248, and 0296150. Th author xprsss his gratitud to NSF for this support. Th contnt of this papr dos not ncssarily rflct th position or th policy of th fdral govrnmnt, and no official ndorsmnt should b infrrd. Rfrncs [1] N. Akhizr and I. Glazman, Thory of Linar Oprators in Hilbrt Spac, Dovr, NY 1993. [2] S. Albvrio, F. Gsztsy, R. Hogh-Krohn, and H. Holdn, Solvabl Modls in Quantum Mchanics, Springr-Vrlag, Nw York, 1988. [3] S. Albvrio and P. Kurasov, Singular Prturbations of Diffrntial Oprators: Solvabl Schrödingr Typ Oprators (London Mathmatical Socity Lctur Not, 271), Cambridg Univ. Prss, 2000. [4] S. Alxandr, Suprconductivity of ntworks. A prcolation approach to th ffcts of disordr, Phys. Rv. B, 27 (1983), 1541-1557. 30

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