APPROXIMATION OF QUANTUM GRAPH VERTEX COUPLINGS BY SCALED SCHRÖDINGER OPERATORS ON THIN BRANCHED MANIFOLDS


 Agnes Hardy
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1 APPROXIMATION OF QUANTUM GRAPH VERTEX COUPLINGS BY SCALED SCHRÖDINGER OPERATORS ON THIN BRANCHED MANIFOLDS PAVEL EXNER AND OLAF POST Abstract. W discuss approximations of vrtx couplings of quantum graphs using familis of thin branchd manifolds. W show that if a Numann typ Laplacian on such manifolds is amndd by suitabl potntials, th rsulting Schrödingr oprators can approximat nontrivial vrtx couplings. Th lattr includ not only th δcouplings but also thos with wavfunctions discontinuous at th vrtx. W work out th xampl of th symmtric δ couplings and conjctur that th sam mthod can b applid to all couplings invariant with rspct to th tim rvrsal. 1. Introduction Th quantum graph modls rprsnt a simpl and vrsatil tool to study numrous physical phnomna. Th currnt stat of art in this fild is dscribd in th rcnt procdings volum [EKK + 08] to which w rfr for an xtnsiv bibliography. On of th big qustions in this ara is th physical maning of quantum graph vrtx coupling. Th gnral rquirmnt of slfadjointnss admits boundary conditions containing a numbr of paramtrs, and on would lik to undrstand how to choos ths whn a quantum graph modl is applid to a spcific physical situation. On natural ida is to approximat th graph in qustion by a family of fat graphs, i.. tublik manifolds built around th graph sklton, quippd with a suitabl scondordr diffrntial oprator. Such systms hav no ad hoc paramtrs and on can try to find what boundary condition aris whn th manifold is squzd to th graph. Th qustion is by no mans asy and th answr dpnds on th typ of th oprator chosn. If it is th Laplacian with Dirichlt boundary conditions on has to mploy an nrgy rnormalisation bcaus th spctral thrshold givn by th lowst transvrs ignvalu blows up to infinity as th tub diamtr tnds to zro. If on chooss th rfrnc point btwn th thrsholds, th limiting boundary conditions ar dtrmind by th scattring on th rspctiv fat star manifold [MV07]. If, on th othr hand, th thrshold nrgy is subtractd, th limit givs gnrically a dcoupld graph, i.. th family of dgs with Dirichlt conditions at thir ndpoints [P05, MV07, DT06]. On can nvrthlss gt a nontrivial coupling in th limit if th tub ntwork xhibits a thrshold rsonanc [G08, ACF07], and morovr, using a mor involvd limiting procss on can gt also boundary conditions with richr spctral proprtis [CE07]. Th cas whn th fat graph supports a Laplacian of Numann typ is bttr undrstood and th limit of all typs of spctra as wll as of rsonancs has bn workd out [FW93, RS01, KuZ01, EP05, EP07, G08, EP08]. Morovr, convrgnc of rsolvnts tc. has bn shown in [Sai00, P06, EP07]. Of cours, no nrgy rnormalisation is ndd in this cas. On th othr hand, th limit yilds only th simplst boundary conditions calld fr or Kirchhoff. Th aim of this papr is to show that on can do bttr in th Numann cas if th Laplacian is rplacd by suitabl familis of Schrödingr oprators with proprly scald potntials. Such approximations hav bn shown to work on graphs thmslvs [E97, Dat: Novmbr 22, 2008 Fil: potapprox.tx. 1
2 2 PAVEL EXNER AND OLAF POST ENZ01], hr w ar going to lift thm to th tublik manifolds 1. First w will show that using potntials supportd by th vrtx rgions of th manifold with th natural scaling, as ε 1 whr ε is th tub radius paramtr, w can gt th socalld δcoupling, th onparamtr family with th wavfunctions continuous vrywhr, including at th vrtx. This shows, in particular, that on cannot achiv such an approximation in a purly gomtric way, with a curvaturinducd potntial of th typ [DEK01], bcaus th lattr scals typically as ε 2. As main rsult in this cas, w show th convrgnc of th spctra and th rsolvnts cf. Thorms On th othr hand, th δcoupling is only a small part in th st of all admissibl couplings; in a vrtx joining n dgs th boundary conditions contain n 2 paramtrs. Hr w us th sminal ida of Chon and Shighara [CS98] applid to th graph cas in [CE04] and gnralisd in [ET06, ET07]. For simplicity w will work out in this papr th xampl of th socalld symmtric δ coupling, in short δ s, a onparamtr family which is a countrpart of δ, by using th rsult of [CE04] and lifting it to th manifold. W show that such a coupling is approximatd with a potntial in th vrtx rgion togthr with potntials at th dgs with compact supports approaching th vrtx, all proprly scald, cf. Thorm 4.7. Th spd with which th potntials ar coming togthr must b slowr than th squzing; th rat btwn th two w obtain is surly not optimal. W hav no doubts that in th sam way on can lift to th manifolds th mor gnral limiting procdur dvisd in [ET07] which givs ris to a n+1 2 paramtr family of boundary conditions, namly thos which ar invariant with rspct to th tim rvrsal. Such an xtnsion would b tchnically dmanding, howvr, and in ordr not to burdn this papr with a complicatd notation and voluminous stimations w postpon it to a latr work. Lt us survy th contnts of th papr. In th nxt sction w dfin th graph and manifold modls and provid ncssary stimats. In Sction 3 w prov th convrgnc in th δcoupling cas. For simplicity w rstrict ourslvs to a starshapd graph with a singl vrtx; th approximation bars a local charactr and xtnds asily to mor complx graphs. Finally in Sction 4 w xtnd th rsult to th δ s coupling cas and commnt on th applicability of th mthod to mor gnral couplings. 2. Th graph and manifold modls 2.1. Graph modl. Lt us start with a simpl xampl of a starshapd mtric graph G = I v having only on vrtx v and dg v adjacnt dgs E = E v of lngth l 0, ], so w can think of E = {1,..., dg v}. W idntify th mtric dg with th intrval I := 0, l orintd in such a way that 0 corrsponds to th vrtx v. Morovr, th mtric graph G = I v is givn by th abstract spac I v := I / whr dnots th disjoint union, and whr th quivalnc rlation idntifis th points 0 I with th vrtx v. Th basic Hilbrt spac is L 2 G := E L 2I with norm givn by f 2 = f 2 G = E l Th dcoupld Sobolv spac of ordr k is dfind as H k maxg := E H k I 0 fs 2 ds. 1 Anothr approach to approximation of nontrivial vrtx conditions was proposd rcntly in [Pa07b, Pa07a]
3 APPROXIMATION OF VERTEX COUPLINGS BY SCHRÖDINGER OPERATORS ON MANIFOLDS 3 togthr with its natural norm. Lt p = {p } b a vctor consisting of th wights p > 0 for E. Th Sobolv spac associatd to th wight p is givn by H 1 p G := { f H 1 max G f Cp }, 2.1 whr f := {f 0} C dg v is th valuation vctor of f at th vrtx v and Cp is th complx span of p. W us th notation f = fvp, i.., f = fvp 2.2 for all E. In particular, if p = 1,..., 1, w arriv at th continuous Sobolv spac H 1 G := H 1 pg. Th standard Sobolv trac stimat g0 2 a g 2 0,l + 2 a g 2 0,l 2.3 for g H 1 0, l and 0 < a l nsurs that H 1 p G is a closd subspac of H1 max G, and thrfor itslf a Hilbrt spac. A simpl consqunc is th following claim. Lmma 2.1. W hav fv 2 p 2 a f 2 G + 2 a f 2 G for f H 1 pg and 0 < a l := min E l, th minimal lngth at th vrtx v. W dfin various Laplacians on th mtric graph via thir quadratic forms. Lt us start with th wightd fr Laplacian G dfind via th quadratic form d = d G givn by df := f 2 G = f 2 I and dom d := H 1 pg for a fixd p th forms and th corrsponding oprators should b lablld by th wight p, of cours, but w drop th indx, in particular, bcaus w ar most intrstd in th cas p = 1,...,1. Not that d is a closd form sinc th norm associatd to th quadratic form d is prcisly th Sobolv norm givn by f 2 H 1 G = f 2 G + f 2 G. Th Laplacian with δcoupling of strngth q is dfind via th quadratic form h = h G,q givn by hf := f 2 G + qv fv 2 and dom h := H 1 pg. 2.4 Th δcoupling is a small prturbation of th fr Laplacian, namly w hav: Lmma 2.2. Th form h G,q is rlativly formboundd with rspct to th fr form d G with rlativ bound zro, i.., for any η > 0 thr xists C η > 0 such that hf df = qv fv 2 η df + C η f 2 G. Proof. It is again a simpl consqunc of Lmma 2.1. Sinc w nd th prcis bhaviour of th constant C η, w giv a short proof hr. From Lmma 2.1 w conclud that hf df qv p adf a f 2 G. for any 0 < a l. St a := min{η p 2 / qv, l } and { qv 2 C η := 2 max η p, qv }, 4 l p 2 thn th dsird stimat follows. On can s that th norms associatd to h and d ar quivalnt and, in particular, stting η = 1/2 in th abov stimat yilds w gt:
4 4 PAVEL EXNER AND OLAF POST Corollary 2.3. Th quadratic form h is closd and obys th stimat df 2hf + C 1/2 f 2 G. Th oprator H = H G,q associatd to h acts as Hf = f on ach dg and satisfis th conditions f 1 0 = f 2 0 =: fv and p f p 1 p 0 = qvfv for any pair 1, 2 of dgs mting at th vrtx v. W us th formal notation H = H G,q = G + qvδ v ; 2.6 not that G is a nonngativ oprator by dfinition. In ordr to compar th fr quadratic form with th graph norm of H w nd th following stimat: Lmma 2.4. W hav f 2 H 1 G = df + f 2 G 2 max{c 1/2, 2} H if 2 G for f dom H dom h = H 1 p G. Proof. Using th stimat of Corollary 2.3, w obtain df + f 2 2 hf + C 1/2 + 1 f 2 2 hf + f 2 + 2C1/2 f 2. Morovr, th first trm can b stimatd as hf + f hf 2 + f 4 = 2 hf i f 2 2 = 2 f, H if 2 Finally, w apply th stimat f H if to obtain th rsult. 2 f 2 H if 2. Rmark 2.5. Not that w hav not said anything about th boundary conditions at th fr nds of th dgs of finit lngth if thr ar any. As w mploy th Sobolv spac H 1 pg for th domain, w implicitly introduc Numann conditions for th oprator, f l = 0. Howvr, on can choos any othr condition at th fr nds, or to construct mor complicatd graphs by putting th star graphs togthr Manifold modl of th fat graph. Lt us now dfin th othr lmnt of th approximation w ar going to construct. For a givn ε 0, ε 0 ] w associat a ddimnsional manifold X ε to th graph G in th following way. To th dg E and th vrtx v w ascrib th Rimannian manifolds X ε, := I εy and X ε,v := εx v, 2.7 rspctivly, whr εy is a manifold Y quippd with mtric h ε, := ε 2 h and εx ε,v carris th mtric g ε,v = ε 2 g v ; hr h and g v ar εindpndnt mtrics on Y and X v, rspctivly. W idntify th boundary componnt X ε,v = ε X v of X ε,v = ε X v with v X ε, = ε v X = {0} εy and call th rsulting manifold X ε. W rfr to th unscald manifold which convntionally mans ε = 1 as to X. In particular, th manifold X consists of th numbr dg v of straight cylindrs 2 with crosssction Y and a vrtx nighbourhood manifold X v containing th boundary componnts Y := Y. Without loss of gnrality, w may assum that ach crosssction Y is connctd, othrwis w rplac th dg by as many dgs as is th numbr of connctd componnts. W dnot th boundary componnt of X v at th dg by X v and th boundary componnt of X at th vrtx v by v X = {0} Y. Not that ths two boundaris ar idntifid in th ntir manifold X. Similarly, w dnot by X ε,v = ε X v and v X ε, = 2 Th straightnss hr rfrs to th intrinsic gomtry only. W do not assum in gnral that th manifolds X ε ar mbddd, for instanc, into a Euclidan spac, s also Rmark 2.6.
5 APPROXIMATION OF VERTEX COUPLINGS BY SCHRÖDINGER OPERATORS ON MANIFOLDS 5 ε v X thir scald vrsions. For convninc, w will always us th εindpndnt coordinats s, y X = I Y and x X v, so that th radiustyp paramtr ε only ntrs via th Rimannian mtrics. Th ntir manifold X ε may or may not hav a boundary X ε, dpnding on whthr thr is at last on finit dg lngth l < or on transvrs manifold Y has a nonmpty boundary. In such a situation, w assum that X ε is opn in X ε = X ε X ε. A particular cas is rprsntd by mbddd manifolds which dsrv a commnt: Rmark 2.6. Not that th abov stting contains th cas of th εnighbourhood of an mbddd graph G R 2, but only up to a longitudinal rror of ordr of ε. Th manifold X ε itslf dos not form an εnighbourhood of a mtric graph mbddd in som ambint spac, sinc th vrtx nighbourhoods cannot b fixd in th ambint spac unlss on allows slightly shortnd dg nighbourhoods. Nvrthlss, introducing εindpndnt coordinats also in th longitudinal dirction simplifis th comparison of th Laplacian on th mtric graph and th manifold, and th rror mad is of ordr of Oε, as w will s in Lmma 2.7 for a singl dg. Th basic Hilbrt spac of th manifold modl is L 2 X ε = L2 I L 2 εy L 2 εx v 2.8 with th norm givn by u 2 X ε = ε d 1 u 2 dy ds + ε d u 2 dx v E X X v whr dx = dy ds and dx v dnot th Rimannian volum masurs associatd to th unscald manifolds X = I Y and X v, rspctivly. In th last formula w hav mployd th appropriat scaling bhaviour, dx ε, = ε d 1 dy ds and dx ε,v = ε d dx v. Dnot by H 1 X ε th Sobolv spac of ordr on, th compltion of th spac of smooth functions with compact support undr th norm givn by u 2 H 1 X = ε du 2 X ε + u 2 X ε. As in th cas of th mtric graphs, w dfin th Laplacian Xε on X ε via its quadratic form d ε u := du 2 X ε = ε u d 1 s, y E X ε d 2 Y u 2h dy ds + ε d 2 du 2 g v dx v X v 2.9 whr u dnots th longitudinal drivativ, u = s u, and du is th xtrior drivativ of u. As bfor, th form d ε is closd by dfinition. Adding a potntial, w dfin th Hamiltonian H ε as th oprator associatd with th form h ε = h Xε,Q ε givn by h ε = du 2 X ε + u, Q ε u Xε whr th potntial Q ε has support only in th unscald vrtx nighbourhood X v and Q ε x = 1 ε Qx 2.10 whr Q = Q 1 is a fixd boundd and masurabl function on X v. Th rason for this particular scaling will bcom clar in th proof of Lmma 3.2. Roughly spaking, it coms from th fact that vol X ε,v is of ordr ε d, whras th d 1dimnsional transvrs volum vol Y ε, is of ordr ε d 1. Th oprators H ε and ε ar associatd to forms h ε and d ε, rspctivly; not that ε = Xε 0 is th usual Numann Laplacian on X ε. As usual th Numann boundary condition occurs only in th oprator domain if X ε. W postpon for a momnt th chck that H ε is rlativly formboundd with rspct to Xε, s Lmma 2.10 blow.
6 6 PAVEL EXNER AND OLAF POST Lt us compar th two cylindrical nighbourhoods, X ε, = I εy and X ε, = I ε εy, on dgs of lngth l > 0 and l ε = 1 εl, rspctivly. Th rsult for th ntir spac X ε thn follows by combining th stimats on th dgs and th fact that th potntial is only supportd on th vrtx nighbourhoods. Th vrification of th conditions of δclosnss in th nxt lmma is straightforward; for mor dtails on δclosnss w rfr to [P06, App. A] or [P08]. Lmma 2.7. Lt H := L 2 X ε, and H := L 2 X ε,. Morovr, dfin J : H H J : H H J f s, y := f1 ε 1 s, y, J us, y := f1 εs, y. Thn th quadratic forms d ε f := f 2 X ε, and d ε u := u 2 Xε, with domd ε = H 1 X ε, and dom d ε = H 1 X ε, ar δ ε clos with δ ε = 2ε/1 ε 1/2 ; namly, w hav J J = id, J J = id, J 1, J 1 + δ ε, J J δ ε and d ε J f, u d ε f, J u δε. In particular, w gt Xε, J Xε, J 2δ ε = Oε. Bfor w chck th closnss assumptions of [P06, Appndix] in th nxt sction, w nd som mor notation and stimats. Th stimats ar alrady providd in [EP05, P06], but w will also nd a prcis control of th dg lngth, whn w approximat th δ s coupling by δcouplings in Sction 4 blow. Thrfor, w prsnt short proofs of th stimat hr. W first introduc th following avraging oprators u := v u dx v and us := us, dy X v Y for u L 2 X ε, whr w us th following symbols 1 1 u dx v := u dx v and ϕ dy := ϕ dy X v vol d X v X v Y vol d 1 Y Y dnoting th normalisd intgrals. For brvity, w also omit th masur and writ X v u tc. In ordr to obtain th blow Sobolv trac stimat 2.11, w nd a furthr dcomposition of th vrtx nighbourhood X v. Rcall that X v has dg vmany boundary componnts isomtric to Y. W assum that ach such boundary componnt has a collar nighbourhood X v, = 0, l Y of lngth l. Not that th scald vrtx nighbourhood X ε,v = εx v is of ordr ε in all dirctions, so that th scald collar nighbourhoods X ε,v, := εx v, ar of lngth εl. W can always assum that such a dcomposition xists, by possibly using a diffrnt cut of th manifold X into X v and X, th pric bing an additional longitudinal rror of ordr ε s Lmma 2.7. Similarly as in 2.3, on can gt th following Sobolv trac stimats for th scald manifolds: u 2 X ε,v εã du 2 X ε,v, + 2 εã u 2 X ε,v, 2.11 u 2 vx ε, a u 2 X ε, + 2 a u 2 X ε, 2.12 for 0 < a, ã l on th vrtx and dg nighbourhood, rspctivly, whr u = s u dnots th longitudinal drivativ. Th unscald vrsions ar obtaind, of cours, by
7 APPROXIMATION OF VERTEX COUPLINGS BY SCHRÖDINGER OPERATORS ON MANIFOLDS 7 stting ε = 1. Morovr, by th CauchySchwarz inquality w gt vol Y u0 2 u 2 X v = u 2 vx. In th following lmma w compar th avraging ovr th boundary of X v with th avraging ovr th whol spac X v : Lmma 2.8. For u H 1 X v, w hav vol X v X v u u 2 vol v X v E X v u u 2 ã + 2 du 2 v X ãλ 2 v v for 0 < ã l = min l, whr λ 2 v dnots th scond i.., first nonzro ignvalu of th Numann Laplacian on X v ; th lattr is dfind convntionally as th oprator associatd to th form d v u := du 2 X v with th domain dom d v := H 1 X v. Proof. Using CauchySchwarz twic and th stimat 2.11 for ach dg and ε = 1, w obtain vol X v X v w 2 vol X v X v w 2 u 2 X v ã dw 2 X v + 2 ã w 2 X v 2.13 for 0 < ã l, using th fact that X v, X v. W apply th abov stimat to th function w = P v u := u u and obsrv that v w 2 X v 1 λ 2 v dw 2 X v 2.14 as on can chck using th fact that that dw = du and that P v is th projction onto th orthogonal complmnt of th first ignfunction½v L 2 X v. W also nd an stimat ovr th vrtx nighbourhood. It will assur that in th limit ε 0, no family of normalisd ignfunctions u ε ε with ignvalus lying in a boundd intrval can concntrat on X ε,v. Lmma 2.9. W hav [ u 2 X ε,v 4ε 2 1 λ 2 v + c vol ã + 2 ] du 2Xε,v + 4εc vol [a u 2Xε,E + 2 ] ãλ 2 v a u 2Xε,E for 0 < a, ã l = min l, whr c vol := c vol v = vol X v / vol X v and X ε,e := dnots th union of all dg nighbourhoods. Proof. W start with th stimat u u 2 X ε,v 2ε d u 2 v X v + u 2 v X v using Lmma 2.8 and th fact that stimatd by vol X v u 2 2 vol X v v u v 2 ã + 2 ãλ 2 v 2ε d 2 λ 2 v du 2 X v + vol X v v u 2 X ε, u is constant. Morovr, th last trm can b v X v u 2 + du 2 X v + X v u 2 vol X v X v u 2 using Sinc X v is isomtric to v X, w can stimat th lattr sum by vol v X X v u 2 u 2 vx a u 2 X E + 2 a u 2 X E du to 2.12 for ε = 1, ach dg and 0 < a l. Hr, X E := X 1,E is th union of th unscald dg nighbourhoods. Th dsird stimat thn follows from th scaling
8 8 PAVEL EXNER AND OLAF POST bhaviour du 2 X ε,v = ε d 2 du 2 X v and w 2 X ε, = ε d 1 w 2 X for w = u or w = u whr u = s u dnots th longitudinal drivativ. W ar now abl to prov th rlativ formbounddnss of th Hamiltonian H ε with rspct to th Laplacian Xε for th indicatd class of potntials. It is again important hr to hav a prcis control of th constants ε η and C η in trms of th various paramtrs of our spacs. This will b of particular importanc whn w dal with th approximation of th δ scoupling by δcouplings with shrinking spacing a = ε α in Sction 4 blow. Lmma To a givn η 0, 1 thr xists ε η > 0 such that th form h ε is rlativly formboundd with rspct to th fr form d ε with rlativ bound η for all ε 0, ε η ], in othr words, thr xists C η > 0 such that h ε u d ε u η d ε u + C η u 2 X ε whnvr 0 < ε ε η, whr th constants ε η and C η ar givn by η [ 1 ε η = ε η Q, l := 4 Q λ 2 v + c 2 ] 1, vol l a l λ 2 v C η = C 4cvol Q η Q, l := 8c vol Q max{, 1 }. 2.15b η l Not that ε η = Ol and C η = Ol 1 as l 0. Proof. Th potntial Q ε = ε 1 Q is by assumption supportd on th vrtx nighbourhood X v, thrfor w hav h ε f d ε f Q u 2 X ε ε,v [ 4 Q {ε 1 λ 2 v + c vol l + 2 } ] du 2Xε,v + ac vol u 2Xε,E l λ 2 v + 8 Q c vol u 2 X a ε,e using Lmma 2.9, for 0 < a l and ã := l. Choosing a = min{l, η4c vol Q 1 } and 0 < ε ε η with ε η as abov, w can stimat th quadratic form contributions by η du 2 X ε,v + u 2 X ε,e η du 2 Xε. Th xprssion for C η thn follows by valuating th constant in front of th rmaining norm. W nd to stimat th fr quadratic form against th form associatd with th Hamiltonian: Corollary Th quadratic form h ε is closd. Morovr, stting η = 1/2, w gt th stimat d ε u 2 h ε u + C 1/2 u 2 X ε which holds providd 0 < ε ε 1/2. As in Lmma 2.4, w can prov th following stimat in ordr to compar th fr quadratic form with th graph norm of H ε : Lmma W hav u 2 H 1 X ε = d εu + u 2 X ε 2 max{ C 1/2, 2} H ε iu 2 X ε for u dom H ε dom h ε = H 1 X ε and 0 < ε ε 0.
9 APPROXIMATION OF VERTEX COUPLINGS BY SCHRÖDINGER OPERATORS ON MANIFOLDS 9 3. Approximation of δcouplings Aftr this prliminaris w can pass to our main problms. Th first on concrns approximation of a δcoupling by Schrödingr oprators with scald potntials supportd by th vrtx rgions. For th sak of simplicity most part of th discussion will b don for th situation with a singl vrtx as dscribd in Sction Quasiunitary oprators. First w dfin quasiunitary oprators mapping from H to H and vic vrsa, as wll as thir analogus on th scals of ordr on, namly H 1 and H 1. Hr, H := L 2 G, H 1 := H 1 G, H := L2 X ε, H 1 := H 1 X ε. 3.1 Morovr, w nd a rlation btwn th diffrnt constants of th graph and th manifold modl introducd abov. Spcifically, w st p := vol d 1 Y 1/2 and qv = Q dx v. 3.2 X v Lt J : H H b givn by Jf := ε d 1/2 Ef ½ with rspct to th dcomposition 2.8. Hr ½ is th normalisd ignfunction of Y associatd to th lowst zro ignvalu, i.. ½y = vol d 1 Y 1/2. In ordr to rlat th Sobolv spacs of ordr on w nd a similar map: w dfin J 1 : H 1 H 1 by J 1 f := ε d 1/2 f ½ fv½v, 3.4 E whr½v is th constant function on X v with valu 1. Not that th lattr oprator is wll dfind: J 1 f 0, y = ε d 1/2 p 1 f 0 = ε d 1/2 fv = J 1 f v x for any x X v du to 3.2 and 2.2, i.., th function J 1 f matchs along th diffrnt componnts of th manifold, thus Jf H 1 X ε. Morovr, fv is dfind for f H 1 G s Lmma 2.1. Th mapping in th opposit dirction, J : H H, is givn by th adjoint, J := J, which mans that J u s = ε d 1/2 ½, u s, Y = ε d 1/2 p us. 3.5 Furthrmor, w dfin J 1 : H 1 H 1 by [ J 1 us := ε d 1/2 ½, u s, Y + χ sp u ]. v u0 3.6 Hr χ is a smooth cutoff function such that χ 0 = 1 and χ l = 0. If w choos th function χ to b picwis affin linar with χ 0 = 1, χ a = 0 andχ l = 0, thn χ 2 I = a/3 a and χ 2 I = a 1. Morovr, J 1 u 0 = ε d 1/2 p u so that v f := J 1 u satisfis f0 Cp, and thrfor f H 1 p G. Not that by construction of th manifold, w hav X v u = u0.
10 10 PAVEL EXNER AND OLAF POST 3.2. Closnss assumptions. Lt us start this subsction with a lowr bound on th oprators H and H ε in trms of th modl paramtrs; for th dfinitions of th constants C 1/2, ε 1/2 and C 1/2 s Lmma 2.2 and Lmma Not that C 1/2 still dpnds on Q and l. Lmma 3.1. For ε 0, ε 1/2 ] th oprators H ε and H ar boundd from blow by λ 0 := C 1/2. Morovr, if all lngths ar finit, i.. l <, and qv 0, thn w hav inf σh qv qv and inf σh ε, vol X E vol X E + ε vol X v whr X E := X is th union of th dg nighbourhoods. Proof. W hav to calculat th maximum of C 1/2 and C 1/2. Du to 3.2 w hav p 2 = vol X v and qv = X v Q dx v Q vol X v so that { C 1/2 max 4c 2 vol Q 2, 2c } vol Q l C { 1/2 = max 64c 2 vol Q 2, 8c } vol Q, 3.7 l whr c vol := vol X v / vol X v. Th spctral stimats thn follow by insrting suitabl tst functions into th Rayligh quotints hf/ f 2 and h ε u/ u 2. For f, w choos th dgwis constant function f x = p. Not that f H 1 pg. On th manifold, w choos th constant u := J 1 f = ε d 1/2½. Th uppr bound on th infimum on th spctrum follows by th rlation l p 2 = vol X using 3.2. Now w ar in position to dmonstrat that th two Hamiltonians ar clos to ach othr. W start with stimats of th idntification oprators and th forms h, h ε in trms of th fr quadratic forms d and d ε : Lmma 3.2. Th idntification oprators J, J = J, J 1, J 1 and th quadratic forms h ε and h fulfil th stimats Jf J 1 f 2 δ 2 ε f 2 H 1 G, J u J 1 u 2 δ 2 ε u 2 H 1 X ε, 3.8a Jf 2 = f 2, J u 2 u 2, 3.8b J Jf = f, hj 1 u, f h ε u, J 1 f δε u H 1 X ε f H 1 G with δ ε = Oε 1/2 as ε 0, bing givn xplicitly by { δε 2 := max 8εcvol ε 2 [, l 0 λ 2 E, 4ε2 1 λ 2 v + c vol 1 + JJ u u 2 δ 2 ε u 2 H 1 X ε, 3.8c 2 ], l 0 λ 2 v 2ε 1 + l 0 2 l 0 λ 2 v, 4εc vol Q 2 l 0 λ 2 v 3.8d }. 3.9 Hr, l 0 := min{1, l } = min {1, l } 1, λ 2 E := min λ 2 and c vol = vol X v / vol X v. Morovr, λ 2 and λ 2 v dnot th scond first nonvanishing ignvalu of th NumannLaplacian on Y and X v, rspctivly. Proof. Th first condition in 3.8a is hr Jf J 1 f 2 X ε = ε vol X v fv 2 εc vol f 2 G + 2 l 0 f 2 G using Lmma 2.1 with a = l 0 and th fact that p 2 = vol X v du to 3.2. Nxt w nd to show th scond stimat in 3.8a. In our situation, w hav J u J 1 u 2 G = εd 1 χ 2 I p 2 u v u0 2 2 ε 1 + du 2 X l E 0 λ 2 v ε,v
11 APPROXIMATION OF VERTEX COUPLINGS BY SCHRÖDINGER OPERATORS ON MANIFOLDS11 using Lmma 2.8 with a = ã = l 0. Morovr, 3.8b and th first quation in 3.8c ar asily sn to b fulfilld. Th scond stimat in 3.8c is mor involvd. Hr, w hav JJ u u 2 = u u 2 X ε, + u 2 X ε,v. Th first trm can b stimatd as in 2.14 by u u 2 X ε, = us us 2 Y ds 1 d Y us I λ 2 I 2Y ds = ε2 λ 2 d Y u 2 X ε,, whr us := us,. Th scond trm can b stimatd by Lmma 2.9, so that { [ δε 2 max 4ε 2 1 λ 2 v + c 2 ] ε 2 vol 1 +, l 0 λ 2 v λ 2 E, 8εc } vol, l 0 which is sufficint for th stimat 3.8c. Lt us finally prov 3.8d in our modl. Not that this stimat diffrs from th ons givn in [P06] by th absnc of th potntial trm Q ε = ε 1 Q thr. In our situation, w hav hj 1 u, f h ε u, J 1 f 2 [ 2ε d 1 p u v u0 χ, f 2 I + qv u Qu,½v v Xv 2 fv ]. 2 Not that th drivativ trms cancl on th dgs du to th product structur of th mtric and th fact that d Y ½ = 0 and th vrtx contribution vanishs du to d Xv½=0. Th first trm can b stimatd by 2ε ã + 2 ãλ 2 v 1 a du 2 X ε,v 2ε 1 + l 0 2 l 0 λ 2 v using CauchySchwarz, Lmma 2.8 and th fact that χ 2 I = 1/a 1/l 0 by our choic of χ. For th scond trm, w us our dfinition qv = X v Q dx v and qv u = v u, Q½v v Xv to conclud qv u Qu,½v v Xv 2 = u, Q Q 2 v X v = u, Pv Q Xv 2 = Pv u, Q Xv 2 1 λ 2 v du 2 X v Q 2 X v whr P v u := u u is th projction onto th orthogonal complmnt of½v. Th v last stimat follows from Collcting th rror trms for th ssquilinar form stimat, w obtain δ 2 ε max { 2ε l l 0 λ 2 v, 4εc vol Q 2 l 0 λ 2 v as lowr bound on δ ε, using also Lmma 2.1 for th stimat on fv 2, and Q 2 X v vol X v Q 2. Now w can prov our main rsult on th approximation of a δcoupling in th manifold modl; for mor dtails on th notion of δclosnss w rfr to [P06, App.]. Th rsolvnt stimat at z = i will b ndd in Sction 4 whn th lowr bound λ 0 dpnds on ε and may tnd to as ε 0. Rcall th dfinition of C 1/2, 0 < ε 1/2 s 2.15 and λ 0 := C 1/2, and that C 1/2 C 1/2. }
12 12 PAVEL EXNER AND OLAF POST Thorm 3.3. For ε 0, ε 1/2 ], th oprators H ε λ 0 and H λ 0 ar 2δ ε clos with δ ε = Oε 1/2 givn in 3.9; in othr words, thr is an idntification oprator J : L 2 G L 2 X ε such that J J = id, id JJ H ε λ 0 1 2δ ε and JH λ 0 1 H ε λ 0 1 J 3 2δ ε. Morovr, for ε 0, ε 1/2 ] w hav th stimat JH i 1 H ε i 1 J 10δε max{ C 1/2, 2}, whr dnots th oprator norm for oprators from L 2 G into L 2 X ε. Proof. Th closnss of th oprators H λ 0 and H ε λ 0 follows from th stimat du 2 X ε + u 2 X ε 2 h ε u + 1 λ 0 u 2 X ε = 2 Hε λ /2 u 2 X ε by Corollary 2.11, and similarly for H on G by Corollary 2.3 and 3.7, togthr with Lmma 3.2. Th rsolvnt stimat can b sn as follows: Lt R := H i 1 and R := H ε i 1, and lt f L 2 G, ũ L 2 X ε. Stting f := R f dom H and u := R ε ũ dom H ε, w hav ũ, JR R ε J f = ũ, Jf u, J f and thrfor = ũ, J J 1 f + h ε u, J 1 f hj 1 u, f + J 1 J u, f i u, J 1 Jf + J 1 J u, f, ũ, JR Rε J f 10δε max{ C 1/2, 2} f ũ using Lmmata 2.4 and 2.12, and th fact that C 1/2 C 1/2. Using th abstract rsults of [P06, App. A] or [P08], w can show th rsolvnt convrgnc and th convrgnc othr functions of th oprator: Thorm 3.4. W hav JH z 1 H ε z 1 J = Oε 1/2, JH z 1 J H ε z 1 = Oε 1/2 3.10a 3.10b for z / [λ 0,. Th rror dpnds only on δ ε, givn in 3.9, and on z. Morovr, w can rplac th function ϕλ = λ z 1 by any masurabl, boundd function convrging to a constant as λ and bing continuous in a nighbourhood of σh. Th following spctral convrgnc is also a consqunc of th Oε 1/2 closnss; for dtails of th uniform convrgnc of sts, i.. th convrgnc in Hausdorffdistanc sns w rfr to [HN99, App. A] or [P08]. Thorm 3.5. Th spctrum of H ε convrgs to th spctrum of H uniformly on any finit nrgy intrval. Th sam is tru for th ssntial spctrum. Proof. Th spctral convrgnc is a dirct consqunc of th closnss, as it follows from th gnral thory dvlopd in [P06, Appndix] and [P08]. For th discrt spctrum w hav th following rsult: Thorm 3.6. For any λ σ disc H thr xists a family {λ ε } ε with λ ε σ disc H ε such that λ ε λ as ε 0. Morovr, th multiplicity is prsrvd. If λ is a simpl ignvalu with normalisd ignfunction ϕ, thn thr xists a family of simpl normalisd ignfunctions {ϕ ε } ε of H ε ε small such that as ε 0. Jϕ ϕ ε Xε 0
13 APPROXIMATION OF VERTEX COUPLINGS BY SCHRÖDINGER OPERATORS ON MANIFOLDS13 W rmark that th convrgnc of highrdimnsional ignspacs is also valid, howvr, it rquirs som tchnicalitis which w skip hr. To summaris, w hav shown that th δcoupling with wightd ntris can b approximatd by a gomtric stting and a potntial locatd on th vrtx nighbourhood. Lt us brifly sktch how to xtnd th abov convrgnc rsults Thorms to mor complicatd vn to noncompact graphs. Dnot by G a mtric graph, givn by th undrlying discrt graph V, E, with : E V V, =, + dnoting th initial and trminal vrtx, and th lngth function l: E 0,, such that ach dg is idntifid with th intrval I = 0, l for simplicity, w assum hr that all lngth ar finit, i.., l <. Lt X ε b th corrsponding approximating manifold constructd from th building blocks X ε, = I εy and X ε,v = εx v as in Sction 2.2. For mor dtails, w rfr to [EP05, P06, EP08, P08]. Sinc a mtric graph can b constructd from a numbr of star graphs with idntifid nd points of th fr nds, w can dfin global idntification oprators. W only hav to assur that th global rror w mak is still uniformly boundd: Thorm 3.7. Assum that G is a mtric graph and X ε th corrsponding approximating manifold constructd according to G. If inf λ 2v > 0, v V sup v V vol X v vol X v <, sup Q Xv <, v V inf λ 2 > 0, E inf l > 0, E thn th corrsponding Hamiltonians H = G + v qvδ v and H ε = Xε + v ε 1 Q v ar δ ε clos, whr th rror δ ε = Oε 1/2 dpnds only on th abov indicatd global constants. 4. Approximation of th δ s couplings Th main aim of this sction is to show how a th symmtrisd δ coupling, or δ s, can b approximatd using manifold modl discussd abov. To this aim w shall us a rsult of [CE04] by which a δ s coupling can b approximatd by mans of svral δcouplings on th sam mtric graph, locatd clos to th vrtx and lift this approximation to th manifold. For th sak simplicity w will again considr th starshap stting with a singl vrtx. W want to strss, howvr, that th mthod w us can b dirctly gnralisd to mor complicatd graphs but also, what is qually important, to othr vrtx couplings, onc thy can b approximatd by combinations of δcouplings on th graph, possibly with an addition of xtra dgs s [ET06, ET07]. Lt thus G = I v0 b a star graph as in Sction 2 whr w dnot th vrtx in th cntr by v 0 and whr w labl th n = dg v dgs by = 1,...,n. Again for simplicity, w assum that all th unscald transvrsal volums p 2 = vol Y ar th sam; without loss of gnrality w may put vol Y = 1. Morovr, w assum that all lngths ar finit, i.. l <, and qual, so w may put l = 1. First w rcall th dfinition of th δ s coupling: th oprator H β, formally writtn as H β = G + βδ v 0, acts as H β f = f on ach dg for functions f in th domain { dom H β := f H 2 maxg 1, 2 : f 1 0 = f 2 0 =: f 0, f 0 = βf 0, } : f l = For th sak of dfinitnss w imposd hr Numann conditions at th fr nds of th dgs, howvr, th choic is not substantial; w could us qually wll Dirichlt or any
14 14 PAVEL EXNER AND OLAF POST othr boundary condition. Th corrsponding quadratic form is givn as h β f = f f 0 2, dom h β = H 1 β maxg if β 0 and h β f = f 2, dom h β = { f H 1 maxg f 0 = 0 } if β = 0; th condition f H 0 is obviously dual to th fr or Kirchhoff vrtx coupling s,.g., [Ku04, Sc ]. Th ngativ spctrum of H β is asily found: Lmma 4.1. If β 0 thn H β 0. On th othr hand, if β < 0 thn H β has xactly on ngativ ignvalu λ = κ 2 whr κ is th solution of th quation cosh κ + βκ sinh κ = dg v Proof. Th nonngativity of H β follows from th quadratic form xprssion for β > 0 and β = 0. W mak th ansatz f s = cosh κ1 s fulfilling automatically th Numann condition at s = 1 and th continuity condition at s = 0 sinc f 0 = κ sinh κ is indpndnt of. Th rmaining condition at zro lads to th abov rlation of κ and β, showing in anothr way that if β 0 thr cannot xist a ngativ ignvalu. Th main ida of th approximation of a δ scoupling by Schrödingr oprators on a manifold is to mploy a combination of δcouplings in an oprator on may call an intrmdiat Hamiltonian H β,a, and thn to us th approximations for δcouplings givn in th prvious sction. In ordr to dfin H β,a, w first modify th discrt structur of th graph G insrting additional vrtics v of dgr 2 on th dg with th distanc a 0, 1 from th cntral vrtx v 0 s Figur 1. Each dg is splittd into two dgs a and 1. W dnot th mtric graph with th additional vrtics v and splittd dgs by G a, i.., V G a = {v 0 } { v = 1,...,n}, EG a = { a, 1 = 1,...n } and l a = a, l 1 = 1 a. This mtrically quivalnt graph G a will b ndd whn associating th corrsponding manifold. Rmark 4.2. It is usful to not that th Laplacians G and Ga associatd to th mtric graphs G and G a ar unitarily quivalnt. Indd, introducing additional vrtics of dgr two dos not chang th original quadratic form d G with th domain H 1 G = dom d associatd to th fr oprator G = H G,0. Figurativly spaking, th fr oprator dos not s ths vrtics of dgr two. W just hav to chang th coordinat on th dg, i.. w can ithr us th original coordinat s 0, l on th dg or w can split th dg into two dgs a and 1 of lngth l a = a and l 1 = l a = 1 a with th corrsponding coordinats. Th cor of th approximation lis in a suitabl, adpndnt choic of th paramtrs of ths δcouplings. Writing th oprator in trms of th formal notation introducd in 2.6, w put H β,a := G + baδ v0 + caδ v, ba = β a 2, ca = 1 a,
15 APPROXIMATION OF VERTEX COUPLINGS BY SCHRÖDINGER OPERATORS ON MANIFOLDS15 to b th intrmdiat Hamiltonian. Notic that th strngth of cntral δcoupling dpnds on β whil th addd δintractions ar attractiv, th sol paramtr bing th distanc a. Th oprator can b dfind via its quadratic form h β,a f := f 2 β a 2f0 1 f a 2, dom h a = H 1 G, a whr H 1 G = H 1 p G with p = 1,...,1, i.. th functions f H1 G ar distinguishd by bing continuous at v 0, f 1 0 = f 2 0 =: f0. Th nxt thorm shows that th intrmdiat Hamiltonian convrgs indd to th coupling with th strngth β on th starshapd graph: δ s Thorm 4.3 Chon, Exnr. W hav H β,a z 1 H β z 1 = Oa as a 0 for z / R, whr dnots th oprator norm on L 2 G. 3 Not that th choic of th paramtrs ba ca of th δintractions as functions of th distanc a follows from a carful analysis of th rsolvnts of H β,a and H β. Each of ths is highly singular as a 0, howvr, in th diffrnc all th singularitis cancl laving us with a vanishing xprssion. Ndlss to say, that such a limiting procss is highly nongnric. a ε = ε α X ε,v0 ε α ε ε ε v 0 a v 1 X ε,ε X ε,v X ε,1 G X ε Figur 1. Th intrmdiat graph pictur usd in th δ sapproximation and th corrsponding manifold modl. Lt us now considr th manifold modl approaching th intrmdiat situation Hamiltonian H β,a in th limit ε 0 with a = a ε = ε α and 0 < α < 1 to b spcifid latr on. Lt X ε b a manifold modl of th graph G as shown in Figur 1. For th additional vrtics of dgr two w choos th vrtx nighbourhoods as a part of th cylindr of lngth ε and distanc of ordr of a ε from th cntral vrtx v 0. Th dg aε =: ε now has th lngth a ε = ε α dpnding on ε. Th fr dg 1 joining v with th fr nd point at s = 1 is again εdpnding, namly it has th lngth 1 a ε = 1 ε α. By th argumnt givn in Lmma 2.7 w can dal with this rror and assum that this dg again has lngth on, th pric bing an xtra rror of ordr Oε α, affcting nithr th final rsult nor th quantitativ rror stimat. Nxt w hav to choos th potntials 3 Th claim mad in [CE04] is only that th norm tnds to zro, howvr, th rat with which it vanishs is obvious from th proof. W rmov th suprfluous dg v from th dfinition of H β,a in that papr. It should also b notd that th proof in [CE04] is givn for star graphs with smiinfinit dgs but th argumnt again modifis asily to th finitlngth situation w considr for convninc hr.
16 16 PAVEL EXNER AND OLAF POST in th vicinity of th vrtics v = v 0 and v = v. Th simplst option is to assum that thy ar constant, Q ε,v x := 1 ε q ε v, x X v vol X v so that X v Q ε,v dx = ε 1 q ε v s 2.10 and 3.2, whr w put q ε v 0 := bε α = βε 2α and q ε v := cε α = ε α. Th corrsponding manifold Hamiltonian and th rspctiv quadratic form ar thn givn by Hε β = Xε ε 1 2α β vol X v0 ε 1 α v 4.3 v0½x E½X and h β ε u = du 2 X ε ε 1 2α β u 2 X vol X ε,v0 ε 1 α u 2 X ε,v, v0 E rspctivly. Not that th unscald vrtx nighbourhood X v of th addd vrtx v has volum 1 by construction. Bfor procding to th approximation itslf, lt us first mak som commnts about th lowr bounds of th oprators H β,a and thir manifold approximations H β ε : Lmma 4.4. If β < 0, thn th spctrum of H β,a is uniformly boundd from blow as a 0, in othr words, thr is a constant C > 0 such that inf σh β,a C as a 0. If β 0, on th othr hand, thn th spctrum of H β,a is asymptotically unboundd from blow, inf σh β,a as a 0. Not that although w know th limit spctrum as a 0 s Lmma 4.1, th rsolvnt convrgnc of Thorm 4.3 dos not ncssarily imply th uniform bounddnss from blow of H β,a s Rmark 4.8. Proof. Lt β < 0. Thn an ignfunction on th original dg has th form { A coshκs + B sinhκs, 0 s a f s = C coshκ1 s, a s 1. for κ > 0, th corrsponding ignvalu bing λ = κ 2. Th Numann condition f 1 = 0 at s = 1 is automatically fulfilld, as wll as th continuity at s = 0 for th diffrnt dgs, sinc f 0 = A is indpndnt of. Th continuity in s = a and th jump condition in th drivativ lad to th systm of quations A coshκa + B sinhκa C coshκ1 a = 0 1 a C cosh κ1 a = κ A sinhκa B coshκa C sinh κ1 a β a 2A = κ B. With th prmutational invarianc in mind, lt us first analys th situation with symmtric cofficints, A, B = B, C = C. Thn B = nb and th corrsponding cofficint matrix for A, B and C vanishs iff β a 2 sinhκa cosh κ1 a aκ cosh κ + nκ κa sinh κ coshκa cosh κ1 a = 0
17 APPROXIMATION OF VERTEX COUPLINGS BY SCHRÖDINGER OPERATORS ON MANIFOLDS17 lading to an ignvalu λ = κa 2 of multiplicity on. It can b sn that κa is boundd, and that th abov quation rducs to 4.2 as a. Th othr ignvalus can b obtaind from B and C as follows: st Θ n := 2πi/n. Thn for k = 1,...,n 1, w hav th cofficints B,k = Θ k n B and C,k = Θn k C, = 1,..., n. Sinc n n 1 B,k = B =1 =0 Θ k n = 0 for k = 1,..., n 1, w finally arriv at a cofficint matrix similar to th prvious on, but with n rplacd by zro. Consquntly, if thr wr additional ngativ ignvalus λ = κa 2, thy would b of multiplicity n 1 and givn by th rlation sinhκa cosh κ1 a κa cosh κ = 0. But this quation has no solutions for 0 < a 1 and κ > 0. W skip th proof of this fact hr. For th scond part, assum that β 0. It is sufficint to calculat th Rayligh quotint for th constant tst function f =½ H 1 G which yilds h β,a f = 1 β f n 2 a a bing of ordr Oa 2 if β < 0 and of ordr Oa 1 if β = 0, ngativ in both cass; rcall that n = dg v. Similarly, w xpct th sam bhaviour for th oprators on th manifold. Lmma 4.5. If β 0, thn th spctrum of Hε β is asymptotically unboundd from blow, i.., inf σhε β as ε 0. Proof. Again, w plug th constant tst function u =½into th Rayligh quotint and obtain h β ε u u = βε 2a + ε a 2 n1 + ε + ε α + ε volx v0 which obviously tnds to as ε 0. Rmark 4.6. As for a countrpart to th othr claim in Lmma 4.4, th proof of th uniform bounddnss from blow as ε 0 for β < 0 sms to nd quit subtl stimats to compar th ffct of th two compting potntials on X ε,v0 and X ε,v having strngth proportional to β ε 2α and ε α, rspctivly. Sinc th positiv contribution Q ε,v0 = β ε 1 2α is mor singular than th ngativ contributions Q ε,v = ε 1 α, w xpct that th thrshold of th spctrum rmains boundd as ε 0. W can now prov our scond main rsult. For th δ s coupling Hamiltonian H β and th approximating oprator H β ε dfind in 4.1 and 4.3, rspctivly, w mak th following claim. Thorm 4.7. Assum that 0 < α < 1/13, thn H β ε i 1 J JH β i 1 0 as ε 0. Proof. Dnot by H β,ε = H β,aε th εdpnding intrmdiat Hamiltonian on th mtric graph with δpotntials of strngth dpnding on ε as dfind bfor. For th corrsponding graph and manifold modl, th lowr bound to lngths dpnds now on ε,
18 18 PAVEL EXNER AND OLAF POST spcifically, l = a ε = ε α. Morovr, from th dfinition of th constants C 1/2 C 1/2 and ε 1/2 in 2.15 and from Lmma 3.2, w conclud that C 1/2 = C 1/2 ε = Oε 4α, ε 1/2 = ε 1/2 ε = Oε 3α and δ = δ ε = Oε 1 5α/2. Not that th dominant trm in th closnssrror δ ε s 3.9 is th last on containing th potntial. From Thorm 3.3 it follows now that H β ε i 1 J JH β,ε i 1 10δε max{ C 1/2 ε, 2} = Oε 1 13α/2. so that Thorm 4.3 yilds th sought conclusion. Not that th xponnt of ε in δ ε C1/2 ε is 1 5α/2 4α = 1 13α/2 > 0 providd 0 < α < 1/13. W can now procd and stat similar rsults as in Thorms for th δ s  approximation by using argumnts similar to thos in [P06, App.] or [P08], whr only nonngativ oprators wr considrd covring, as usual, oprators boundd uniformly from blow by a suitabl shift. In our prsnt situation, w can only guarant th rsolvnt convrgnc at nonral points lik z = i. Nvrthlss, th argumnts in [P06, App.] or [P08] can b usd to conclud th convrgnc of suitabl functions of oprators as wll as th convrgnc of th dimnsion of spctral projctions. Rmark 4.8. Not that th asymptotic lowr unbounddnss of H β ε and of th intrmdiat oprator H β,ε for β 0 dscribd in Lmmata 4.4 and 4.5 is not a contradiction to th fact that th limit oprator H β is nonngativ. For xampl, th spctral convrgnc of Thorm 3.5 holds only for compact intrvals I R. In particular, σh β I = implis that σh β ε I = and σhβ,ε I = providd ε > 0 is sufficintly small. This spctral convrgnc mans that th ngativ spctral branchs of H β ε all hav to tnd to. Acknowldgmnt. O.P. njoyd th hospitality in th Dopplr Institut whr a part of th work was don. Th rsarch was supportd by th Czch Ministry of Education, Youth and Sports within th projct LC Rfrncs [ACF07] S. Albvrio, C. Cacciapuoti, and D. Finco, Coupling in th singular limit of thin quantum wavguids, J. Math. Phys , [CE07] C. Cacciapuoti and P. Exnr, Nontrivial dg coupling from a Dirichlt ntwork squzing: th cas of a bnt wavguid, J. Phys. A , L511 L523. [CE04] T. Chon and P. Exnr, An approximation to δ couplings on graphs, J. Phys. A , no. 29, L329 L335. [CS98] T. Chon and T. Shighara, Ralizing discontinuous wav functions with rnormalizd shortrang potntials, Phys. Ltt. A , [DEK01] P. Duclos, P. Exnr, and D. Krjčiřík, Bound stats in curvd quantum layrs, Comm. Math. Phys , no. 1, [DT06] G. F. Dll Antonio and L. Tnuta, Quantum graphs as holonomic constraints, J. Math. Phys , no. 7, , 21. [E97] P. Exnr, A duality btwn Schrödingr oprators on graphs and crtain Jacobi matrics, Ann. Inst. H. Poincaré Phys. Théor , no. 4, [EKK + 08] P. Exnr, J. P. Kating, P. Kuchmnt, T. Sunada, and A. Tplayav ds., Analysis on graphs and its applications, Proc. Symp. Pur Math., vol. 77, Providnc, R.I., Amr. Math. Soc., [ENZ01] P. Exnr, H. Nidhardt, and V.A. Zagrbnov, Potntial approximations to δ : an invrs Klaudr phnomnon with normrsolvnt convrgnc, Comm. Math. Phys , no. 3, [EP05] P. Exnr and O. Post, Convrgnc of spctra of graphlik thin manifolds, Journal of Gomtry and Physics ,
19 APPROXIMATION OF VERTEX COUPLINGS BY SCHRÖDINGER OPERATORS ON MANIFOLDS19 [EP07], Convrgnc of rsonancs on thin branchd quantum wav guids, J. Math. Phys , [EP08], Quantum ntworks modlld by graphs, Quantum FwBody Systms, AIP Conf. Proc., vol. 998, Amr. Inst. Phys., Mlvill, NY, 2008, pp [ET06] P. Exnr and O. Turk, Approximations of prmutationsymmtric vrtx couplings in quantum graphs, Quantum graphs and thir applications, Contmp. Math., vol. 415, Amr. Math. Soc., Providnc, RI, 2006, pp [ET07], Approximations of singular vrtx couplings in quantum graphs, Rv. Math. Phys , no. 6, [FW93] M. I. Fridlin and A. D. Wntzll, Diffusion procsss on graphs and th avraging principl, Ann. Probab , no. 4, [G08] D. Grisr, Spctra of graph nighborhoods and scattring, Proc. London Math. Soc , no. 3, [HN99] I. Hrbst and S. Nakamura, Schrödingr oprators with strong magntic filds: Quasipriodicity of spctral orbits and topology, Amrican Mathmatical Socity. Transl., Sr. 2, Am. Math. Soc , [Ku04] P. Kuchmnt, Quantum graphs: I. Som basic structurs, Wavs Random Mdia , S107 S128. [KuZ01] P. Kuchmnt and H. Zng, Convrgnc of spctra of msoscopic systms collapsing onto a graph, J. Math. Anal. Appl , no. 2, [MV07] S. Molchanov and B. Vainbrg, Scattring solutions in ntworks of thin fibrs: small diamtr asymptotics, Comm. Math. Phys , no. 2, [Pa07a] B. Pavlov, Numann Schrödingr 2D junction: collapsing on a quantum graph: a gnralizd Kirchhoff boundary condition, Prprint [Pa07b], A stargraph modl via oprator xtnsion, Math. Proc. Cambridg Philos. Soc , [P05] O. Post, Branchd quantum wav guids with Dirichlt boundary conditions: th dcoupling cas, Journal of Physics A: Mathmatical and Gnral , no. 22, [P06], Spctral convrgnc of quasiondimnsional spacs, Ann. Hnri Poincaré , no. 5, [P08], Spctral analysis on graphlik spacs, Habilitation thsis in prparation [RS01] J. Rubinstin and M. Schatzman, Variational problms on multiply connctd thin strips. I. Basic stimats and convrgnc of th Laplacian spctrum, Arch. Ration. Mch. Anal , no. 4, [Sai00] Y Saito, Th limiting quation for Numann Laplacians on shrinking domains., Elctron. J. Diffr. Equ , 25 p. Dpartmnt of Thortical Physics, NPI, Acadmy of Scincs, Řž nar Pragu, and Dopplr Institut, Czch Tchnical Univrsity, Břhová 7, Pragu, Czchia addrss: Institut für Mathmatik, HumboldtUnivrsität zu Brlin, Rudowr Chauss 25, Brlin, Grmany addrss:
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