A SIMPLICITY CRITERION FOR SYMMETRIC OPERATOR ON A GRAPH

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1 Methods of Functionl Anlysis nd Topology Vol. 20 (2014), no. 2, pp A SIMPLICITY CRITERION FOR SYMMETRIC OPERATOR ON A GRAPH E. N. ASHUROVA, A. N. KANDAGURA, AND I. I. KARPENKO To Professor Yu. S. Smoilenko on the occsion of his 70th birthdy Abstrct. In the present pper we show tht the topology of the underlying grph determines the domin nd deficiency indices of certin ssocited miniml symmetric opertor. We obtine criterion of simplicity for the miniml opertor ssocited with the grph. Introduction Anumberofuthorsintheirrecentppers, see, e.g., [9,1,11,10,4,13,5,6], consider Lplce opertor (Lplcin) defined on metric grph. Such grph, provided tht it is equipped with differentil opertor cting on the edges nd subject to certin mtching conditions t the vertices, is clled [10] quntum grph. Quntum grphs ply rôle of nturl intuitive models in mthemtics, physics, chemistry, nd engineering, when one considers the phenomenon of wve propgtion (e.g., electromgnetic, coustic, etc.) in some qusi one dimensionl system. Mtching conditions which determine the domin of the quntum grph comprise of certin conditions for the vlue of the function nd its norml derivtives t ll grph vertices. These conditions cn be either locl (linking the vlues of the function nd its norml derivtives t precisely one vertex) or non-locl (when they link together these vlues pertining to more thn one vertex). In the pper of Yu. Ershov nd A. V. Kiselev [3] the method of boundry triples ws successfully pplied to the study of n inverse spectrl problem on grph with locl mtching conditions. This method ws suggested nd developed by V. I. nd M. L. Gorbchuk [7], A. N. Kochubei [8], V. A. Derkch nd M. M. Mlmud [2]. Within its frmework the corresponding self-djoint differentil opertor is considered s proper extension of some symmetric opertor with equl deficiency indices. One might rgue tht this method cn be employed quite efficiently in the study of spectr of quntum grphs with non locl mtching conditions s well. Under the dditionl ssumption tht the symmetric opertor in question is simple (i.e., hs no reducing subspce such tht on it the opertor induces self djoint opertor), the Weyl-Titchmrsh mtrix function llows to investigte ll of the spectrum of lmost solvble extensions of this opertor. In the present pper we show tht the topology of the underlying grph determines the domin nd deficiency indices of certin miniml symmetric opertor which we cll ssocited to the grph nd which plys specil rôle in its spectrl nlysis (Proposition 1). Ershov nd Kiselev [3] formulte some conditions sufficient for miniml opertor corresponding to quntum grph to be simple. Proposition 1 obtined by us provides criterion of simplicity for the miniml opertor ssocited to the grph Mthemtics Subject Clssifiction. Primry 47A10; Secondry 47A55. Key words nd phrses. Lplce opertor, quntum grph, symmetric opertor, deficiency indices. 117

2 118 E. N. ASHUROVA, A. N. KANDAGURA, AND I. I. KARPENKO 1. Opertor A min ssocited to the grph Γ Assume tht the edge set E = {e k } n of metric grph Γ = (V,E) consists of finite intervls e k = [x 2k 1,x 2k ] R of lengths l k = x 2k x 2k 1, nd tht the vertex set V is certin prtition of the set {x j } 2n j=1. Throughout the text of the pper we use the terminology nd nottion of [12]. The Hilbert spce L 2 (Γ) corresponding to the grph Γ is the spce of mesurble squre summble functions defined on ll grph edges nd equipped with the stndrd norm f 2 L = 2(Γ) f 2 L 2(e k ), e k E i.e., it is nothing but n orthogonl sum of L 2 (e k ) spces, L 2 (Γ) = e E L 2(e k ). The Sobolev spce W 2,2 (Γ) = W 2,2 (e k ) is introduced in n nlogous fshion. e k E Here the spce W 2,2 (e k ) consists of functions defined on the edge e k such tht these functions re bsolutely continuous together with their first derivtives, nd f e k (x) L 2 (e k ). For functions f W 2,2 (Γ) we define the norml derivtive n f(x j ) t the endpoints of ech intervl e k, { f n f(x j ) = (x j ), if x j is the left endpoint of the intervl, f (x j ), if x j is the right endpoint of the intervl. Then it ppers nturl to introduce the following nottion for the function f W 2,2 (Γ) t ny vertex of the grph: f(v k ) = f(x j ), n f(v k ) = n f(x j ). x j V k x j V k We ssign mrk (either of the symbols δ or δ, lso referred to s type) to every interior vertex of the grph Γ, i.e., to ny vertex for which the degree is greter thn one, deg V k > 1). The grph obtined s result of this procedure will be referred to s mrked nd denoted Γ δ. A mrked grph Γ δ defines the liner mnifold D(Γ δ ) := { f W 2,2 (Γ) f is continuous t vertices of δ type, n f is continuous t the vertices of δ type If V k is vertex of δ-type then evidently f(v k ) = deg V k f(x j ), x j V k. In the sme wy, for vertex V k of δ -type n f(v k ) = deg V k n f(x j ), x j V k. In the Hilbert spce L 2 (Γ) = n L 2 (e k ) we now define the opertor A min, which cts on every grph edge s the differentil expression (1) A min = d2 dx 2 on the domin dom(a min ) which is the set of functions f D(Γ δ ) subject to conditions (2) f(v k ) = 0, n f(v k ) = 0 ( k). This opertor A min nd the grph Γ δ will henceforth be clled ssocited. For instnce, if A min is ssocited to the grph Γ δ in which every vertex is of δ type, then the domin of the opertor consists of functions f D(Γ δ ) which stisfy the following conditions: f(x k ) = 0 ( x k ), n f(x k ) = 0 ( V i ). x k V i HenceforthwewilllwysssumethtthegrphΓ δ ndtheopertora min ressocited. Note tht A min is closed symmetric opertor. Adjoint to it, A mx := A min, is defined by the sme differentil expression on the domin D(Γ δ ). }.

3 A SIMPLICITY CRITERION FOR SYMMETRIC OPERATOR ON A GRAPH 119 Bck to the symmetric opertor, we formulte the following Proposition 1. Deficiency indices of the opertor A min ssocited to the grph Γ δ re equl to the number of grph vertices. Proof. Let n be the number of grph edges, m the number of grph vertices. For Imλ 0 the generl solution of the eqution (3) (A mx λi)f = 0 on every edge is of the form f k (x) = C + k cos λx+c k sin λx, i.e., is determined by 2n prmeters {C + k, C k, k = 1,n.}. Moreover, t every vertex nd for ll x i,x j V k the following mtching conditions hold: f(x i ) = f(x j ), if V k is vertex of δ-type, n f(x i ) = n f(x j ), if V k is vertex of δ -type. The totl number of these reltions pertining to ech vertex is exctly deg V k 1. Then for the number of free prmeters of the generl solution of (3) one obtins m m 2n (deg V k 1) = 2n+m deg V k. Since due to the hndshkes lemm m deg V k = 2n, the dimension of ker(a mx λi) is exctly m. 2. Criterion of simplicity for the opertor A min The question of whether the opertor A min ssocited to the grph Γ δ is simple cn be reduced to the question of whether this opertor hs ny point spectrum. Detiled proof of the fct tht A min is simple provided tht its point spectrum is empty is contined in [3]. As it will be shown below, the property of simplicity for the opertor A min is completely determined by the structure nd mrking of the metric grph Γ δ. Lemm 1. If the grph Γ δ contins loop, the opertor A min ssocited to the grph Γ δ is not simple. Proof. In order to simplify the required clcultions, ccept without loss of generlity the following convention. Assumethttheleftendpointofechintervl [x 2k 1,x 2k ], k = 1,n is shifted to the point zero, wheres the right endpoint is then t l k = x 2k x 2k 1. Let now grph Γ δ hve loop e k = [0,l k ] ttched to the vertex V k. 0 l k V k Solutions to the eqution A min f = λf re subject to conditions (2) t the vertex V k. Moreover, the conditions corresponding to the type of the vertex (δ or δ ) hve to be stisfied: f(0) = f(l k ) = 0, f (0) f (l k ) = 0, if V k is of δ-type, f (0) = f (l k ) = 0, f(0)+f(l k ) = 0, if V k is of δ -type. Let λ 0. Then on e k the solution ssumes the form f k (x) = C + k cosµx+c k sinµx, where µ = λ.

4 120 E. N. ASHUROVA, A. N. KANDAGURA, AND I. I. KARPENKO Tking into ccount the bove mentioned conditions, we obtin the following reltions for C + k,c k : C + k = 0, C k sinµl k = 0, C k (1 cosµl k) = 0, if V k is δ-vertex, C k = 0, C+ k sinµl k = 0, C + k (1+cosµl k) = 0, if V k is δ -vertex. Obviously, in either cse there exists non trivil solution on the edge e k for µ = 2kπ/l k, k Z, if V k is δ-vertex, µ = (2k +1)π/l k, k Z, if V k is δ -vertex. Hence the point spectrum of A min is non empty nd this opertor is not simple. Lemm 2. The opertor A min ssocited to the grph Γ δ is not simple if the grph Γ δ contins one of the following subgrphs, ll vertices of which re of δ -type: (i) cycle with n even number of edges; (ii) subgrph consisting of two cycles with odd numbers of edges ech which re connected by chin or hve exctly one common vertex. Proof. (i) If the grph Γ δ contins cycle with n even number of edges nd ll of the cycle s vertices re of δ -type, then on this cycle there exists non-trivil solution to the eqution A min f = 0. Here is n exmple of the cycle of four edges, on which four constnts re shown forming such solution: Therefore, the opertor A min is not simple. (ii) If the grph hs subgrph consisting of two cycles with odd numbers of edges nd ll vertices of which re of δ -type, then on these edges there lso exists nontrivil solution to A min f = 0 nd hence the opertor A min is not simple. The figure below demonstrtes n exmple of such eigenfunction which is collection of constnts, defined on grph with two cycles of three edges ech. This exmple dmits nturl generliztion to the cse of rbitrry cycles of odd lengths. 2 2 Lemm 3. The eqution A min f = λf, λ 0, hs non-trivil solution on cycle of the grph Γ δ iff ll the lengths of this cycles edges re pirwise rtionlly dependent. Proof. Let the cycle C = e 1,e 2,...,e n belong to the grph Γ δ. Consider the vertex V k of the cycle C which is incident to the edges e k 1 nd e k. V k 1 l k l k V k V k+1

5 A SIMPLICITY CRITERION FOR SYMMETRIC OPERATOR ON A GRAPH 121 1) Suppose tht V k is the vertex of δ-type. Then eigenvectors re subject to the following conditions: f k 1 (l k 1 ) = f k (0) = 0, (4) f k 1(l k 1 )+f k(0) = 0. Since λ 0, solutions on these edges ssume the form f j (x) = C + j cosµx+c j sinµx, where µ = λ, j = k 1,k. Due to (4) one obtins the following reltions for the coefficients: Therefore, (5) C + k = 0, C + k 1 cosµl k 1 +C k 1 sinµl k 1 = 0, C + k 1 sinµl k 1 C k 1 cosµl k 1 +C k = 0. f k 1 (x) = C k sinµ(x l k 1), f k (x) = C k sinµx, where µ = λ. Note tht from the expressions obtined it follows tht on the two neighboring edges of the cycle the solution to the eqution A min f = λf, λ 0, is either non-trivil on both or is necessrily trivil on both. Two vertices, djcent to V k, cn lso be of either δ or δ -type. Consider ll possible situtions one by one. 1.1) If both V k 1 nd V k+1 re of δ-type, the mtching conditions t them re of the form f k 1 (0) = 0, f k (l k ) = 0, tht is, tking into ccount (5), C k sinµl k 1 = 0, C k sinµl k = ) If V k 1 is δ-vertex, wheres V k+1 is of δ -type, f k 1 (0) = 0, f k (l k) = 0, which yields the following conditions: C k sinµl k 1 = 0, C k cosµl k = ) Let the vertex V k 1 be of δ -type, the vertex V k of δ-type. Then the conditions t these vertices led to the following reltions: C k cosµl k 1 = 0, C k sinµl k = ) If the vertices V k 1 nd V k re of δ -type, the corresponding mtching conditions led to: C k cosµl k 1 = 0, C k cosµl k = 0. It is quite obvious tht in ech of these four cses the condition, necessry nd sufficient for existence of non-trivil solution, is rtionl dependence of the lengths l k 1 nd l k. 2) Hving ssumed tht V k is vertex of δ -type, one gets the following solutions in plce of (5): f k 1 (x) = C + k cosµ(x l k 1), f k (x) = C + k cosµx. Suppose tht V k 1 nd V k+1 re δ-vertices. Then mtching conditions t these vertices yield C + k cosµl k 1 = 0, C + k cosµl k = 0. Therefore in this cse s well the solution is non-trivil iff the lengths l k 1 nd l k re rtionlly dependent. The sme result will hold in ll other possible cses for the vertices V k 1 nd V k+1. Now let some pir of the cycles edges hve rtionlly independent lengths. This mens, tht there hs to exist pir of neighboring rtionlly independent edges. Indeed,

6 122 E. N. ASHUROVA, A. N. KANDAGURA, AND I. I. KARPENKO if e s,e s+1,...,e s+k is the mximl sequence of pirwise rtionlly dependent edges, then for ll i = 0,k l s+i = α i l s, α i Q. Hence, k is less then the length of the cycle nd the edges e s+k nd e s+k+1 re rtionlly independent. On these edges one gets the trivil solution only of the eqution A min f = λf, λ 0, which tking into ccount the remrk mde bove leds to the trivil solution only on ll edges forming the cycle. Lemm 4. If n edge of the grph Γ δ is incident to boundry vertex, then on this edge the eqution A min f = λf dmits only the trivil solution for ny λ. Proof. Suppose tht the edge e 1 is incident to the boundry vertex V 1. Without loss of generlity we cn ssume tht the vertex V 1 is the left endpoint of the edge e 1. Then on this edge one gets the Cuchy problem with zero initil dt which hs no non-trivil solutions. Corollry 1. If the grph Γ δ is tree, the opertor A min is simple. The proof follows immeditely from Lemm 4. Lemm 5. If n edge of the grph Γ δ is incident to vertex of δ-type, then on this edge the eqution A min f = 0 dmits only the trivil solution. Proof. LetV k bevertexofδ-typendlettheedgee k beincidenttothisvertex. Without loss of generlity one ssumes tht the vertex V k is the left endpoint of the edge e k. Then the generl solution on this edge is f k (x) = C k x. If the right endpoint of the edge e k is vertex of δ-type, the equlity C k l k = 0 must hold; if on the other hnd the right endpoint of the edge e k is vertex of δ -type, one hs C k = 0. In both cses obviously f k (x) = 0. These results llow to receive the criterion of simplicity for symmetric opertor ssocited with grph. Theorem 1. The opertor A min ssocited with the grph Γ δ is not simple iff Γ δ contins t lest one of the following subgrphs: 1) loop; 2) cycle, ll edges of which hve pirwise rtionlly dependent lengths; 3) cycle with n even number of edges, ll vertices of which re of δ -type; 4) subgrph with ll vertices of δ -type, consisting of two cycles with odd numbers of edges ech which re connected by chin or hve exctly one common vertex. Proof. First observe tht the opertor A min is simple iff it is simple on every connected component of the grph. Therefore henceforth without loss of generlity consider connected grph Γ δ. If the grph Γ δ contins the subgrph of the form (1) (4) of Theorem, then ccording to Lemms 1 3 the opertor A min ssocited with the grph Γ δ is not simple. Now ssume tht the grph Γ δ doesn t contin the subgrphs of the form (1) (4) of Theorem. Let us nlyze solutions to the eqution A min f = λf on the grph Γ δ. ) First consider solutions for λ 0. Since the lengths of t lest two edges of every cycle belonging to the grph Γ δ re rtionlly independent, by Lemm 3 one gets only the trivil solution on every cycle. After removl of ll cycles, the grph breks up into collection of trees. On ech of these the solution must be trivil by Corollry 1. b) Consider the cse λ = 0. By Lemm 5, the eqution A min f = 0 dmits only the trivil solution on ll edges incident to vertices of δ-type. Therefore we cn ssume tht the grph Γ δ breks up into connected components hving vertices of δ -type only. Cutting wy in ccordnce with Lemm 4 ll edges contining pendnt vertices, one ends up with isolted cycles with odd numbers of edges (see the conditions (3) nd (4) of Theorem). Observe tht the cycles cnnot hve djcent edges since otherwise one could

7 A SIMPLICITY CRITERION FOR SYMMETRIC OPERATOR ON A GRAPH 123 single out cycle hving n even number of edges, which is impossible by ssumptions of Theorem. Consider solution to the eqution A min f = 0 on cycle with n odd number of edges n. This solution must be of the form f k (x) = C k on ech of the edges of this grph, nd constnts C j t every vertex V k re subject to the following conditions: C k 1 +C k = 0, k = 2,n, C 1 +C n = 0. It is esy to see tht this system of liner equtions dmits no solution but the trivil one. Hence for λ = 0 there exist no non trivil solutions of the eqution A min f = λf s well. Therefore the opertor A min ssocited to the grph Γ δ hs no point spectrum nd is thus simple [3]. We re grteful to A. V. Kiselev for suggesting the problem nd useful dvices. References 1. J. Bomn, P. Kursov, Symmetries of quntum grphs nd the inverse scttering problem, Adv. Appl. Mth. 35 (2005), V. A. Derkch, M. M. Mlmud, Generlised resolvents nd the boundry vlue problems for Hermitin opertors with gps, J. Funct. Anl. 95 (1991), Yu. Ershov, A. V. Kiselev, Trce formule for grph Lplsins with pplictions to recovering mtching conditions, Methods Funct. Anl. Topology 18 (2012), no. 4, P. Exner, P. Sheb, Free quntum motion on brnching grph, Rep. Mth. Phys. 28 (1989), N. I. Gersimenko, B. S. Pvlov, Scttering problem on noncompct grphs, Theor. Mth. Phys. 74 (1988), N. I. Gersimenko, Inverse scttering problem on noncompct grphs, Theor. Mth. Phys. 75 (1988), V. I. Gorbchuk nd M. L. Gorbchuk, Boundry Vlue Problems for Opertor Differentil Equtions, Kluwer Acdemic Publishers, Dordrecht Boston London, (Russin edition: Nukov Dumk, Kiev, 1984) 8. A. N. Kochubei, On chrcteristic functions of symmetric opertors nd their extensions, Izv. Akd Nuk Armenin SSR, Ser. Mt. 15 (1980), no. 3, (Russin) 9. V. Kostrykin, R. Schrder, Kirchhoff s rule for quntum wires, J. Phys. 32 (1999), P. Kuchment, Quntum grphs: n introduction nd brief survey, rxiv: v1 [mthph] 23 Feb P. Kursov, Grph Lplcins nd topology, Ark. Mt. 46 (2008), Yu. P. Mosklev, Yu. S. Smoilenko, Introduction into the Spectrl Theory of Grphs, Centre of eductionl literture, Kyiv, (Russin) 13. K. Ruedenberg, C. W. Scherr, Free-electron network model for conjugted systems. I. Theory, J. Chem. Phys. 21 (1953), Tvrid Ntionl V. I. Verndsky University, 4 Acdemicin Verndsky Ave., Simferopol, 95007, Ukrine E-mil ddress: work1991@ukr.net Tvrid Ntionl V. I. Verndsky University, 4 Acdemicin Verndsky Ave., Simferopol, 95007, Ukrine E-mil ddress: kndgur@yndex.u Tvrid Ntionl V. I. Verndsky University, 4 Acdemicin Verndsky Ave., Simferopol, 95007, Ukrine E-mil ddress: i krpenko@ukr.net Received 28/10/2013; Revised 12/11/2013