Approximation of a general singular vertex coupling in quantum graphs

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1 Approximation of a general singular vertex coupling in quantum graphs Taksu Cheon a, Pavel Exner b,c, Onřej Turek b, a Laboratory of Physics, Kochi University of Technology Tosa Yamaa, Kochi , Japan b Doppler Institute for Mathematical Physics an Applie Mathematics, Czech Technical University Břehová 7, 59 Prague, Czech Republic c Department of Theoretical Physics, Nuclear Physics Institute, Czech Acaemy of Sciences 2568 Řež near Prague, Czech Republic Department of Mathematics, Faculty of Nuclear Sciences an Physical Engineering, Czech Technical University Trojanova 3, 2 Prague, Czech Republic Abstract The longstaning open problem of approximating all singular vertex couplings in a quantum graph is solve. We present a construction in which the eges are ecouple; an each pair of their enpoints is joine by an ege carrying a δ potential an a vector potential couple to the loose eges by a δ coupling. It is shown that if the lengths of the connecting eges shrink to zero an the potentials are properly scale, the limit can yiel any prescribe singular vertex coupling, an moreover, that such an approximation converges in the norm-resolvent sense. Key wors: quantum graphs, bounary conitions, singular vertex coupling, quantum wires PACS: w, 3.65.Db, 73.2.Hb. Introuction While the origin of the iea to investigate quantum mechanics of particles confine to a graph was conceive originally to aress to a particular physical problem, namely the spectra of aromatic hyrocarbons [], the motivation was quickly lost an for a long time the problem remaine rather an obscure textbook example. This change in the last two ecaes when the progress of microfabrication techniques mae graph-shape structures of submicron sizes technologically important. This generate an intense interest to investigation of quantum graph moels which went beyon the nees of practical applications, since these moels prove to be an excellent laboratory to stuy various properties of quantum systems. The literature on quantum graphs is nowaays huge; we limit ourselves to mentioning the recent volume [2] where many concepts are iscusse an a rich bibliography can be foun. The essential component of quantum graph moels is the wavefunction coupling in the vertices. While often the most simple matching conitions ubbe free, Kirchhoff, or Neumann or the slightly more general δ coupling in which the functions are continuous in the vertex are aresses: taksu.cheon@kochi-tech.ac.jp Taksu Cheon, exner@ujf.cas.cz Pavel Exner, turekon@fjfi.cvut.cz Onřej Turek Preprint submitte to Elsevier August 9, 29

2 use, these cases represent just a tiny subset of all amissible couplings. The family of the latter is etermine by the requirement that the corresponing Hamiltonian is a self-ajoint operator, or in physical language, that the probability current is conserve at the vertices. It is not ifficult to fin all the amissible conitions mathematically; if the vertex joins n eges they contain n 2 free parameters, an with exception of the one-parameter subfamily mentione above they are all singular in the sense that the wavefunctions are iscontinuous at the vertex. What is much less clear is the physical meaning of such conitions. It is longstaning open problem whether an in what sense one can approximate all the singular couplings by regular ones epening on suitable parameters, an the aim of the present paper is to answer this question by presenting such a construction, minimal in a natural sense using n 2 real parameters, an to show that the closeness is achieve in the norm-resolvent sense, so the convergence of all types of the spectra an the corresponing eigenprojections is guarantee. The key iea comes from a paper of one of us with Shigehara [3] which showe that a combination of regular point interactions on a line approaching each other with the coupling scale in a particular way w.r.t. the interaction istance can prouce a singular point interaction. Later it was emonstrate [4] that the convergence in this moel is norm-resolvent an the scaling choice is highly non-generic. The iea was applie by two of us to the simplest singular coupling, the so-calle δ s, in [5] an was emonstrate to work; the question was how much it can be extene. Two other of us examine it [6] an foun that with a larger number of regular interactions one can eal with families escribe by 2n parameters, an changing locally the approximating graph topology one can eal with all the couplings invariant with respect to the time reversal which form an n 2 -parameter subset. It was clear that to procee beyon the time-reversal symmetry one has to involve vector potentials similarly as it is was one in the simplest situation in [7]. In this paper we present such a construction which contains parameters breaking the symmetry an which at the same time is more elegant than that of [6] in the sense that the neee ornamentation of the graph is minimal: we isconnect the n eges at the vertex an join each pair of the so obtaine free ens by an aitional ege which shrinks to a point in the limit. The number of parameters leans on the ecomposition n 2 = n 2 n 2, where the first summan, n, correspons to δ couplings of the outer ege enpoints with those of the ae shrinking ones. The secon summan can be consiere as n 2 times two parameters: one is a δ potential place at the ege, the other is a vector potential supporte by it. Our result shows that any singular vertex coupling can be approximate by a graph in which the vertex is replace by a local graph structure in combination with local regular interactions an local magnetic fiels. This opens way to constructing structure vertices tailore to the esire conuctivity properties, even tunable ones, if the interactions are controlle by gate electroes, however, we are not going to elaborate such approximations further in this paper. We have to note for completeness that the problem of unerstaning vertex couplings has also other aspects. The approximating object nees not to be a graph but can be another geometrical structure. A lot of attention was pai to the situation of fat graphs, or networks of this tubes built aroun the graph skeleton. The two approaches can be combine, for instance, by lifting the graph results to fat graphs. In this way approximations to δ an δ s couplings by suitable families of Schröinger operators on such manifols with Neumann bounaries were recently emonstrate in [8]. The results of this paper can be similarly lifte to manifols; that will be the subject of a subsequent work. Let us review briefly the contents of the paper. In the next section we gather the neee 2

3 preliminary information. We review the information about vertex couplings an erive a new parametrization of a general coupling suitable for our purposes. In Section 3 we escribe in etail the approximation sketche briefly above an show that on a heuristic level it converges to a chosen vertex coupling. Finally, in the last section we present an prove our main result showing that the sai convergence is not only formal but it is vali also in the norm-resolvent sense. 2. Vertex coupling in quantum graphs Let us first recall briefly a few basic notions; for a more etaile iscussion we refer to the literature given in the introuction. The object of our interest are Schröinger operators on metric graphs. A graph is conventionally ientifie with a family of vertices an eges; it is metric if each ege can be equippe with a istance, i.e. to be ientifie with a finite or semi-infinite interval. We regar such a graph Γ with eges E,..., E n as a configuration space of a quantum mechanical system, i.e. we ientify the orthogonal sum H = n j= L2 E j with the state Hilbert space an the wave function of a spinless particle living on Γ can be written as the column Ψ = ψ, ψ 2,..., ψ n T with ψ j L 2 E j. In the simplest case when no external fiels are present the system Hamiltonian acts as H Γ Ψ j = ψ j, with the omain consisting of functions from W 2,2 Γ := n j= W2,2 E j. Not all such functions are amissible, though, in orer to make the operator self-ajoint we have to require that appropriate bounary conitions are satisfie at the vertices of the graph. We restrict our attention to the physically most interesting case when the bounary conitions are local, coupling values of the functions an erivatives is each vertex separately. Our aim is explain the meaning of a general vertex coupling using suitable approximations; the local character means that we can investigate how such a system behaves in the vicinity of a single vertex. A prototypical example of this situation is a star graph with one vertex in which a finite number of semi-infinite eges meet; this is the case we will mostly have in min in the following. Let us thus consier a graph vertex V of egree n, i.e. with n eges connecte at V. We enote these eges by E,..., E n an the components of the wave function values at them by ψ x,..., ψ n x n. We choose the coorinates at the eges in such a way that x j for all j =,..., n, an the value x j = correspons to the vertex V. For notational simplicity we put Ψ V = ψ,..., ψ n T an Ψ V = ψ,..., ψ n T. Since our Hamiltonian is a secon-orer ifferential operator, the sought bounary conitions will couple the above bounary values, their most general form being AΨ V BΨ V =, where A an B are complex n n matrices. To ensure self-ajointness of the Hamiltonian, which is in physical terms equivalent to conservation of the probability current at the vertex V, the matrices A an B cannot be arbitrary but have to satisfy the following two conitions, ranka B = n, the matrix AB is self-ajoint, 2 where A B enotes the n 2n matrix with A, B forming the first an the secon n columns, respectively, as state for the first time by Kostrykin an Schraer [9]. The relation together 3

4 with conitions 2 for brevity, we will write &2 escribe all possible vertex bounary conitions giving rise to a self-ajoint Hamiltonian; we will speak about amissible bounary conitions. On the other han, it is obvious that the formulation &2 is non-unique in the sense that ifferent pairs A, B, A 2, B 2 may efine the same vertex coupling, as A, B can be equivalently replace by CA, CB for any regular matrix C C n,n. To overcome this ambiguity, Harmer [], an inepenently Kostrykin an Schraer [] propose a unique form of the bounary conitions, namely U IΨ V iu IΨ V =, 3 where U is a unitary n n matrix. Note that in a more general context such conitions were known before [2], see also [3]. The natural parametrization 3 of the family of vertex couplings has several avantages in comparison to &2, besies its uniqueness it also makes obvious how large the family is: since the unitary group Un has n 2 real parameters, the same is true for vertex couplings in a quantum graph vertex of the egree n. Of course, this fact is also clear if one interprets the couplings from the viewpoint of self-ajoint extensions [4]. On the other han, among the isavantages of the formulation 3 one can mention its complexity: vertex couplings that are simple from the physical point of view may have a complicate escription when expresse in terms of the conition 3. As an example, let us mention in the first place the δ-coupling with a parameter α R, characterize by relations ψ j = ψ k =: ψ, j, k =..., n, for which the matrix U use in 3 has entries given by U jk = n j= ψ j = αψ, 4 2 n iα δ jk, 5 δ jk being the Kronecker elta. When we substitute 5 into 3 an compare with 4 rewritten into a matrix form, we observe that the first formulation is not only more complicate with respect to the latter, but also contains complex values whereas the latter oes not. This is a reason why it is often better to work with simpler expressions of the type &2. Another aspect of this parametrization ifference concerns the meaning of the parameters. Since the n 2 ones mentione earlier are encapsulate in a unitary matrix, it is ifficult to unerstan which role each of them plays. On the other han, both formulations &2 an 3 have a common feature, namely that they have a form insensitive to a particular ege numbering. If the eges are permute one has just to replace the matrices A, B an U by Ã, B an Ũ, respectively, obtaine by the appropriate rearrangement of rows an columns. This may hie ifferent ways in which the eges are couple; it is easy to see that a particular attention shoul be pai to singular situations when the matrix U has eigenvalues equal to ±. Since the type of the coupling will be important for the approximation we are going to construct, we will rewrite the vertex coupling conitions in another form which is again simple an unique but requires an appropriate ege numbering. This will be one in Theorem 2., before 4

5 stating it we introuce several symbols that will be employe in the further text, namely C k,l the set of complex matrices with k rows an l columns, ˆn the set {, 2,..., n}, I n the ientity matrix n n. To be precise, let us remark that the term numbering with respect to the eges connecte in the graph vertex of the egree n means strictly numbering by the elements of the set ˆn. Theorem 2.. Let us consier a quantum graph vertex V of the egree n. i If m n, S C m,m is a self-ajoint matrix an T C m,n m, then the equation I m T Ψ V = S T I n m Ψ V 6 expresses amissible bounary conitions. This statement hols true for any numbering of the eges. ii For any vertex coupling there exist a number m n an a numbering of eges such that the coupling is escribe by the bounary conitions 6 with the uniquely given matrices T C m,n m an self-ajoint S C m,m. iii Consier a quantum graph vertex of the egree n with the numbering of the eges explicitly given; then there is a permutation Π S n such that the bounary conitions may be written in the moifie form I m T for Ψ V = ψ Π. ψ Πn Ψ V = S Ψ T I n m V 7, Ψ V = ψ Π. ψ Πn where the self-ajoint matrix S C m,m an the matrix T C m,n m epen unambiguously on Π. This formulation of bounary conitions is in general not unique, since there may be ifferent amissible permutations Π, but one can make it unique by choosing the lexicographically smallest permutation Π. Proof. The claim iii is an immeiate consequence of ii using a simultaneous permutation of elements in the vectors Ψ V an Ψ V, so we have to prove the first two. As for i, we have to show that the vertex coupling with matrices S I n T A = T I n m an B =,, conform with 2. We have S I m T rank T I n m I m S T = rank I n m T 5 = n

6 an S I n T T I n m = S the latter matrix is self-ajoint since S = S, thus 2 is satisfie. Now we procee to ii. Consier a quantum graph vertex of the egree n with an arbitrary fixe vertex coupling. Let Ψ V an Ψ V enote the vectors of values an erivatives of the wave function components at the ege ens; the orer of the components is arbitrary but fixe an the same for both vectors. We know that the coupling can be escribe by bounary conitions with some A, B C n,n satisfying 2. Our aim is to fin a number m n, a certain numbering of the eges an matrices S an T such that the bounary conitions are equivalent to 6. Moreover, we have to show that such a number m is the only possible an that S, T epen uniquely on the ege numbering. When proceeing from to 6, we may use exclusively manipulations that o not affect the meaning of the coupling, namely simultaneous permutation of columns of the matrices A, B combine with corresponing simultaneous permutation of components in Ψ V an Ψ V, multiplying the system from left by a regular matrix. We see from 6 that m is equal to the rank of the matrix applie at Ψ V. We observe that the rank of this matrix, as well as of that applie at Ψ V, is not influences by any of the manipulations mentione above, hence it is obvious that m = rankb an that such a choice is the only possible, i.e. m is unique. Since rankb = m with m {,..., n}, there is an m-tuple of linearly inepenent columns of the matrix B; suppose that their inices are j,..., j m. We permute simultaneously the columns of B an A so that those with inices j,..., j m are now at the positions,..., m, an the same we o with the components of the vectors Ψ V, Ψ V. Labelling the permute matrices A, B an ; vectors Ψ V, Ψ V with tiles, we get à ΨV B Ψ V =. 8 Since rank B = rankb = m, there are m rows of B that are linearly inepenent, let their inices be i,..., i m, an n m rows that are linear combinations of the preceing ones. First we permute the rows in 8 so that those with inices i,..., i m are put to the positions,..., m; note that it correspons to a matrix multiplication of the whole system 8 by a permutation matrix which is regular from the left, i.e. an authorize manipulation. In this way we pass from à an B to matrices which we enote as Ǎ an ˇB; it is obvious that this operation keeps the first m columns of the matrix ˇB linearly inepenent. In the next step we a to each of the last n m rows of Ǎ Ψ ˇB Ψ = such a linear combination of the first m rows that all the last n m rows of ˇB vanish. This is possible, because the last n m lines of ˇB are linearly epenent on the first m lines. It is easy to see that it is an authorize operation, not changing the meaning of the bounary conitions; the resulting matrices at the LHS will be enote as ˆB an Â, i.e. Â Ψ V ˆB Ψ V =. 9 From the construction escribe above we know that the matrix ˆB has a block form, ˆB ˆB = ˆB 2, 6

7 where ˆB C m,m an ˆB 2 C m,n m ; the square matrix ˆB C m,m is regular, because its columns are linearly inepenent. We procee by multiplying the system 9 from the left by the matrix ˆB I n m, arriving at bounary conitions A A 2 I m B Ψ A 2 A V 2 22 Ψ V =, where B 2 = ˆB ˆB 2. Bounary conitions are equivalent to, therefore they have to be amissible. In other A A wors, the matrices 2 I m B an 2 have to satisfy both the conitions 2, A 2 A 22 which we are now going to verify. Let us begin with the secon one. We have A A 2 A 2 A 22 I m B 2 = A A 2 B 2 A 2 A 22 B 2 an this matrix is self-ajoint if an only if A A 2 B 2 is self ajoint an A 2 A 22 B 2 =. We infer that A 2 = A 22 B 2, hence conition acquires the form A A 2 I m B Ψ A 22 B 2 A V 2 Ψ 22 V =. The first one of the conitions 2 says that A rank A 2 I m B 2 A 22 B 2 A 22 = n, hence rank A 22 B 2 A 22 = n m. Since A22 B 2 A 22 = A22 B 2 In m we obtain the conition ranka 22 = n m, i.e. A 22 must be a regular matrix. It allows us to multiply the equation from the left by the matrix I m A 2 A 22 A, 22 which is obviously well-efine an regular; this operation leas to the conition A A 2 B 2 I m B Ψ B 2 I n m V 2 Ψ V =. If follows from our previous consierations that the square matrix A A 2 B 2 is self-ajoint. If we enote it as S, rename the block B 2 as T an transfer the term containing Ψ V to the right han sie, we arrive at bounary conitions I m T Ψ V = S Ψ T I n m V. 2 7

8 The orer of components in Ψ V an Ψ V etermines just the appropriate numbering, in other wors, the vectors Ψ V an Ψ V represent exactly what we unerstoo by Ψ V an Ψ V in the formulation of the theorem. Finally, the uniqueness of the matrices S an T with respect to the choice of the permutation Π is a consequence of the presence of the blocks I m an I n m. First of all, the block I n m implies that there is only one possible T, otherwise the conitions for ψ m,..., ψ n woul change, an next, the block I m together with the uniqueness of T implies that there is only one possible S, otherwise the conitions for ψ,..., ψ m woul change. Remark 2.2. The expression 7 implies, in particular, that if B has not full rank, the number of real numbers parametrizing the vertex coupling is reuce from n 2 to at most m2n m = n 2 n m 2, where m = rankb. Another reuction can come from a lower rank of the matrix A. Remark 2.3. The proceure of permuting columns an applying linear transformations to the rows of the system has been one with respect to the matrix B, but one can start by same right from the matrix A as well. In this way we woul obtain similar bounary conitions as 6, only the vectors Ψ V an Ψ V woul be interchange. Theorem 2. can be thus formulate with Equation 6 replace by I m T Ψ V = S T I n m Ψ V. For completeness sake we a that another possible forms of Equation 6 in Theorem 2. are S I m T T I n m Ψ V Ψ V = an I m T Ψ V S T I n m Ψ V = ; having the stanarize form AΨ V BΨ V =, last two formulations may be sometimes more convenient than 6. Obviously, an analogous remark applies to Equation 7. Remark 2.4. A formulation of bounary conitions with a matrix structure singling out the regular part as in 7 has been erive in a ifferent way by P. Kuchment [5]. Recall that in the setting analogous to ours he state existence of an orthogonal projector P in C n with the complementary projector Q = I P an a self-ajoint operator L in QC n such that the bounary conitions may be written in the form PΨ V = QΨ V LQΨ V =. 3 Let us briefly explain how P. Kuchment s form iffers from 7. When transforme into a matrix form, 3 consists of two groups of n linearly epenent equations. If we then naturally extract a single group of n linearly inepent ones, we arrive at a conition with a structure similar to, i. e. the upper right submatrix staning at Ψ V is generally a nonzero matrix m n m. In other wors, whilst P. Kuchment aime to ecompose the bounary conitions with respect to 8

9 two complementary orthogonal projectors, our aim was to obtain a unique matrix form with as many vanishing terms as possible; the form 6 turne out to have a highly suitable structure for solving the problem of approximations that we are going to analyze in the rest of the paper. To conclue this introuctory section, let us summarize main avantages an isavantages of the conitions 6 an 7. They are unique an exhibit a simple an clear corresponence between the parameters of the coupling an the entries of matrices in 6, furthermore, the matrices in 6 are relatively sparse. On the negative sie, the structure of matrices in 6 epens on rankb an the vertex numbering is not fully permutable. 3. The approximation arrangement We have argue above that ue to a local character one can consier a single-vertex situation, i.e. star graph, when asking about the meaning of the vertex coupling. In this section we consier such a quantum graph with general matching conitions an show that the singular coupling may be unerstoo as a limit case of certain family of graphs constructe only from eges connecte by δ-couplings, δ-interactions, an supporting constant vector potentials. Following the above iscussion, one may consier the bounary conitions of the form 6, renaming the eges if necessary. It turns out that for notational purposes it is avantageous to aopt the following convention on a shift of the column inices of T: Convention 3.. The lines of the matrix T are inexe from to m, the columns are inexe from m to n. Now we can procee to the escription of our approximating moel. Consier a star graph with n outgoing eges couple in a general way given by the conition 7. The approximation in question looks as follows cf. Fig.: We take n halflines, each parametrize by x [,, with the enpoints enote as V j, an put a δ-coupling to the eges specifie below with the parameter v j at the point V j for all j ˆn. Certain pairs V j, V k of halfline enpoints will be joine by eges of the length 2, an the center of each such joining segment will be enote as W { j,k}. For each pair { j, k}, the points V j an V k, j k, are joine if one of the following three conitions is satisfie keep in min Convention 3.: j ˆm, k m, an T jk or j m, k ˆm, an T k j, 2 j, k ˆm an l m T jl T kl, 3 j, k ˆm, S jk, an the previous conition is not satisfie. At each point W { j,k} we place a δ interaction with a parameter w { j,k}. From now on we use the following convention: the connecting eges of the length 2 are consiere as compose of two line segments of the length, on each of them the variable runs from corresponing to the point W { j,k} to corresponing to the point V j or V k. On each connecting segment escribe above we put a vector potential which is constant on the whole line between the points V j an V k. We enote the potential strength between the points W { j,k} an V j as A j,k, an between the points W { j,k} an V k as A k, j. It follows from the continuity that A k, j = A j,k for any pair { j, k}. 9

10 V j j V k W{j,k} k Figure : The scheme of the approximation. All inner links are of length 2. Some connection links may be missing if the conitions given in the text are not satisfie. The quantities corresponing to the inex pair { j, k} are marke, an the grey line symbolizes the vector potential A j,k. The choice of the epenence of v j, w { j,k} an A j,k on the parameter is crucial for the approximation an will be specifie later. It is useful to introuce the set N j ˆn containing inices of all the eges that are joine to the j-th one by a connecting segment, i.e. N j ={k ˆm S jk } {k ˆm l m T jl T kl } {k m T jk } for j ˆm 4 N j ={k ˆm T k j } for j m The efinition of N j has these two trivial consequences, namely k N j j N k 5 j m N j ˆm 6 For the wave function components on the eges we use the following symbols: the wave function on the j-th half line is enote by ψ j, the wave function on the line connecting points V j an V k has two components: the one on the line between W { j,k} an V j is enote by ϕ j,k, the one on the half between the mile an the enpoint of the k-th half line is enote by ϕ k, j. We remin once more the way in which the variable x of ϕ j,k an ϕ k, j is consiere: it grows from at the point W { j,k} to at the point V j or V k, respectively. Next we escribe how the δ couplings involve look like; for simplicity we will refrain from inicating in the bounary conitions the epenence of the parameters u, v j, w { j,k} on the istance. The δ interaction at the ege connecting the j-th an k-th half line of course, for j, k ˆn such that k N j only is expresse through the conitions ϕ j,k = ϕ k, j =: ϕ { j,k}, ϕ j,k ϕ k, j = w { j,k} ϕ { j,k}, 7

11 the δ coupling at the enpoint of the j-th half line j ˆn means ψ j = ϕ j,k for all k N j, ψ j k N j ϕ j,k = v jψ j. 8 Further relations which will help us to fin the parameter epenence on come from Taylor expansion. Consier first the case without any ae potential, ϕ j,k = ϕ { j,k} ϕ j,k O2, ϕ j,k = ϕ j,k O, j, k ˆn. 9 To take the effect of ae vector potentials into account, the following lemma will prove useful: Lemma 3.2. Let us consier a line parametrize by the variable x, L, L, { }, an let H enote a Hamiltonian of a particle on this line interacting with a potential V, H = 2 x 2 V, 2 sufficiently regular to make H self-ajoint. We enote by ψ s,t the solution of Hψ = k 2 ψ with the bounary values ψ s,t = s, ψ s,t = t. Consier the same system with a vector potential A ae, again sufficiently regular; the Hamiltonian is consequently given by H A = i 2 x A V. 2 Let ψ s,t A enote the solution of H Aψ = k 2 ψ with the same bounary values as before, i.e. ψ s,t A = s, ψ s,t A = t. Then the function ψ s,t A can be expresse as Proof. The aim is to prove that implies ψ s,t A x = ei x Azz ψ s,t x for all x, L. ψ s,t Vψ s,t = k 2 ψ s,t ψ s,t = s ψ s,t = t i x A 2 e i x Azz ψ s,t V e i x Azz ψ s,t = k 2 e i x Azz ψ s,t an ψ s,t A = s, ψs,t A = t. The last part is obvious, since the exponential factor involve is equal to one, hence it suffices to prove the isplaye relation. It is straightforwar that the Hamiltonian H A acts generally as H A = 2 x 2iA 2 x ia A 2 V. We substitute e i x Azz ψ s,t for ψ, obtaining [ HA e i x Azz ψ s,t] x = 2 x e i x 2 Azz ψ s,t x 2iAx e i x Azz ψ s,t x ia x Ax 2 Vx e i x Azz ψ s,t x. x

12 Now we express the erivatives applying the formula x Azz = Ax. Most of the terms then cancel, it remains only [ HA e i x Azz ψ s,t] x = e i x Azz ψ s,t x Vx ψ s,t x. Due to the assumption ψ s,t Vψ s,t = k 2 ψ s,t, we have [ HA e i x Azz ψ s,t] x = k 2 e i x Azz ψ s,t x, what we have set out to prove. The lemma says that aing a vector potential on an ege of a quantum graph has a very simple effect of changing the phase of the wave function by the value x Azz. We will work in this paper with the special case of constant vector potentials on the connecting segments of the lengths 2, hence the phase shift will be given here as a prouct of the value A an the length in question. Lemma 3.2 has the following very useful consequence. Corollary 3.3. Consier a quantum graph vertex with n outgoing eges inexe by,..., n an parametrize by x, L j. Suppose that there is a δ coupling with the parameter α at the vertex, an moreover, that there is a constant vector potential A j on the j-th ege for all j ˆn. Let ψ j enote the wave function component on the j-the ege. Then the bounary conitions acquire the form x ψ j = ψ k =: ψ for all j, k ˆn, 22 n n ψ j = α i A j ψ, 23 j= where ψ j, ψ j, etc., stan for the one-sie right limits at x =. In other wors the effect of the vector potentials on the bounary conitions corresponing to a pure δ coupling is the following: the continuity is not affecte, the coupling parameter is change from α to α i n j= A j. Proof. Consier first the situation without any vector potentials. If ψ j, j ˆn, enote the wave function components corresponing to this case, the bounary conitions expressing the δ coupling have the form 4, i.e. j= ψ j = ψ k =: ψ for all j, k ˆn, nj= ψ j = αψ. 24 If there are vector potentials on the eges, A j on the j-th ege, one has in view of the previous lemma, ψ j x = e ia jx ψ j x, i.e. ψ j x = e ia jx ψ j x, ψ j x = e ia j x ψ j x = e ia jx ψ j x ia j e ia jx ψ j x. x 2

13 Thence we express ψ j an ψ j : they are equal to substituting them to 24 we obtain ψ j = ψ j, ψ j = ψ j ia j ψ j ; ψ j = ψ k =: ψ for all j, k ˆn, n ψ j ia j ψ j = αψ. j= The first line expresses the continuity of the wavefunction in the vertex supporting the δ coupling in the same way as in the absence of vector potentials, whereas the secon line shows how the conition for the sum of the erivatives is change. With the continuity in min, we may replace ψ j by ψ obtaining n n ψ j = α i A j ψ, which finishes the proof. j= Recall that approximating we are constructing supposes that constant vector potentials are ae on the joining eges. If an ege of the length 2 joins the enpoints of the j-th an k-th half line, there is a constant vector potential of the value A j,k on the part of the length closer to the j-th half line an a constant vector potential of the value A k, j = A j,k on the part of the length closer to the k-th half line. With regar to Lemma 3.2, the impact of the ae potentials consists in phase shifts by A j,k an A k, j. Let us inclue this effect into the corresponing equations, i.e. into 9: j= ϕ j,k = e ia j,k ϕ { j,k} ϕ j,k O2, ϕ j,k = eia j,k ϕ j,k O, j, k ˆn. 25 The system of equations 7, 8, an 25 escribes the relations between values of wave functions an their erivatives at all the vertices. Next we will eliminate the terms with the auxiliary functions ϕ { j,k} an express the relations between 2n terms ψ j, ψ j, j ˆn. We begin with the first one of the relations 25 together with the continuity requirement 8, which yiels ϕ j,k = e ia j,k ψ j ϕ { j,k} O The same relation hols with j replace by k, summing them together an using the secon of the relations 7 we get 2 w{ j,k} ϕ{ j,k} = e ia j,k ψ j e ia k, j ψ k O 2. We express ϕ { j,k} from here an substitute into the first of the equations 25; using at the same time the first relation of 8 we get ψ j = e ia j,k e ia j,k ψ j e ia k, j ψ k O 2 ϕ j,k 2 w O 2, { j,k} 3

14 an consiering the secon of the equations 25, we have ψ j = ψ j e ia j,k A k, j ψ k O 2 2 w { j,k} ϕ j,k O2. Since the values of vector potentials are suppose to have the antisymmetry property, A k, j = A j,k, we may simplify the above equation to ψ j = ψ j e 2iA j,k ψ k O 2 2 w { j,k} ϕ j,k O2. 27 Summing the last equation over k N j yiels #N j ψ j = ψ j 2 w k N { j,k} j e2ia j,k ψ k 2 w k N { j,k} j ϕ j,k k N j O2 O 2 2 w k N { j,k} j #N j enotes the carinality of N j, an with the help of the secon of the relations 8 we arrive at the final expression, ψ j = v j #N j 2 w k N { j,k} ψ j eia j,k Ak, j ψ k 2 w j k N { j,k} j O2 O w k N { j,k} j Our objective is to choose v j, w { j,k} an A j,k in such a way that in the limit the system of relations 28 with j ˆn tens to the system of n bounary conitions 7. The lines of 7 are of two types, let us recall: ψ j n l=m m T jl ψ l = S jk ψ k j ˆm 29 = k= m T k j ψ k ψ j j = m,..., n. 3 k= We point out here with reference to 4 that these relations may be written also with the summation inices running through the restricte sets, namely ψ j l N j \ ˆm m T jl ψ l = S jk ψ k j ˆm 3 k= = T k j ψ k ψ j j = m,..., n, 32 k N j since for any pair j ˆm, l {m,, n} the implication T jl l N j hols, see also Eqs. 5, 6. 4

15 When looking for a suitable epenence of v j, w { j,k} an A j,k on, we start with Eq. 28 in the case when j m. Our aim is to fin conitions uner which 28 tens to 32 as. It is obvious that the sufficient conitions are lim v j #N j 2 w { j,k} R\{}, 33 lim k N j 2 w { j,k} R\{} k N j, 34 e 2iA j,k 2 w { j,k} v j #N j h N j 2 w { j,h} = T k j k N j. 35 Now we procee to the case j ˆm. Our approach is base on substitution of 28 into the left-han sie of 3 an a subsequent comparison of the right-han sies. The substitution is straightforwar, ψ j T jl ψ l = v j #N j 2 w l N j \ ˆm h N { j,h} ψ j e2ia j,k ψ k 2 w j k N { j,k} j T jl v l #N l 2 w l N j \ ˆm h N {l,h} ψ l ψ e2ial,k k 2 w l k N {l,k} l O n O T jl 2 w { j,k} O O 2 w {l,h}, 36 k N j l=m then we apply two ientities, which can be easily proven, namely i e2ia j,k ψ k = 2 w k N { j,k} j k N j ˆm e 2iA j,k ψ k 2 w { j,k} l N j \ ˆm e 2iA j,l ψ l 2 w { j,l}, h N l ii ψ T e2ial,k k jl 2 w k N {l,k} l e 2iA l, j = T jl 2 w {l, j} ψ j l N j \ ˆm l N j \ ˆm k N j ˆm l N k \ ˆm T jl e 2iA l,k 2 w {l,k} ψ k 5

16 an obtain ψ j l N j \ ˆm T jl ψ l = v j #N j e 2iA l, j T jl 2 w h N { j,h} 2 w j l N j \ ˆm {l, j} ψ j e 2iA j,k e 2iA l,k T jl 2 w k N j ˆm { j,k} 2 w l N k \ ˆm {l,k} ψ k e 2iA j,l T jl 2 w { j,l} v l #N l 2 w {l,h} ψ l l N j \ ˆm O O 2 w { j,k} k N j h N l k N j ˆm l N k N j \ ˆm O 2 w {l,k}. 37 As we have announce above, the goal is to etermine terms v j, w { j,k} an A j,k such that if, the right-han sie of 37 tens to the eight-han sie of 3 for all j ˆm. We observe that this will be the case provie v j #N j e 2iA j,k 2 w { j,k} e 2iA j,l T jl 2 w { j,l} lim T e2ial, j jl = S j j, 2 w h N { j,h} 2 w j l N {l, j} j 38 e 2iA l,k T jl = S jk k N j ˆm, 2 w l N k \ ˆm {l,k} 39 lim v l #N l 2 w {l,h} = l N j\ ˆm, 4 h N l 2 w { j,k} R k N j, 4 2 w {l,k} R k N j ˆm, l N k N j \ ˆm. 42 It is easy to see that the set of equations 4 for j ˆm, l N j \ ˆm is equivalent to the set 35 for j m, k N j. Similarly, Eq. 42 for j ˆm, k N j ˆm, l N k N j \ ˆm is a weaker set of conitions than 34 with j m, k N j. Finally, Eq. 4 reuces for k N j \ ˆm to 34, thus it suffices to consier 4 with k N j ˆm. The proceure of etermination of v j, w { j,k} an A j,k will procee in three steps, at the en we a the fourth step involving the verification of the limit conitions 33, 34, an 4 restricte to k N j ˆm. Step I. We use Eq. 4 to fin an expression for w { j,l} an A j,l when j ˆm an l N j \ ˆm. We begin with rearranging Eq. 35 into the form = e 2iA j,l T jl 2 w { j,l} v l #N l 2 w {l,h} l N j \ ˆm h N l

17 Since all the terms except e 2iA j,l an T jl are real, we can obtain immeiately a conition for A j,l : We put T e 2iA jl / T jl if Re T jl, j,l = T jl / T jl if Re T jl < ; it is easy to see that such a choice ensures that the expression e 2iA j,l T jl is always real. The vector potential strength may be then chosen as follows, A j,l = 2 arg T jl if Re T jl, arg T jl π if Re T jl < 2 for all j ˆm, l N j \ ˆm. We remark that this choice is obviously not the only one possible. Note that in this situation, namely if j ˆm an l N j \ ˆm, the potentials o not epen on. Taking 44 into account, Eq. 43 simplifies to = T jl 2 w { j,l} v l #N l 2 w {l,h} l m, j N l ; 45 h N l note that j ˆm l N j \ ˆm l m j N l. The symbol here has the following meaning: if c C, then { c if Re c, c = c if Re c <. Summing 45 over j N l we get = T jl 2 w j N { j,l} v l #N l 2 w l j N {l,h}, l i.e. h N l T hl = T jl v l #N l. 2 w h N l j N { j,l} l j N l We have to istinguish here two situations: i If h N l T hl, one obtains h N = l T hl 2 w h N {l,h} h N l T hl v l #N l, l an the substitution of the left-han sie into the right-han sie of 45 leas to the formula for /2 w { j,l}, namely 44 v l #N l = T jl 2 w { j,l} h N l T hl j ˆm, l N j \ ˆm. We observe that the sum in the enominator may be compute over the whole set ˆm as well, since h N l T hl =, which slightly simplifies the formula, 2 w { j,l} = T jl v l #N l m 7 h= T hl j ˆm, l N j \ ˆm.

18 From here one can easily express w { j,l}, if v l is known. However, it turns out that v l, l m can be chosen almost arbitrarily, the only requirements are to keep the expression v l #N l nonzero an to satisfy 33, 34 an 4. The simplest choice possible is to efine v l by the expression v l #N l m h= T hl =, which simplifies the expressions for other parameters. Here we obtain alreay expressions for v l an w { j,l} if l m, viz v l = #N l m h= T hl l m, 46 w { j,l} = 2 T jl j ˆm, l N j \ ˆm. 47 ii If h N l T hl =, it hols necessarily v l #N l =, an consequently, v l = #N l. Note that this equation may be obtaine from Eq. 46 by putting formally m h= T hl =. It is easy to check that w { j,l} given by Eq. 47 satisfies 43 in the case h N l T hl = as well. Summing these facts up, we conclue that Eqs. 46, 47 hol universally regarless whether h N l T hl equals zero or not. We woul like to stress that the freeom in the choice of v l is a consequence of the fact mentione in Remark 2.2, namely that the number of parameters of a vertex coupling ecreases with the ecreasing value of rankb. Step II. Equation 39 together with the results of Step I will be use to etermine w { j,k} an A j,k in the case when j ˆm an k N j ˆm. From 39 we have e2ia j,k = S jk 2 w { j,k} l N k \ ˆm T jl e 2iA l,k 2 w {l,k} ; the pairs l, k appearing in the sum are of the type examine in Step I, i.e. k ˆm, l N k \ ˆm. Thus one may substitute from 46 an 47 to obtain e2ia j,k = S jk T jl T kl. 2 w { j,k} l N k \ ˆm We observe that the summation inex may run through the whole set ˆn\ ˆm, because l m l N k T kl =. This allows one to obtain a more elegant formula. In a similar way as above, we fin the expression for A j,k, A j,k = n n 2 arg S jk T jl T kl for Re S jk T jl T kl 48a l=m l=m an A j,k = 2 arg S jk n l=m T jl T kl π for Re S jk n l=m T jl T kl <, 48b 8

19 an for w { j,k}, = S jk 2 w { j,k} n l=m T jl T kl. 49 Step III. Substitution of the results of Steps I an II into Eq. 38 provies an expression for v j in the case when j ˆm. A simple calculation gives v j = S j j #N m j S jk n T jl T kl n T jl T jl. 5 k= l=m l=m Since S is a self-ajoint matrix, the term S j j is real, thus the whole right-han sie is a real expression. Step IV. Finally, we verify conitions 33, 34, an 4, the last one being restricte to k N j ˆm. This step consists in trivial substitutions: 33 : lim v j #N j 2 w { j,k} = lim = R\{} j m, k N j 34 : lim 2 w { j,k} = lim 4 : lim 2 w { j,k} = lim j ˆm, k N j ˆm. 4. The norm-resolvent convergence T k j = T k j R\{} j m, k N j, n n S jk T jl T kl = T jl T kl R In the previous section we have shown that any vertex coupling in the center point of a star graph may be regare as a limit of a certain family of graphs supporting nothing but δ couplings, δ interactions an constant vector potentials. The parameter values of all the δ s an vector potentials have been erive using a metho evise originally in [3, 7] for the case of a generalize point interaction on the line. The aim of this section is to give a clear meaning to this convergence. Specifically, we are going to show that the Hamiltonian of the approximating system converges to the Hamiltonian of the approximate system in the norm-resolvent sense, with the natural consequences for the convergence of eigenvalues, eigenfunctions, etc. We enote the Hamiltonian of the star graph Γ with the coupling 6 at the vertex as H A referring to the approximate system, an H Ag will stan for the approximating family of graphs that has been constructe in the previous section. Symbols R A k 2 an R Ag k2 will then enote the resolvents of H A an H Ag at the points k 2 from the resolvent set. Neeless to say, the operators act on ifferent spaces: R A k 2 on L 2 G, where G = R n correspons to the star graph Γ, an R Ag k2 on L 2 G, where l=m l=m G = R n, n j= N j. 5 Our goal is to compare these resolvents. In orer to o that, we nee to ientify R A k 2 with the orthogonal sum R A k2 = R A k 2, 52 9

20 where is a zero operator acting on the space L 2, n j= N j which is remove in the limit. Then both the operators R A k2 an R Ag k2 are efine as acting on functions from the set G which are vector functions with n n j= N j components; we will inex the components by the set I = ˆn {l, h l ˆn, h N l }. 53 In this setting we are able to state now the main theorem of this section an the whole paper. Theorem 4.. Let v j, j ˆn,, w { j,k} j ˆn, k N j an A j,k epen on accoring to 5, 46, 49, 47, 48 an 44, respectively. Then the family H Ag converges to H A in the norm-resolvent sense as. Proof. We have to compare the resolvents R A k2 an R Ag k2. It is obviously sufficient to check the convergence in the Hilbert-Schmit norm, R Ag k2 R A k2 as, 2 in other wors, to show that the ifference of the corresponing resolvent kernels enote as G Ag, k an G A, k, respectively, tens to zero in L 2 G G. Recall that these kernels, or Green s functions, are in our case matrix functions with n n j= N j n nj= N j entries. We will inex the entries by pairs of inices taken from the set I cf. 53. The proof is ivie into three parts. In the first an the secon part we will erive the resolvent kernels G Ag, k an G A, k, respectively, in the last part we compare them an emonstrate the norm-resolvent convergence. I. Resolvent of the approximate Hamiltonian Let us construct first G A k for the star-graph Hamiltonian with the conition at the vertex. We begin with n inepenent halflines with Dirichlet conition at its enpoints; Green s function for each of them is well-known to be G i x, y = sinh x < e x > where x < := min{x, y}, x > := max{x, y}, an we put i = k assuming conventionally Re >. The sought Green s function is then given by Krein s formula [6, App. A], R A k 2 = R Hl k 2, n λ jl k 2 φ l k 2, L φ jk 2, 54 2 R n j,l= where R Hl k 2 acts on each half line as an integral operator with the kernel G i, an for φ j k 2 one can choose any elements of the eficiency subspaces of the largest common restriction; we will work with φ j k 2 x = δ jle x j where the symbol x stans here for the vector x,..., x n l R n. Then we have R A k 2 ψ. ψ n x. x n = G i x, y ψ y y n. λ jl k 2 e y l, ψ l y l L 2 R j,l= G i x n, y n ψ n y n y n 2. e x j.,

21 which shoul be satisfie for any ψ,..., ψ n T n j= L2 R. We observe that for all j ˆn, the j-th component on the right han sie epens only on the variable x j. That is why one can consier the components as functions of one variable; we enote them as g j x j, j ˆn, in other wors, g x R A k 2 ψ. ψ n x. x n The functions g j are therefore given explicitly by g j x j = G i x j, yψ j yy =: n λ jl k 2 l=. g n x n. e y ψ l yy e x j. Since the resolvent maps the whole Hilbert space into the omain of the operator, these functions have to satisfy the bounary conitions at the vertex, where n A jh g h h= A = n h= B jh g h = for all j ˆn, 55 S I m T T I n m, B = Using the explicit form of G i x h, y an the equality G x h,y x h an g h = n λ hl k 2 l= g h = e y ψ h yy n λ hl k 2 l= e y ψ l yy. xh = = e y, we fin e y ψ l yy. Substituting from these two relations into 55 we get a system of equations, n n n A jh λ hl k 2 B jl B jh λ hl k 2 e y ψ l yy = l= h= with j ˆn. We require that the left-han sie vanishes for any ψ, ψ 2,..., ψ n ; this yiels the conition AΛ B BΛ =. From here it is easy to fin the matrix Λk 2 : we have A BΛ = B, an therefore Λk 2 = A B B. Substituting the explicit forms of A an B into the expression for Λ, we obtain S I Λk 2 m T I m T = T I n m, or explicitly Λk 2 = h= S I m TT S I m TT T T S I m TT T S I m TT T 2

22 provie that S I m TT is well efine. To check that the matrix S I m TT is regular, we notice that S I m TT = S I m TT, 56 where the matrix I m TT is positive efinite an thus regular, an the value may be chosen arbitrarily with the only restriction Re >. Consequently, it suffices to choose Re big enough to make the matrix I m TT ominate over S, which ensures the regularity of S I m TT. Having foun the coefficients λ jl k 2, we have fully etermine the Green s function G A i of the approximate system. Recall that it is an n n matrix-value function the j, l-th element of which is given by G A i, jl x, y = δ sinh x < e x > jl λ jl k 2 e x e y ; 57 we use the convention that x is from the j-th halfline an y from the l-th one. The kernel of the operator R A k2 is accoring to 52 given simply by G A, G A i = i, 58 i.e. all entries of G A, i except for those inexe by j, l ˆn vanish. II. Resolvents of the approximating family of Hamiltonians Next we will pass to resolvent construction for the approximating family of operators H Ag. As a starting point we consier n inepenent halflines an n j= N j lines of the length with constant vector potentials A j,l, both halflines an lines of the finite length are suppose to have Dirichlet enpoints. We know that the Green s function is G i x, y = sinh x < e x > in the case of the halflines. The Green s function in the case of the lines of the length will be foun in two steps. We begin with a line without vector potential an with Dirichlet enpoints; the Green s function can be easily erive being equal to G i x, y = sinh x < sinh x > sinh The Hamiltonian of a free particle on a line segment acts as 2 x 2, if a vector potential A is ae it changes to i x A 2. Using Lemma 3.3 it is easy to check that where U is the unitary operator acting as i 2 x A = U 2 U, 59 x 2 Uψx = e iax ψx. If we enote H = 2 an H x 2 A = i x A 2, we see that H A λ = UH U λ = UH λ U = U H λ U,. 22

23 so the corresponing resolvents are relate by the relation analogous to 59. This yiels HA λ ψ x = UH A λ U ψ x = e iax thus the sought integral kernel is equal to = G A i x, y = eiax sinh x < sinh x > sinh G i x, y e iay ψyy e iax sinh x < sinh x > sinh e iay. e iay ψyy, Now we can procee to the erivation of the complete resolvent R Ag k2 which will be one again by means of the Krein s formula. The situation here is more complicate than in the case of the approximate system; recall that R Ag k2, as well as H A k2, acts on the larger Hilbert space L 2 G, where G has been efine in 5. Moreover, the application of Krein s formula means that we have to connect all the line segments using the appropriate bounary conitions, i.e. we must change bounary conitions at n 2 n j= N j enpoints, specifically n belonging to n half lines an 2 n j= N j belonging to n j= N j segments of the length. Thus the inex set for the inices in the sum on the right han sie of the formula has n 2 n j= N j elements; we will inex them by the set Î = ˆn { l, h l ˆn, h Nl } { l, h l ˆn, h Nl }. The elements of ˆn correspon to change bounary conitions at the enpoints of the half lines, an the elements of the type l, h an l, h h N l correspon to change bounary conitions at the enpoints of the segments of the length which are connecte to the enpoint of the l-th half line. If we enote by the symbol R Dc k2 the resolvent of the system of the n n j= N j ecompose eges with Dirichlet bounary conitions at the enpoints, Krein s formula for this pair of operators has the form R Ag k2 = R Dc k2 λ JL k2 φ L k 2, L 2 G φ J k2. 6 J,L Î The role of the superscript in the lamba symbols is to istinguish them from λ jl that have been use in Eq. 54 for the resolvent of the approximate system. The functions φ J J Î may be chosen, as before in the case of the approximate system, as any elements of the corresponing eficiency subspaces of the largest common restriction. Note that each function φ J has n n j= N j components inexe by elements of the set I = ˆn {l, h l ˆn, h N l }. It turns out that a suitable choice is φ j k 2 x L = δ j Le x j for j ˆn, L I, φ l,h k 2 x L = eia l,hx l,h δ l,h L sinh x l,h for l ˆn, h N l, L I φ l,h k 2 x L = eia l,hx l,h δ l,h L sinh x l,h for l ˆn, h N l, L I, 6 where the symbol x enotes the vector from G with the components inexe by I. We remark that if J ˆn, φ J is inepenent of an equal to the corresponing function chosen above in the case of the approximate system. 23

24 If we apply the operator 6 to an arbitrary Ψ n j= L2 G, we obtain a vector function with n n j= N j components inexe by I, we enote them by g j j ˆn an g l,h with l ˆn, h N l. As in the case of the approximate system, a component g J epens on x J only, thus each g J can be consiere as a function of a single variable. A calculation leas to the following explicit expressions for g j, j ˆn an g l,h, l ˆn, h N l ; for better clarity we istinguish the integral variables on R an on, by a tile, i.e. y R, ỹ,. g j x j = G i x j, yψ j y y n l = h N l n λ j j k2 j = λ jl h k 2 λ jl h k 2 e y ψ j y y e x j e ia l,h ỹ sinh ỹ ψ l,h ỹ ỹ e ia l,h ỹ sinh ỹ ψ l,h ỹ ỹ e x j. 62a g l,h x l,h = G A l,h i x l,h, ỹψ l,h ỹ ỹ n e ia l,hx l,h sinh x l,h λ k 2 l,h j n l = h N l λ l,h l h k 2 j = e y ψ j y y e ia l,h ỹ sinh ỹ ψ l,h ỹ ỹ ] λ k 2 e ia l,h l h l,h ỹ sinh ỹ ψ l,h ỹ ỹ n e ia l,hx l,h sinh x l,h λ k 2 e y ψ l,h j j y y n l = h N l λ l,h l h k 2 λ l,h l h k 2 j = e ia l,h ỹ sinh ỹ ψ l,h ỹ ỹ ] e ia l,h ỹ sinh ỹ ψ l,h ỹ ỹ. 62b By efinition the function g J J I belongs to the omain of the operator H Ag, in particular, it has to satisfy the bounary conitions at the points where the eges are connecte by δ interactions an δ couplings. Step by step we will write own now all these bounary conitions; this will lea to the explicit expressions for the coefficients λ JL k2. Step. The continuity at the points W { j,k} means g l,h = g h,l 63 24