Inverse scattering problem for quantum graph vertices
IInverse scattering problem for quantum graph vertices
Taksu Cheon ∗ , Pavel Exner † , Ondˇrej Turek ∗ ∗ Laboratory of Physics, Kochi University of Technology, Tosa Yamada, Kochi 782-8502, Japan † Doppler Institute for Mathematical Physics and Applied Mathematics,Czech Technical University, Bˇrehov´a 7, 11519 Prague, Czechia (Dated: October 29, 2018)We demonstrate how the inverse scattering problem of a quantum star graph can be solved bymeans of diagonalization of Hermitian unitary matrix when the vertex coupling is of the scaleinvariant (or F¨ul˝op-Tsutsui) form. This enables the construction of quantum graphs with desiredproperties in a tailor-made fashion. The procedure is illustrated on the example of quantum verticeswith equal transmission probabilities.
PACS numbers: 03.65.-w, 03.65.Nk, 73.63.Nm
The interest in the inverse scattering problem for quan-tum graphs [1–3] is two-fold. These graphs are a primeexample of solvable systems possessing nontrivial physi-cal properties [4]. At the same time, it is also importantfor its relevance as the design principle of nanowire-basedsingle electron devices.In this letter, we consider the inverse scattering prob-lem on a star graph with scale invariant vertex coupling[5], which is an important subset among all the couplingspreserving the probability current [6]; we note that thecorresponding scattering matrix is energy independent.A star graph with a number of half-line edges connectedat a single point is the elementary building block of ageneric graph. In general, the inverse problem for stargraphs is easy to solve [7] but the solution does not givemuch insight into the physical meaning of the coupling.Here we present an alternative approach for star graphswith scale invariant coupling. We shall give the solutionto the corresponding inverse scattering problem in theform of eigenvalue problem of a Hermitian unitary ma-trix. In particular, we consider quantum vertices withequal transmission probabilities, a subclass of the scaleinvariant case, and we show that the problem boils downto a search for a special class of Hermitian unitary matri-ces which can be regarded as a generalization of complexHadamard matrices. Intriguing designs emerge for therealization of quantum devices with such properties.Note that any quantum-graph vertex coupling behaveseffectively as a scale-invariant one in both the high andlow energy limits [8], which gives a hint that our analysismight have an extension to the general case.Consider thus a star graph vertex of degree n , with n half-line edges sticking out of a point-like node. The scaleinvariant subfamily of the general coupling conditions ischaracterized by a complex matrix T of size ( n − m ) × m ,where m can take an integer value m = 1 , , ..., n − (cid:18) I ( m ) T (cid:19) Ψ (cid:48) = (cid:18) − T † I ( n − m ) (cid:19) Ψ , (1)where I ( l ) signifies the identity matrix of size l × l , and the boundary-value vectors Ψ and Ψ (cid:48) are defined byΨ = ψ (0)... ψ n (0) , Ψ (cid:48) = ψ (cid:48) (0)... ψ (cid:48) n (0) , (2)with ψ i ( x i ) and ψ (cid:48) i ( x i ) being the wave function and itsderivative on i -th edge [9]. The coordinates x i on the i -th half-line are labeled outwardly from the vertex, whichcorresponds to x i = 0 for all i . To cast the coupling intothe form (1) one has, in general, to renumber suitablythe edges. The quantum particle coming in from the j -th edge and scattered off the vertex is described by thescattering wave function on the i -th line, ψ ( j ) i ( x ), whichis of the form ψ ( j ) i ( x ) = δ i,j e − i kx + S i,j e i kx . (3)Let us express S in terms of T . From (3) we haveΨ(0) = I + S and Ψ (cid:48) (0) = k ( − I + S );we substitute into (1), which leads to the equation (cid:18) I ( m ) TT † − I ( n − m ) (cid:19) S = (cid:18) I ( m ) T − T † I ( n − m ) (cid:19) . (4)It is easy to observe from (4) that S = X − m Z m X m , (5)with the matrices X m , Z m defined by X m = (cid:18) I ( m ) TT † − I ( n − m ) (cid:19) , Z m = (cid:18) I ( m ) − I ( n − m ) (cid:19) . (6)We see that (5) can be viewed as a diagonalizationformula of Hermitian unitary matrix S with a diagonal-izing matrix of specific block diagonal form, X m , whichgives a prescription to obtain T that defines the boundarycondition from the scattering matrix S . In other words,solution of the our inverse scattering problem is given interms of a diagonalization.In practice, the procedure of obtaining T by a diago-nalization of S can be cumbersome for large n , and there a r X i v : . [ qu a n t - ph ] A p r is an alternative simpler way. A calculation shows that(5) can be rewritten in the form S = − I ( n ) + 2 (cid:18) I ( m ) T † (cid:19) (cid:16) I ( m ) + T T † (cid:17) − (cid:0) I ( m ) T (cid:1) . (7)Let us divide S into four submatrices S , S , S and S of size m × m , m × ( n − m ), ( n − m ) × m and ( n − m ) × ( n − m ), respectively, as follows S = (cid:18) S S S S (cid:19) . (8)The block matrices S ij express in terms of T as S = − I ( m ) + 2 (cid:0) I ( m ) + T T † (cid:1) − , S = 2 (cid:0) I ( m ) + T T † (cid:1) − T ,and S = I ( n − m ) − (cid:0) I ( n − m ) + T † T (cid:1) − . From here,one gets easily T = (cid:16) I ( m ) + S (cid:17) − S = S † (cid:16) I ( n − m ) − S (cid:17) − . (9)Hence, the algorithm to obtain the matrix T character-izing the vertex is the following:1. Take the scattering matrix S and set m = rank( S + I ( n ) ).2. Decompose S according to (8). If necessary, changethe numbering of the incoming edges so that I ( m ) + S is regular (it is always possible).3. Calculate T using (9).We remark that the matrix T obtained by the algorithmabove naturally depends on the numbering of the edgeswe choose.In the rest of the paper we demonstrate how the matrix T is used for understanding the meaning of the couplingand for the construction of the vertex with prescribedscattering properties.Asking about the meaning of the coupling, recall a star-shaped network with a potential at the node can tendin the zero-diameter limit to the star graph with δ cou-pling [10]. More complicated couplings can be obtainedby using δ vertices as building blocks, applying localizedmagnetic fields to achieve phase change if necessary; theaim is to devise a design principle to construct an ar-bitrary coupling condition. Here the matrix T derivedabove plays an important role. In the scale-invariantcase (1) with the elements of T = { t ij } , i = 1 , ..., m and j = m + 1 , .., n , given, the scheme works as follows[11]:(i) Take the endpoints of the n edges, numbered by j = 1 , , ..., n , and connect them in pairs ( i, j ) by internaledges of length d/r ij , except when r ij = 0 in which casethe pair remains unconnected. Apply a vector potential A ij on the segment ( i, j ) to produce extra phase shift χ ij between the endpoints when its value is nonzero. Placethe δ potential of strength v i at each endpoint i . (ii) The length ratio r ij used above and the phase shift χ ij are determined from the non-diagonal elements of thematrix Q defined by Q = (cid:18) TI ( n − m ) (cid:19) (cid:0) − T † I ( m ) (cid:1) = (cid:18) − T T † T − T † I ( m ) (cid:19) (10)using the relation r ij e i χ ij = Q ij ( i (cid:54) = j ). This means thatwe have r ij e i χ ij = t ij for i ≤ m, j > m , and r ij e i χ ij = (cid:80) l>m t il t ∗ jl for i, j ≤ m ; for i, j > m we have r ij = 0 andnaturally also χ ij = 0.(iii) The δ coupling strength v i is given by the diagonalelements of the matrix V defined by V = 1 d (2 I ( n ) − J ( n ) ) R, (11)where R is the matrix whose elements equal the absolutevalues of the matrix elements of Q , i.e. R = { r ij } = {| Q ij |} ; the n × n matrix J ( n ) has all the elements equalto one. This means that we have v i = d (cid:0) − (cid:80) l ≤ m r li (cid:1) for i > m , and v i = d (cid:0) (cid:80) l>m [ r il − r il ] − (cid:80) l ( (cid:54) = i ) ≤ m r il (cid:1) for i ≤ m . The described way to fine tuning of the lengthsand δ coupling strengths is devised to counter the genericopaqueness brought in with every addition of a vertex ora connecting edge into the graph.The wave function φ ( x )= φ i,j ( x ) on any internal edgewith indices ( i, j ) has to satisfy the relation (cid:18) φ (cid:48) (0) e iχ φ (cid:48) ( dr ) (cid:19) = − rd (cid:18) F ( dr )) − G ( dr ) G ( dr ) − F ( dr ) (cid:19) (cid:18) φ (0) e iχ φ ( dr ) (cid:19) , (12)with F ( x ) = x cot x and G ( x ) = x cosec x . Combining(12) with the condition at the i -th endpoint where wehave the δ -potential of strength v i , ψ (cid:48) i (0) + (cid:88) j (cid:54) = i φ (cid:48) ij (0) = v i ψ i (0) (13)we obtain the relations between the boundary values ψ i = ψ i (0) and ψ (cid:48) i = ψ (cid:48) i (0) in the form dψ (cid:48) i = v i d + (cid:88) l (cid:54) = i r il F il ψ i − (cid:88) l (cid:54) = i e i χ ij r il G il ψ l , (14)where the obvious notations F ij = dr il cot dr il and G ij = dr il cosec dr il have been adopted. Note that the equation(14) is exact and does not involve any approximation.For small values of the length parameter d we have F ij =1 + O ( d ) and G ij = 1 + O ( d ); then we can show by astraightforward computation in the manner of [9] that theshrinking limit d → Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ FIG. 1. Finite approximation to the reflectionless scale in-variant vertices corresponding to (16) (left) and (18) (right)constructed according to (10)–(11). Dotted line indicates thepresence of a non-zero phase shift χ ij . more easily by the current method. Here, we will illus-trate the application of the procedure on the followingexemplary problem. Let us look at this question: Canone construct a quantum vertex for which the particleincoming from any line is transmitted to all other lineswith equal probability? At first, we should ask about theexistence condition for the scattering matrix of the form S = 1 √ d + n − d e i φ · · · e i φ n e i φ d · · · e i φ n ... . . . ... e i φ n − · · · − d e i φ n − n e i φ n · · · e i φ nn − − d , (15)with a non-negative real parameter d .These matrices have been recently examined in [13]. Ithas been proved that the parameter d is always boundedfrom above by n − n ≤ d ∈ [0 , n − n isodd. By contrast, if n is even, then one can construct an S for infinitely many values of d , in particular for any d from the interval [ n − , n − d ∈ [0 , d = 0.Vertices yielding such scattering matrices are known tobe useful in investigation of semiclassical properties ofquantum-graph spectra [12]. Let us adopt a result from[13] which says that such an S can exist only for even n . We limit ourselves for simplicity to real S ; we remarkthat a real matrix of the type (15) with zero diagonalis called symmetric conference matrix , and is known toexist for n = 2 , , , , , , , , ... .Here we inspect the example of n = 6 when S is given by S = 1 √ (cid:18) I (3) − J (3) − I (3) + J (3) − I (3) + J (3) − I (3) + J (3) (cid:19) . (16)Applying (9), the corresponding T is easily calculated, T = − γI (3) + (1 + γ ) J (3) (17)where γ = ( √ − / r = r = r = 4 + 3 γ,r = r = r = 1 ,r = r = r = r = r = r = 1 + γ , r = r = r =0, e i χ = e i χ = e i χ = − e i χ ij = 1 for all others, and v = v = v = − γ +1 d , v = v = v = − γ +1 d . Thefinite graph approximation is schematically illustrated inthe left side of Figure 1.Second example is the equal-scattering graph, in whichthe scattering is uniform to all the edges including theone of the incoming particle. Such a matrix, called symmetric Hadamard matrix , is known to exist for n =2 , , , , , , , ... . An example of such S for n = 8is given by S = 1 √ (cid:18) I (4) − J (4) − I (4) + J (4) − I (4) + J (4) − I (4) + J (4) (cid:19) . (18)The matrix T specifying the vertex coupling is found tobe equal to T = σ − σ + 1 I (4) + 1 σ + 1 J (4) (19)where σ = √ − r = r = r = r = r = r = 1 + σ , r = r = r = r = σ σ , r = r = r = r = r = r = r = r = r = r = r = r = σ , r = r = r = r = r = r = 0, e i χ = e i χ = e i χ = e i χ = e i χ = e i φ = e i χ = − e i χ ij = 1 for all others, and v = v = v = v = − σ +3 d , v = v = v = v = − σ +1 d . The finite graph approx-imation is schematically illustrated in the right side ofFigure 1.Finally, let us discuss the convergence of the the de-scribed finite-size graph approximation. In Figure 2, wedisplay scattering matrix elements of the finite graph con-structed to approximate the equal-scattering reflection-less matrix (16). They are calculated directly from (14).The scale of the wave length k is given by 1 /d . The ap-proximation can be seen to be quite good below kd < . S having interesting specifications other than |S j1 | k S S S S C |S j4 | k S S S S C FIG. 2. Scattering probabilities as functions of incomingmomentum k (in the unit of 1 /d ) of finite quantum graph ap-proximating the equal-transmitting reflectionless vertex rep-resented in the left side of Figure 1. those examined here should follow. Also, a study of thebound state spectra is one thing we have completely ne-glected in this letter; applications to non-quantum waves,including particularly electromagnetic and water wavesshould be another interesting subject.In our finite approximation of star graph with no inter-nal edges, we have actually studied the low-energy prop-erties of graphs with internal edges all of which are con-nected to the external ones, which we might term depth-one graphs. The examination of depth-two graphs andbeyond seems to be a natural future direction. Our re-sult showing the full solution to the inverse scatteringproblem is, in a sense, a partial fulfillment of the hopethat quantum graph somehow could be a solvable model and useful design tool at the same time.We thank Prof. U. Smilansky, Prof. L. Feher, Prof.I. Tsutsui, Prof. T. Kawai and Prof. T. Shigehara forstimulating discussions. This research was supported bythe Japanese Ministry of Education, Culture, Sports, Sci-ence and Technology under the Grant number 21540402,and also by the Czech Ministry of Education, Youth andSports within the projects LC06002 and P203-11-0701. [1] M. Harmer, J. Phys. A: Math. Theor. , 4875 (2005).[2] J. Boman and P. Kurasov, Adv. Appl. Math. , 58(2005).[3] B. Gutkin and U. Smilansky, J. Phys. A: Math. Gen. Analysis on Graphs and Applications ,AMS Proc. of Symposia in Pure Math. Ser., vol. 77, Prov-idence, R.I., 2008, and references therein .[5] T. F¨ul˝op, I. Tsutsui, Phys. Lett.