Inverse scattering on the quantum graph for graphene
Kazunori Ando, Hiroshi Isozaki, Evgeny Korotyaev, Hisashi Morioka
IINVERSE SCATTERING ON THE QUANTUM GRAPH FORGRAPHENE
KAZUNORI ANDO, HIROSHI ISOZAKI, EVGENY KOROTYAEV,AND HISASHI MORIOKA
Abstract.
We consider the inverse scattering on the quantum graph associ-ated with the hexagonal lattice. Assuming that the potentials on the edges arecompactly supported and symmetric, we show that the S-matrix for all energiesin any given open set in the continuous spectrum determines the potentials. Introduction
In this paper, we are concerned with a family of one-dimensional Schr¨odingeroperators − d /dz + q e ( z ) defined on the edges of the hexagonal lattice assumingthe Kirchhoff condition on the vertices. Here, z varies over the interval (0 , e ∈ E , E being the set of all edges of the hexagonal lattice. The followingassumptions are imposed on the potentials. (Q-1) q e ( z ) is real-valued, and q e ∈ L (0 , (Q-2) q e ( z ) = 0 on (0 , except for a finite number of edges. (Q-3) q e ( z ) = q e (1 − z ) for z ∈ (0 , (cid:98) H E = (cid:110) − d dz + q e ( z ) ; e ∈ E (cid:111) is self-adjoint with essential spectrum σ e ( (cid:98) H E ) = [0 , ∞ ). There exists a discrete (butinfinte) subset T ⊂ R such that σ e ( (cid:98) H E ) \ T is absolutely continuous.We can thendefine Heisenberg’s S-matrix S ( λ ) for λ ∈ (0 , ∞ ) \ T . The following two theoremsare the main purpose of this paper. Theorem 1.1.
Assume (Q-1), (Q-2) and (Q-3). Then, given any open interval I ⊂ (0 , ∞ ) \ T , and the S-matrix S ( λ ) for all λ ∈ I , one can uniquely reconstructthe potential q e ( z ) for all e ∈ E . Under our assumptions (Q-1), (Q-2), (Q-3), S ( λ ) is meromorphic in the complexdomain { Re λ > } with possible branch points at T . Therefore, the assumptionof Theorem 1.1 is equivalent to the condition that we are given S ( λ ) for all λ ∈ (0 , ∞ ) \ T . One can also deal with perturbation of periodic edge potentials. Date : February 11, 2021.2000
Mathematics Subject Classification.
Primary 81U40, Secondary 47A40.
Key words and phrases.
Schr¨odinger operator, lattice, quantum graph, S-matrix, inversescattering. a r X i v : . [ m a t h - ph ] F e b KAZUNORI ANDO, HIROSHI ISOZAKI, EVGENY KOROTYAEV, AND HISASHI MORIOKA
Theorem 1.2.
Assume (Q-1) and (Q-3). Assume that we are given a real q ( z ) ∈ L (0 , satisfying q ( z ) = q (1 − z ) and q e ( z ) = q ( z ) on (0 , except for a finitenumber of edges e ∈ E . Given an open interval I ⊂ σ e ( (cid:98) H E ) \ T and the S-matrix S ( λ ) for all λ ∈ I , one can uniquely reconstruct the potential q e ( z ) for all edges e ∈ E . It is well-known that there is a close connection between the Laplacian on thequantum graph and that on the associated vertex set (see e.g. [9], [16], [36], [43],[15]). Therefore, the basic results on the spectral theory for the quantum graphare derived from those for the associated discrete Laplacian. Sections 2, 3 and4 are devoted to this transfer. In particular, we show that the S-matrix for thewhole quantum graph determines the Dirchlet-to-Neumann map in a finite regionon which perturbations are confined (Theorem 4.3). The inverse problem is solvedin § M, dµ ), let L ( M ; C n ; dµ ) be the space of the C n -valuedfunctions on M . It is often denoted by L ( M ; C n ) or L ( M ) when n = 1. ForBanach spaces X and Y , let B ( X ; Y ) be the set of all bounded operators from X to Y , and B ( X ) = B ( X ; X ). 2. Quantum graph
Vertex Laplacian.
We follow the standard formulation of metric graph (seee.g. [36] or [43]) . In R , let p (1) = (1 , p (2) = (2 , v = (cid:16) , − √ (cid:17) , v = (cid:16) , √ (cid:17) , and v ( n ) = n v + n v for n = ( n , n ). We define the vertex set V by V = ∪ i =1 V i , V i = { p ( i ) + v ( n ) ; n ∈ Z } . Let I : L loc ( V ; C ) → L loc ( Z ; C ) be defined by(2.1) I : (cid:98) f ( v ) → ( I (cid:98) f )( n ) = (cid:32) (cid:98) f ( n ) (cid:98) f ( n ) (cid:33) = (cid:32) (cid:98) f ( p (1) + v ( n )) (cid:98) f ( p (2) + v ( n )) (cid:33) . For the figure of hexagonal lattice, see e.g. [36] or [6].
NVERSE SCATTERING ON THE QUANTUM GRAPH FOR GRAPHENE 3
We often write (cid:98) f ( n ) instead of ( I (cid:98) f )( n ). The Laplacian is defined by (cid:0) (cid:98) ∆ V (cid:98) f (cid:1) ( n ) = 13 (cid:32) (cid:98) f ( n , n ) + (cid:98) f ( n − , n ) + (cid:98) f ( n , n − (cid:98) f ( n , n ) + (cid:98) f ( n + 1 , n ) + (cid:98) f ( n , n + 1) (cid:33) , which is self-adjoint on L ( V ) equipped with the inner product ( (cid:98) f , (cid:98) g ) = 3 (cid:88) n ∈ Z (cid:98) f ( n ) · (cid:98) g ( n ) . Define the discrete Fourier transform U V : L ( Z ; C ) → L ( T ; C ) by( U V (cid:98) f )( x ) = √ π ) − (cid:88) n ∈ Z e in · x (cid:98) f ( n ) , x ∈ T = R / (2 π Z ) . Then on L ( T ; C ), U V ( − ∆ V ) U ∗V is the operator of multiplication by(2.2) H ( x ) = − (cid:18) e ix + e ix e − ix + e − ix (cid:19) . The edge set E consists of the segments e of length 1 with end points in V , endowedwith arclength metric, as well as the identification with the interval (0 ,
1) : e = { (1 − z ) e (0) + z e (1) ; 0 ≤ z ≤ } , where e (0) , e (1) ∈ V . We put E v = E v (0) ∪ E v (1) , E v ( i ) = { e ∈ E ; e ( i ) = v } , i = 0 , . For a function (cid:98) f on an edge e ∈ E v , we define (cid:98) f (cid:48) ( v ) to be the derivative at v along e . A function (cid:98) f = { (cid:98) f e } e ∈E defined on E is said to satisfy the Kirchhoff condition if (K-1) (cid:98) f is continuous on E . (K-2) (cid:98) f e ∈ C ([0 , on each edge e ∈ E , and (cid:80) e ∈E v (cid:98) f (cid:48) e = 0 at any vertex v ∈ V . Edge Laplacian.
We consider 1-dimensional Schr¨odinger operators h (0) e = − d /dz , h e = h (0) e + q e ( z )on L e = L (0 , L ( E ) of C -valued L -functions (cid:98) f = (cid:8) (cid:98) f e (cid:9) e ∈E on the edge set E : L ( E ) = ⊕ e ∈E L e equipped with the inner product(2.3) ( (cid:98) f , (cid:98) g ) L ( E ) = (cid:88) e ∈E ( (cid:98) f e , (cid:98) g e ) L (0 , . Define the Hamiltonian(2.4) (cid:98) H E : (cid:98) u = { (cid:98) u e } e ∈E → { h e (cid:98) u e } e ∈E with domain D ( (cid:98) H E ) consisting of (cid:98) u e ∈ H (0 , satisfying the Kirchhoff condition(K-1), (K-2) and (cid:80) e ∈E (cid:107) h e (cid:98) u e (cid:107) L (0 , < ∞ . Then, (cid:98) H E is self-adjoint in L ( E ).When q e = 0, (cid:98) H E is denoted by (cid:98) H (0) E or − (cid:98) ∆ E , i.e. (cid:0) − (cid:98) ∆ E (cid:98) u (cid:1) e ( z ) = − d dz (cid:98) u e ( z ) , e ∈ E . We call it edge Laplacian . Let q E be the multiplication operator defined by (cid:0) q E (cid:98) f (cid:1) e ( z ) = q e ( z ) (cid:98) f e ( z ) , e ∈ E . Note that the degree of each vertex in V is 3. This is the Sobolev space of order 2.
KAZUNORI ANDO, HIROSHI ISOZAKI, EVGENY KOROTYAEV, AND HISASHI MORIOKA
Then (cid:98) H E = (cid:98) H (0) E + q E . We put(2.5) (cid:98) R (0) E ( λ ) = ( (cid:98) H (0) E − λ ) − , (cid:98) R E ( λ ) = ( (cid:98) H E − λ ) − . Let − ( d /dz ) D be the Laplacian on (0 ,
1) with boundary condition u (0) = u (1) = 0. Let φ e ( z, λ ) , φ e ( z, λ ) be the solutions of(2.6) (cid:0) − d /dz + q e ( z ) − λ (cid:1) φ = 0with initial data (cid:40) φ e (0 , λ ) = 0 ,φ (cid:48) e (0 , λ ) = 1 , (cid:40) φ e (1 , λ ) = 0 ,φ (cid:48) e (1 , λ ) = − . In the following, we assume that λ (cid:54)∈ ∪ e ∈E σ ( − ( d /dz ) D + q e ( z )) , which guarantees that φ e (1 , λ ) (cid:54) = 0 and φ e (0 , λ ) (cid:54) = 0. If w, v ∈ V are two endponts of an edge e ∈ E , we define ψ wv ( z, λ ) by ψ wv ( z, λ ) = (cid:40) φ e ( z, λ ) , if e (0) = v,φ e ( z, λ ) , if e (0) = w. Note that by the assumption (Q-3), we have φ e ( z, λ ) = φ e (1 − z, λ ), hence ψ wv (1 , λ ) = ψ vw (1 , λ ) . Definition 2.1.
We define the reduced vertex Laplacian (cid:98) ∆ V ,λ on V by(2.7) (cid:0) (cid:98) ∆ V ,λ (cid:98) u (cid:1) ( v ) = 13 (cid:88) w ∼ v ψ wv (1 , λ ) (cid:98) u ( w ) , v ∈ V for (cid:98) u ∈ L loc ( V ), where w ∼ v means that there exists an edge e ∈ E such that v, w are end points of e . We also define a scalar multiplication operator: (cid:0) (cid:98) Q V ,λ (cid:98) u (cid:1) ( v ) = (cid:98) Q v,λ ( v ) (cid:98) u ( v ) , where(2.8) (cid:98) Q v,λ ( v ) = 13 (cid:88) w ∈E v ψ (cid:48) wv (1 , λ ) ψ wv (1 , λ ) . The resolvent r e ( λ ) = ( − ( d /dz ) D + q e ( z ) − λ ) − is written as( r e ( λ ) (cid:98) f )( v ) = (cid:90) z φ e ( z, λ ) φ e ( t, λ ) φ e (1 , λ ) (cid:98) f ( t ) dt + (cid:90) z φ e ( z, λ ) φ e ( t, λ ) φ e (0 , λ ) (cid:98) f ( t ) dt. We put Φ e ( λ ) (cid:98) f = ddz (cid:0) r e ( λ ) (cid:98) f (cid:1)(cid:12)(cid:12)(cid:12) z =0 = (cid:90) φ e ( t, λ ) φ e (0 , λ ) (cid:98) f ( t ) dt, Φ e ( λ ) (cid:98) f = − ddz (cid:0) r e ( λ ) (cid:98) f (cid:1)(cid:12)(cid:12)(cid:12) z =1 = (cid:90) φ e ( t, λ ) φ e (1 , λ ) (cid:98) f ( t ) dt, and define an operator (cid:98) T V ( λ ) : L loc ( E ) → L loc ( V ) by (cid:0) (cid:98) T V ( λ ) (cid:98) f (cid:1) ( v ) = 13 (cid:16) (cid:88) e ∈E v (1) Φ e ( λ ) (cid:98) f e + (cid:88) e ∈E v (0) Φ e ( λ ) (cid:98) f e (cid:17) , v ∈ V . (2.9) NVERSE SCATTERING ON THE QUANTUM GRAPH FOR GRAPHENE 5
Let (cid:98) u = { (cid:98) u e } e ∈E be a solution to the equation ( (cid:98) H E − λ ) (cid:98) u = (cid:98) f . On each edge e ∈ E , it is written as (cid:98) u e ( z, λ ) = Φ e ( λ ) ∗ c e (1 , λ ) + Φ e ( λ ) ∗ c e (0 , λ ) + r e ( λ ) (cid:98) f e = φ e ( t, λ ) φ e (1 , λ ) c e (1 , λ ) + φ e ( t, λ ) φ e (1 , λ ) c e (0 , λ ) + r e ( λ ) (cid:98) f e (2.10)with some constants c e (0 , λ ), c e (1 , λ ). Then, the condition (K-1) is satisfied if andonly if for two edges e , e (cid:48) ∈ E and p, q = 0 , c e ( p, λ ) = c e (cid:48) ( q, λ ) if e ( p ) = e (cid:48) ( q ). Lemma 2.2.
Let (cid:98) u (cid:12)(cid:12) V be the restriction of (cid:98) u on V . Then the condition (K-2) isrewritten as (cid:16) − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ (cid:17) (cid:98) u (cid:12)(cid:12) V = (cid:98) T V ( λ ) (cid:98) f . (2.11)This lemma is well-known. In fact, (K-2) is rewritten as − (cid:88) e ∈E v (0) φ e (1 , λ ) c e (1 , λ ) − (cid:88) e ∈E v (1) φ e (0 , λ ) c e (0 , λ ) − (cid:88) e ∈E v (0) φ (cid:48) e (0 , λ ) φ e (0 , λ ) c e (0 , λ ) + (cid:88) e ∈E v (1) φ (cid:48) e (1 , λ ) φ e (1 , λ ) c e (1 , λ )= (cid:88) e ∈E v (1) Φ e ( λ ) (cid:98) f e + (cid:88) e ∈E v (0) Φ e ( λ ) (cid:98) f e , which implies (2.11). Therefore, (cid:98) u (cid:12)(cid:12) V should be written as(2.12) (cid:98) u (cid:12)(cid:12) V = (cid:16) − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ (cid:17) − (cid:98) T V ( λ ) (cid:98) f . Here, we must be careful about the operator (cid:0) − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ (cid:1) − . For λ (cid:54)∈ R , theoperator − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ has complex coefficients, hence is not self-adjoint. Therefore,the existence of its inverse is not obvious. We discuss the validity of (2.12) inSubsection 3.1. For the moment, we admit it as a formal formula.Noting that (cid:98) T V ( λ ) ∗ : L loc ( V ) → L loc ( E ) is written as (see (2.10))( (cid:98) T V ( λ ) ∗ (cid:98) u ) e ( z ) = Φ e ( λ ) ∗ (cid:98) u ( e (1)) + Φ e ( λ ) ∗ (cid:98) u ( e (0)) , we have the following lemma by (2.11). Let r E ( λ ) ∈ B ( L ( E )) be defined by r E ( λ ) (cid:98) f = r e ( λ ) (cid:98) f e , on e . Lemma 2.3.
The resolvent of (cid:98) H E is written as (2.13) (cid:98) R E ( λ ) = (cid:98) T V ( λ ) ∗ (cid:16) − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ (cid:17) − (cid:98) T V ( λ ) + r E ( λ ) . For the unperturbed case (cid:98) q E = 0, we put the superscript (0) for every term.Then, we have(2.14) φ (0) e ( z ) = sin √ λz √ λ , φ (0) e ( z ) = sin √ λ (1 − z ) √ λ . Therefore by (2.7) and (2.8),(2.15) (cid:16) (cid:98) ∆ (0) V ,λ (cid:98) u (cid:17) ( v ) = √ λ sin √ λ (cid:88) w ∈E v (cid:98) u ( w ) = √ λ sin √ λ (cid:0) (cid:98) ∆ V (cid:98) u (cid:1) ( v ) , KAZUNORI ANDO, HIROSHI ISOZAKI, EVGENY KOROTYAEV, AND HISASHI MORIOKA (2.16) (cid:98) Q (0) v,λ = √ λ sin √ λ cos √ λ. Lemma 2.3 then implies the following formula(2.17) (cid:98) R (0) E ( λ ) = (cid:98) T (0) V ( λ ) ∗ sin √ λ √ λ (cid:0) − (cid:98) ∆ V + cos √ λ (cid:1) − (cid:98) T (0) V ( λ ) + r (0) E ( λ ) . Resolvent estimates
Limiting absorption principle.
In our previous work [3], we proved resol-vent estimates of vertex Laplacian − ∆ V in weighted L spaces or Besov spacesof C -valued functions. By virtue of the formulas (2.13) and (2.17), the resolventestimates of edge Laplacian are derived from those of vertex Laplacian using thespace of L ( e )-valued functions on the edge set E defined as follows.For e ∈ E , we put c ( e ) = 12 ( e (0) + e (1)) . Letting r − = 0, r j = 2 j ( j ≥ (cid:98) B ( E ) (cid:51) (cid:98) f ⇐⇒ (cid:107) (cid:98) f (cid:107) (cid:98) B ( E ) = ∞ (cid:88) j =0 r / j (cid:16) (cid:88) r j − ≤| c ( e ) |
2, and weak ∗ -limitin B ( B ( V ); B ∗ ( V )) of (cid:0) − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ ± i (cid:1) − . The arguments in § B ( E ) or B ∗ ( E ). The limiting absorption principle isthen extended to the edge Laplacian in the following way. NVERSE SCATTERING ON THE QUANTUM GRAPH FOR GRAPHENE 7
Theorem 3.1. (1) For any compact interval I in (0 , ∞ ) \T , there exists a constant C > such that for any λ ∈ I and (cid:15) > (cid:107) ( (cid:98) H E − λ ∓ i(cid:15) ) − (cid:107) B ( B ( E ); B ∗ ( E )) ≤ C. (2) For any λ ∈ (0 , ∞ ) \ T and s > / , there exists a strong limit (3.3) s − lim (cid:15) ↓ ( (cid:98) H E − λ ∓ i(cid:15) ) − := ( (cid:98) H E − λ ∓ i − ∈ B (cid:0)(cid:98) L ,s ( E ); (cid:98) L , − s ( E ) (cid:1) , and for any (cid:98) f ∈ (cid:98) L ,s ( E ) , ( (cid:98) H E − λ ∓ i − (cid:98) f is an (cid:98) L , − s ( E ) -valued strongly continuousfunction of λ .(3) For any (cid:98) f , (cid:98) g ∈ (cid:98) B ( E ) and λ ∈ (0 , ∞ ) \ T , there exists a limit (3.4) lim (cid:15) ↓ (cid:0) ( (cid:98) H E − λ ∓ i(cid:15) ) − (cid:98) f , (cid:98) g (cid:1) := (cid:0) ( (cid:98) H E − λ ∓ i − (cid:98) f , (cid:98) g (cid:1) , and (cid:0) ( (cid:98) H E − λ ∓ i − (cid:98) f , (cid:98) g (cid:1) is a continuous function of λ . Analytic continuation of the resolvent.
It is well-known that for theSchr¨odinger operator − ∆ + V ( x ) in R d , where V ( x ) has compact support, theboundary value of the resolvent ( − ∆ + V ( x ) − λ − i − has a meromorphic con-tinuation into the lower half plane { Re λ > , Im λ < } as an operator from thespace of compactly supported L ( R d ) functions to L loc ( R d ). This is proven byconsidering the free case, i.e. the operator (cid:90) R d e ix · ξ (cid:101) f ( ξ ) | ξ | − ζ dξ = (cid:90) ∞ (cid:82) S d − e irω · x (cid:101) f ( rω ) dωr − ζ r d − dr ( (cid:101) f ( ξ ) being the Fourier transfrom of f ) for Im ζ >
0, deforming the path of inte-gration into the lower half-plane, and then applying the perturbation theory. Thismethod also works for the discrete case, and one can show that the resolvents ofthe vertex Hamiltonian and the edge Hamiltonian defined for { Re λ > , Im λ > } can be continued meromorphically into the lower half-plane { Re λ > , Im λ < } with possible branch points on T , when the perturbation is compactly supported.3.3. Spectral representation.
We can then construct the spectral representationof the edge Laplacian. Letting P V ,j ( x ) be the eigenprojection associated with theeigenvalue λ j ( x ) of H ( x ), we put D (0) ( λ ± i
0) = sin √ λ √ λ U V I (cid:0) − (cid:98) ∆ V + cos √ λ ± i (cid:1) − I ∗ U V ∗ = sin √ λ √ λ (cid:88) j =1 λ j ( x ) + cos √ λ ∓ iσ ( λ )0 P V ,j ( x ) , (3.5)where σ ( λ ) = 1 if λ > , sin √ λ > σ ( λ ) = − λ > , sin √ λ <
0. We also put(3.6) Φ (0) ( λ ) = U V I (cid:98) T (0) V ( λ ) . By (2.17), (cid:98) R (0) E ( λ ± i
0) is rewritten as(3.7) (cid:98) R (0) E ( λ ± i
0) = Φ (0) ( λ ) ∗ D (0) ( λ ± i (0) ( λ ) + r (0) E ( λ ) . We put M λ = ∪ j =1 M λ,j , M λ,j = { x ∈ T d ; λ j ( x ) + cos √ λ = 0 } , KAZUNORI ANDO, HIROSHI ISOZAKI, EVGENY KOROTYAEV, AND HISASHI MORIOKA ( ϕ, ψ ) λ,j = (cid:90) M λ,j ϕ ( x ) ψ ( x ) dS j , dS j = | sin √ λ |√ λ dM λ,j |∇ x λ j ( x ) | , where dM λ,j is the induced measure on M λ,j . For (cid:98) f ∈ B ( E ), we define (cid:98) F (0) j ( λ ) (cid:98) f by (cid:98) F (0) j ( λ ) (cid:98) f = (cid:0) P V ,j ( x )Φ (0) ( λ ) (cid:98) f (cid:1)(cid:12)(cid:12) M λ , i.e. the restriction of P V ,j ( x )Φ (0) ( λ ) (cid:98) f to M λ , and(3.8) (cid:98) F (0) ( λ ) = (cid:0) (cid:98) F (0)1 ( λ ) , (cid:98) F (0)2 ( λ ) (cid:1) , h λ = L (cid:0) M λ (cid:1) = ⊕ j =1 L (cid:0) M λ,j ; dS j (cid:1) , H = L (cid:0) (0 , ∞ ) , h λ ; dλ (cid:1) . Noting that (cid:98) F (0) ( λ ) ∈ B ( B ( E ); h λ ), the spectral representation associated with (cid:98) H E is constructed by the perturbation method. Define (cid:98) F ( ± ) ( λ ) by(3.9) (cid:98) F ( ± ) ( λ ) = (cid:98) F (0) ( λ ) (cid:16) − q E (cid:98) R E ( λ ± i (cid:17) ∈ B ( B ( E ) ; h λ ) . Then we have(3.10) 12 πi (cid:16)(cid:0) (cid:98) R E ( λ + i − (cid:98) R E ( λ − i (cid:1) (cid:98) f , (cid:98) g (cid:17) = ( (cid:98) F ( ± ) ( λ ) (cid:98) f , (cid:98) F ( ± ) ( λ ) (cid:98) g ) h λ . We can prove (3.10) first for (cid:98) H (0) E by (3.7), and then for (cid:98) H E by using the resolventequation (see Lemma 7.8 in [3]). Theorem 3.2. (1) The operator (cid:98) F ( ± ) is uniquely extended to a partial isometrywith initial set H ac ( (cid:98) H E ) and final set H annihilating H p ( (cid:98) H E ) , the point spectralsubspace for (cid:98) H E .(2) It diagonalizes (cid:98) H E : (cid:0) (cid:98) F ( ± ) (cid:98) H E (cid:98) f (cid:1) ( λ ) = λ (cid:0) (cid:98) F ( ± ) (cid:98) f (cid:1) ( λ ) , ∀ (cid:98) f ∈ D ( (cid:98) H E ) . (3) The adjoint operator (cid:98) F ( ± ) ( λ ) ∗ ∈ B ( h λ ; B ∗ ( E )) is an eigenoperator in the sensethat ( (cid:98) H E − λ ) (cid:98) F ( ± ) ( λ ) ∗ φ = 0 , ∀ φ ∈ h λ . (4) For (cid:98) f ∈ H ac ( (cid:98) H E ) , the inversion formula holds: (cid:98) f = (cid:90) ∞ (cid:98) F ( ± ) ( λ ) ∗ (cid:0) (cid:98) F ( ± ) (cid:98) f (cid:1) ( λ ) dλ. The crucial step for the inverse scattering procedure is Theorem 3.9 below, whichcan be proven by the same argument as in [3]. We do not repeat the whole proce-dure, but explain important intermediate steps. Let us prepare a lemma.
Lemma 3.3.
For a solution (cid:98) u of the equation ( (cid:98) H E − λ ) (cid:98) u = (cid:98) f satisfying the Kirchhoffcondition, we have the inequality C − λ (cid:107) (cid:98) u (cid:107) B ∗ ( E ) ≤ (cid:107) (cid:98) u (cid:12)(cid:12) V (cid:107) B ∗ ( V ) ≤ C λ (cid:107) (cid:98) u (cid:107) B ∗ ( E ) , and the equivalence (cid:98) u ∈ B ∗ ( E ) ⇐⇒ (cid:98) u (cid:12)(cid:12) V ∈ B ∗ ( V ) . NVERSE SCATTERING ON THE QUANTUM GRAPH FOR GRAPHENE 9
Proof.
Note that (cid:98) u e ( z ) is written as in (2.10). Since φ e ( t, λ ) and φ e ( t, λ ) arelinearly independent, there exists a constant C λ > e such that C − λ (cid:107) (cid:98) u e (cid:107) L e ≤ | c e (0) | + | c e (1) | ≤ C λ (cid:107) (cid:98) u e (cid:107) L e . The lemma then follows from this inequality. (cid:3) (I)
Rellich type theorem.
We define exterior and interior domains E ext,R and E int,R in E by E ext,R (cid:51) e ⇐⇒ | c ( e ) | ≥ R, E int,R (cid:51) e ⇐⇒ | c ( e ) | < R. Theorem 3.4.
Let λ ∈ (0 , ∞ ) \ T (0) , and suppose (cid:98) u ∈ (cid:98) B ∗ ( E ) satisfies (cid:98) H (0) E (cid:98) u = λ (cid:98) u in E ext,R , and the Kirchhoff condition for some R > . Then (cid:98) u = 0 on E ext,R for some R > .Proof. By Lemma 3.3, (cid:98) u (cid:12)(cid:12) V ∈ B ∗ ( V ). Since ( − ∆ V + cos √ λ ) (cid:98) u (cid:12)(cid:12) V = 0 near infinity,by Theorem 5.1 in [3], (cid:98) u (cid:12)(cid:12) V = 0 near infinity. This proves Theorem 3.4. (cid:3) We say that the operator (cid:98) H E − λ has the unique continuation property on E whenthe following assertion holds: If (cid:98) u satisfies ( (cid:98) H E − λ ) (cid:98) u = 0 on E and (cid:98) u = 0 on E ext,R for some R >
0, then (cid:98) u = 0 on E . The following lemma can be checked easily. Lemma 3.5.
For the hexagonal lattice in R , the unique continuation propertyholds. (II) Radiation condition . The radiation condition for the vertex Laplacian wasintroduced in [3] for the distinction between ( − ∆ V − λ − i − and ( − ∆ V − λ + i − .Hence it is extended to the edge Laplacian. Note that for the edge Laplacian, onemust replace λ in the definition (6.2) of [3] by cos √ λ . See [6] for details. Theorem 3.6.
Let λ ∈ (0 , ∞ ) \ T and (cid:98) f ∈ B ( E ) .(1) The solution (cid:98) u ∈ (cid:98) B ∗ ( E ) of the equation ( − (cid:98) ∆ E + q E − λ ) (cid:98) u = (cid:98) f satisfying theoutgoing or incoming radiation condition is unique.(2) ( (cid:98) H E − λ − i − (cid:98) f satisfies the outgoing radiation condition, and ( (cid:98) H E − λ + i − (cid:98) f satisfies the incoming radiation condition. (III) Singularity expansion . Asymptotic behavior at infinity of the resolvent isclosely related to the far-field behavior of the scattering waves. For the case ofscattering on perturbed lattices, instead of observing the asymptotic expansion of( (cid:98) H E − λ ∓ i − at infinity of the edge space E , it is more convenient to consider thesingularities of its Fourier transform in B ∗ . For f, g ∈ B ∗ ( E ), we use the notation f (cid:39) g in the following sense: f (cid:39) g ⇐⇒ f − g ∈ B ∗ ( E ) . We use the same notation (cid:39) for B ∗ ( V ).Since r E ( λ ) is bounded on L ( E ), we have r E ( λ ) (cid:98) f (cid:39) (cid:98) f ∈ L ( E ).Then, by (2.10) and (2.12), the singularities of ( (cid:98) H E − λ ∓ i − appear from (cid:16) − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ (cid:17) − (cid:98) T V ( λ ) (cid:98) f , which were studied in [3]. Therefore, in view of [3]Theorem 7.7, we have for f ∈ (cid:0) B ( T ) (cid:1) U V I (cid:0) − (cid:98) ∆ V + cos √ λ ± i (cid:1) − I ∗ U ∗V f (cid:39) (cid:88) j =1 λ j ( x ) + cos √ λ ∓ iσ ( λ )0 (cid:0) P V ,j ( x ) f (cid:1)(cid:12)(cid:12) M λ,j . (3.11) We denote the right-hand side as (cid:0) − (cid:98) ∆ V + cos √ λ ± i (cid:1) − f (cid:12)(cid:12) M λ . We can then prove the following theorem for (cid:98) H (0) E by using (3.11), and for (cid:98) H E bythe formula (3.9) and the resolvent equation. Theorem 3.7.
For any λ ∈ (0 , ∞ ) \ T and (cid:98) f ∈ B ( E ) , we have (cid:98) R E ( λ ± i (cid:98) f (cid:39) sin √ λ √ λ Φ (0) ( λ ) ∗ (cid:0) − (cid:98) ∆ V + cos √ λ ± i (cid:1) − (cid:0) U V ∗ (cid:98) F ( ± ) ( λ ) (cid:98) f (cid:1)(cid:12)(cid:12) M λ . Helmholtz equation and S-matrix.
Theorem 3.7 enables us to characterizethe solution space to the Helmholtz equation.
Lemma 3.8.
Let λ ∈ (0 , ∞ ) \ T and (cid:98) f ∈ B ( E ) . Then (3.12) { (cid:98) u ∈ (cid:98) B ∗ ( E ) ; ( (cid:98) H E − λ ) (cid:98) u = 0 } = (cid:98) F ( − ) ( λ ) ∗ h λ . Theorem 3.9.
For any incoming data φ in ∈ L ( M λ ) , there exist a unique solution (cid:98) u ∈ (cid:98) B ∗ ( E ) of the equation ( (cid:98) H E − λ ) (cid:98) u = 0 and an outgoing data φ out ∈ L ( M λ ) satisfying (cid:98) u (cid:39) − Φ (0) ( λ ) ∗ (cid:88) j =1 λ j ( x ) + cos √ λ + i σ ( λ ) (cid:0) P V ,j ( x ) φ inj (cid:1)(cid:12)(cid:12) M λ,j + Φ (0) ( λ ) ∗ (cid:88) j =1 λ j ( x ) + cos √ λ − i σ ( λ ) (cid:0) P V ,j ( x ) φ outj (cid:1)(cid:12)(cid:12) M λ,j . (3.13) The mapping S ( λ ) : φ in → φ out is the S-matrix, which is unitary on h λ . We omit the proof of Lemma 3.8 and Theorem 3.9, since they are almost thesame as that of Theorem 7.15 of [3].As is proven in [31], using the wave operator (cid:99) W ± = s − lim t →±∞ e it (cid:98) H E e − it (cid:98) H (0) E (cid:98) P ac ( (cid:98) H (0) E ) , where (cid:98) P ac ( (cid:98) H (0) E ) is the projection onto the absolutely continuous subspace of (cid:98) H (0) E ,one can define the scattering operator (cid:98) S = ( (cid:99) W (+) ) ∗ (cid:99) W ( − ) , which is unitary. Define S by S = (cid:98) F (0) (cid:98) S ( (cid:98) F (0) ) ∗ . The S-matrix S ( λ ) and the scattering amplitude A ( λ ) are defined by S ( λ ) = 1 − πiA ( λ ) , (3.14) A ( λ ) = (cid:98) F (+) ( λ ) q E (cid:98) F (0) ( λ ) . NVERSE SCATTERING ON THE QUANTUM GRAPH FOR GRAPHENE 11
Then S ( λ ) is unitary on h λ , and for λ ∈ (0 , ∞ ) \ T ( Sf )( λ ) = S ( λ ) f ( λ ) , f ∈ H . Since the resolvent has a meromorphic extension into the lower half-plane { Re λ > , Im λ < } with possible branch points on T , the formula (3.14) implies that theS-matrix S ( λ ) is also meromorphic in the same domain.4. From S-matrix to interior D-N map
Boundary value problem.
For a subgraph Ω = {V Ω , E Ω } ⊂ {V , E} and v ∈ V , v ∼ Ω means that there exist a vertex w ∈ V Ω and an edge e ∈ E such that v ∼ w , e (0) = v or e (1) = v . For a connected subgraph Ω ⊂ {V , E} , we define asubset ∂ Ω = {V ∂ Ω , E ∂ Ω } ⊂ {V , E} by V ∂ Ω = { v (cid:54)∈ V Ω ; v ∼ Ω } , E ∂ Ω = { e ∈ E ; e (0) ∈ V ∂ Ω or e (1) ∈ V ∂ Ω } . We then put Ω = Ω ∪ ∂ Ω and ◦ V Ω = V Ω , ∂ V Ω = V ∂ Ω , which are called the set of interior vertices and the set of boundary vertices of Ω,respectively. We put V Ω = ◦ V Ω ∪ ∂ V Ω . As for the edges, we simply put E Ω = E Ω ∪ E ∂ Ω . We then define the edge Dirichlet Laplacian (cid:98) ∆ E Ω by (cid:98) ∆ E Ω u e ( z ) = d dz u e ( z ) , e ∈ E Ω whose domain D ( (cid:98) ∆ E Ω ) is the set of all u = { u e } e ∈E Ω ∈ H ( E Ω ) satisfying u ( v ) = 0at any boundary vertex v ∈ ∂ V Ω and the Kirchhoff condition at any interior vertex v ∈ ◦ V Ω . By the standard argument, (cid:98) ∆ E Ω is self-adjoint.The vertex Dirichlet Laplacian on V Ω is defined in the same way as in (2.7) : (cid:0) (cid:98) ∆ V Ω ,λ (cid:98) u (cid:1) ( v ) = 1deg V Ω ( v ) (cid:88) w ∼ v,w ∈V Ω ψ wv (1 , λ ) (cid:98) u ( w ) , v ∈ V Ω . Recall that for a domain
W ⊂ V , we definedeg W ( v ) = (cid:93) { w ∈ W ; w ∼ v } , v ∈ ◦ W ,(cid:93) { w ∈ ◦ W ; w ∼ v } , v ∈ ∂ W . (See (2.6) of [4]). We impose the Dirichlet boundary condition for the domain D ( (cid:98) ∆ V Ω ,λ ) : (cid:98) u ∈ D ( (cid:98) ∆ V Ω ,λ ) ⇐⇒ (cid:98) u ∈ (cid:96) ( V Ω ) ∩ { (cid:98) u ; (cid:98) u ( v ) = 0 , v ∈ ∂ V Ω } . As in §
3, we first define the vertex Dirichlet Laplacian for the case without potentialand then add the pontntial (cid:98) Q V ,λ as a perturbation. By modifying the inner product, − (cid:98) ∆ V Ω ,λ + (cid:98) Q V ,λ is self-adjoint. The normal derivative at the boundary associatedwith (cid:98) ∆ V Ω ,λ is defined by(4.1) (cid:0) ∂ ν (cid:98) ∆ V Ω ,λ (cid:98) u (cid:1) ( v ) = − V Ω ( v ) (cid:88) w ∼ v,w ∈ ◦ V Ω ψ wv (1 , λ ) (cid:98) u ( w ) . (c.f. (2.7) of [4]). Note that in the right-hand side, w is taken only from ◦ V Ω .Let us give an example of interior and exterior domains as well as their boundariesfor the case of hexagonal lattice. We identify R with C and put ω = e πi/ = (1 + √ i ) /
2. Let D be the hexagon with center at the origin and vertices ω n , ≤ n ≤ − ω and 1 + ω , we put D k(cid:96) = D + k (2 − ω ) + (cid:96) (1 + ω ) , which denotes the translation of D by k (2 − ω ) and (cid:96) (1 + ω ). For an integer L ≥ D L = ∪ | k |≤ L, | (cid:96) |≤ L D k(cid:96) . As is illustrated in Figure 1, we take an interior domain Ω int in such a way that ◦ V Ω int = V ∩ D L , ◦ E Ω int = E ∩ D L . In Figure 1, ∂ V Ω int is denoted by white dots. Figure 1.
Boundary of a domain in the hexagonal latticeThe exterior domain Ω ext is defined similarly. We then put V int = V Ω int , E int = E Ω int , V ext = V Ω ext , E ext = E Ω ext , for the sake of simplicity. Note that V = V int ∪ V ext , ∂ V int = ∂ V ext , E = E int ∪ E ext , E int ∩ E ext = ∅ . We define the edge Dirichlet Laplacians on E int , E ext , which are denoted by (cid:98) ∆ int, E , (cid:98) ∆ ext, E : (cid:98) ∆ int, E = (cid:98) ∆ E int , (cid:98) ∆ ext, E = (cid:98) ∆ E ext . NVERSE SCATTERING ON THE QUANTUM GRAPH FOR GRAPHENE 13
Let us note that σ e (cid:0) − (cid:98) ∆ E (cid:1) = σ e (cid:0) − (cid:98) ∆ ext, E (cid:1) . We assume that the support of the potential lies strictly inside of E int . Namelyintroducing a set: (cid:103) E int = { e ∈ E int ; e (0) (cid:54)∈ ∂ V int , e (1) (cid:54)∈ ∂ V int } , we assume(4.2) supp q E ⊂ (cid:103) E int . The formal formulas (2.13), (2.17) are also valid for boundary value problemsof edge Laplacians. For the case of the exterior problem, the resolvent of − (cid:98) ∆ ext, E is written by (2.17) with (cid:98) H (0) E replaced by − (cid:98) ∆ ext, E . In our previous work [4], westudied the spectral properties of the vertex Laplacian in the exterior domain byreducing them to the whole space problem. Therefore, all the results for the edgeLaplacian in the previous section also hold in the exterior domain. In particular,we have • Rellich type theorem (Theorem 3.4), • Limiting absorption principle (Theorem 3.1), • Spectral representation (Theorem 3.2), • Resolvent expansion (Theorem 3.7), • Exapansion of solutions to the Helmholtz equation (Theorem 3.9), • S-matrix (Theorem 3.9)in the exterior domain E ext . In fact, Theorem 3.4 holds without any change. Usingthe formula (2.10) and the limiting absorption principle for (cid:98) ∆ ext proven in Theorem7.7 in [3], one can extend Theorem 3.1 for the exterior domain. The radiationcondition is also extended to the exterior domain. Then, the remaining theorems(Theorems 3.2, 3.9) are proven by the same argument.4.2. Exterior and interior D-N maps.
We consider the edge model for theexterior problem. Let (cid:98) u ( ± ) = { (cid:98) u ( ± ) e } e ∈E ext be the solution to the equation(4.3) ( − (cid:98) ∆ ext, E − λ ) (cid:98) u = 0 , in ◦ E ext , (cid:98) u = (cid:98) f , on ∂ E ext , satisfying the radiation condition (outgoing for (cid:98) u (+) and incoming for (cid:98) u ( − ) ). Then,the extrior D-N map Λ ( ± ) ext, E ( λ ) is defined by(4.4) Λ ( ± ) ext, E ( λ ) (cid:98) f ( v ) = − ddz (cid:98) u ( ± ) e ( v ) , v ∈ ∂ V ext , where e is the edge having v as its end point. Here, to compute ddz (cid:98) u ( ± ) e ( v ), wenegelect the original orientation of e . Namely, we parametrize e by z ∈ [0 ,
1] sothat v ∈ ∂ V corresponds to z = 0, and define ddz (cid:98) u ( ± ) e ( v ) = ddz (cid:98) u ( ± ) e ( z ) (cid:12)(cid:12) z =0 .For the case of the interior problem, the Dirichlet boundary value problem forthe edge Laplacian(4.5) ( − (cid:98) ∆ int, E + q e − λ ) (cid:98) u = 0 , in ◦ E int , (cid:98) u = (cid:98) f , on ∂ V int is formulated as above. Note that the spectrum of − (cid:98) ∆ int, E + q E is discrete. In thefollowing, we assume that(4.6) λ (cid:54)∈ σ ( − (cid:98) ∆ int, E + q E ) . The D-N map Λ int, E ( λ ) is defined by(4.7) Λ int, E ( λ ) (cid:98) f ( v ) = ddz (cid:98) u e ( v ) , v ∈ ∂ V int , where e is the edge having v as its end point and (cid:98) u = { (cid:98) u ( ± ) e } e ∈E int is the solutionto the equation (4.5). The same remark as above is applied to ddz (cid:98) u e ( v ) , v ∈ ∂ V int .The D-N maps are also defined for vertex operators. Let us slightly change thenotation. For a subset V D ⊂ V and v ∈ V D , let( (cid:98) ∆ (0) V (cid:98) u )( v ) = 13 (cid:88) w ∼ v (cid:98) u ( w ) , ( (cid:98) ∆ (0) V D (cid:98) u )( v ) = 1deg V D ( v ) (cid:88) w ∼ v,w ∈V D (cid:98) u ( w ) . By this definition, we have (see (2.15))(4.8) (cid:98) ∆ (0) V ,λ = √ λ sin √ λ (cid:98) ∆ (0) V . For the exterior and interior domains Ω ext and Ω int defined in the previous section, (cid:98) ∆ (0) V D is denoted by (cid:98) ∆ ext, V and (cid:98) ∆ int, V , respectively: (cid:98) ∆ ext, V = (cid:98) ∆ (0) V ext , (cid:98) ∆ int, V = (cid:98) ∆ (0) V int . Now, consider the exterior boundary value problem(4.9) (cid:0) − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ (cid:1)(cid:98) u = 0 , in ◦ V ext , (cid:98) u = (cid:98) f , on ∂ V ext . Note that by (4.8) and (2.16) this is equivalent to ( − (cid:98) ∆ (0) V + cos √ λ ) (cid:98) u = 0 , in ◦ V ext , (cid:98) u = (cid:98) f , on ∂ V ext . Let (cid:98) u ( ± ) ext, V be the solution of this equation satisfying the radiation condition. Then,taking account of (4.1) and (4.8), we define the exterior D-N map by (cid:98) Λ ( ± ) ext, V ( λ ) (cid:98) f = − sin √ λ √ λ ∂ ν (cid:98) ∆ V ext,λ (cid:98) u ( ± ) ext, V = ∂ ν (cid:98) ∆ ext, V (cid:98) u ( ± ) ext, V = 1deg V ext ( v ) (cid:88) w ∼ v,w ∈ ◦ V ext (cid:98) u ( ± ) ext, V ( w ) . (4.10)We also consider the interior boundary value problem(4.11) (cid:0) − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ (cid:1)(cid:98) u = 0 , in ◦ V int , (cid:98) u = (cid:98) f , on ∂ V int . NVERSE SCATTERING ON THE QUANTUM GRAPH FOR GRAPHENE 15
Taking account of (4.2), we define the interior D-N map by (cid:98) Λ int, V ( λ ) (cid:98) f ( v ) = sin √ λ √ λ ∂ ν (cid:98) ∆ V int,λ (cid:98) u int, V = − ∂ ν (cid:98) ∆ int, V (cid:98) u int, V = − V int ( v ) (cid:88) w ∼ v,w ∈ ◦ V int (cid:98) u int, V ( w ) . (4.12)Note that by virtue of Lemma 2.2, if (cid:98) u satisfies the edge Schr¨odinger equation( (cid:98) H E − λ ) (cid:98) u = 0 and the Kirchhoff condition, (cid:98) u (cid:12)(cid:12) V satisfies the vertex Schr¨odingerequation (cid:0) − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ (cid:1)(cid:98) u (cid:12)(cid:12) V = 0. Therefore, if the exterior boundary value problem(4.3) for the edge model is solvable, so is the exterior boundary value problem (4.9)for the vertex model. The same remark applies to the interior boundary valueproblem.If ϕ ( z ) satisfies − ϕ (cid:48)(cid:48) ( z ) − λϕ ( z ) = 0 in (0 , ϕ (1) = ϕ (0) cos √ λ + ϕ (cid:48) (0) sin √ λ √ λ , Since the D-N map for the vertex model is computed by (cid:98) u (cid:12)(cid:12) V , where (cid:98) u is the solutionto the edge Schr¨odinger equation, this implies, by (4.4), (4.7), (4.10) and (4.12), thefollowing formulas between the D-N maps of edge-Laplacian and vertex Laplacian. Lemma 4.1.
The following equalities hold: (cid:98) Λ ( ± ) ext, V ( λ ) = cos √ λ − sin √ λ √ λ Λ ( ± ) ext, E ( λ ) , λ ∈ (0 , ∞ ) \ T , (cid:98) Λ int, V ( λ ) = − cos √ λ − sin √ λ √ λ Λ int, E ( λ ) , λ ∈ C \ σ ( − (cid:98) ∆ int, E + q E ) . Therefore, the D-N map for the edge model and the D-N map for the vertexmodel determine each other.4.3.
Relations between S-matices and D-N maps.
We show that the S-matrices for the vertex Laplacian and the edge Laplacian coincide.In [3], Theorem 7.15, we have proven the following theorem, which is the counterpart of Theorem 3.9 for the discrete Laplacian − (cid:98) ∆ V at the energy − cos √ λ : Forany incoming data φ in ∈ L ( M λ ), there exist a unique solution (cid:101) u V ∈ (cid:98) B ∗ ( V ) of theequation ( − (cid:98) ∆ V + cos √ λ ) (cid:101) u V = 0and an outgoing data (cid:101) φ out ∈ L ( M λ ) satisfying I (cid:101) u V (cid:39) − (cid:88) j =1 λ j ( x ) + cos √ λ + i σ ( λ ) (cid:0) P V ,j ( x ) φ inj (cid:1)(cid:12)(cid:12) M λ,j + (cid:88) j =1 λ j ( x ) + cos √ λ − i σ ( λ ) (cid:0) P V ,j ( x ) (cid:101) φ outj (cid:1)(cid:12)(cid:12) M λ,j . (4.13)in the sense that the difference of both sides is in B ∗ ( T ; C ). The mapping (cid:101) S ( λ ) : φ in → (cid:101) φ out is the S-matrix of − (cid:98) ∆ V at the energy − cos √ λ , which is unitary on h λ . By virtue of Lemma 3.3, we see that (cid:101) u E := Φ (0) ( λ ) ∗ (cid:101) u V has the properties inTheorem 3.9, hence by the uniqueness u = (cid:101) u E . Therefore, (cid:101) φ ( out ) = φ ( out ) , whichimplies S ( λ ) = (cid:101) S ( λ ). We have thus proven the followin theorem. Theorem 4.2.
The S-matrix for the vertex Schr¨odinger operator at the energy − cos √ λ coincides with that of the edge Schr¨odinger operator at the energy λ . In [4], we have proven that for the vertex Laplacian the scattering matrix andthe interior D-N map determine each other. By virtue of Theorems 4.1 and 4.2, wehave the following theorem.
Theorem 4.3.
For the edge Laplacian on the hexagonal lattice, the S-matrix andthe D-N map in the interior domain determine each other. Inverse scattering
Hexagonal parallelogram.
We are now in a position to consider the inversescattering problem. Note here that although the choice of fundamental domainof the lattice L is not unique, different choices give rise to unitarily equivalentHamiltonians. In this section, we take v , v and p (1) , p (2) as in (5.1) and (5.2) tomake use of our previous results in [3], [4]. We identify R with C , and put ω = e πi/ . For n = n + in ∈ Z [ i ] = Z + i Z , let L = { v ( n ) ; n ∈ Z [ i ] } , v ( n ) = n v + n v , (5.1) v = 1 + ω, v = √ i, (5.2) p = ω − = ω , p = 1 , and define the vertex set V by V = V ∪ V , V i = p i + L . By virtue of Theorem 4.3, given an S-matrix and a bounded domain E int , we cancompute the D-N map associated with E int . The problem is now reduced to thereconstruction of the potentials on the edges from the knowledge of the D-N mapfor the vertex Schr¨odinger operator defined on V int , the set of the vertices in E int .As V int , we use the following domain which is different from the one in Figure1. Let D be the Wigner-Seitz cell of V . It is a hexagon having 6 vertices ω k , ≤ k ≤
5, with center at the origin. Take D N = { n ∈ Z [ i ] ; 0 ≤ n ≤ N, ≤ n ≤ N } ,where N is chosen large enough, and put D N = ∪ n ∈ D N (cid:16) D + v ( n ) (cid:17) . This is a parallelogram in the hexagonal lattice (see Figure 2). The interior angleof each vertex on the periphery of D N is either 2 π/ π/
3. Let A be the setof the former, and for each z ∈ A , we assign a new edge e z,ζ , and a new vertex ζ = t ( e z,ζ ) on its terminal point, hence ζ is in the outside of D N . LetΩ = { v ∈ V ; v ∈ D N } NVERSE SCATTERING ON THE QUANTUM GRAPH FOR GRAPHENE 17
Figure 2.
Hexagonal parallelogram ( N = 2)be the set of vertices in the inside of the resulting graph. The boundary ∂ Ω = { t ( e z,ζ ) ; z ∈ A} is divided into 4 parts, called top, bottom, right, left sides, whichare denoted by ( ∂ Ω) T , ( ∂ Ω) B , ( ∂ Ω) R , ( ∂ Ω) L , i.e.( ∂ Ω) T = { α , · · · , α N } , ( ∂ Ω) B = { ω + k (1 + ω ) ; 0 ≤ k ≤ N } , ( ∂ Ω) R = { N (1 + ω ) + k √ i ; 1 ≤ k ≤ N } ∪ { N (1 + ω ) + N √ i + 2 ω } , ( ∂ Ω) L = { ω } ∪ { β , · · · , β N } , where α k = β N + 2 ω + k (1 + ω ) and β k = − k √ i for 0 ≤ k ≤ N .5.2. Special solutions to the vertex Schr¨odinger equation.
Taking N largeenough so that D N contains all the supports of the potentials q e ( z ) in its interior,we consider the following Dirichlet problem for the vertex Schr¨odinger equation(5.3) ( − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ ) (cid:98) u = 0 , in ◦ Ω , (cid:98) u = (cid:98) f , on ∂ Ω . Let Λ (cid:98) Q be the associated D-N map. The key to the inverse procedure is thefollowing partial data problem. Lemma 5.1. (1) Given a partial Dirichlet data (cid:98) f on ∂ Ω \ ( ∂ Ω) R , and a partialNeumann data (cid:98) g on ( ∂ Ω) L , there is a unique solution (cid:98) u on ◦ Ω ∪ ( ∂ Ω) R to the equation (5.4) ( − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ ) (cid:98) u = 0 , in ◦ Ω , (cid:98) u = (cid:98) f , on ∂ Ω \ ( ∂ Ω) R ,∂ D N ν (cid:98) u = (cid:98) g, on ( ∂ Ω) L . (2) Given the D-N map Λ (cid:98) Q , a partial Dirichlet data (cid:98) f on ∂ Ω \ ( ∂ Ω) R and apartial Neumann data (cid:98) g on ( ∂ Ω) L , there exists a unique (cid:98) f on ∂ Ω such that (cid:98) f = (cid:98) f on ∂ Ω \ ( ∂ Ω) R and Λ (cid:98) Q (cid:98) f = (cid:98) g on ( ∂ Ω) L . For the proof, see [4], Lemma 6.1.Now, for 0 ≤ k ≤ N , let us consider a diagonal line A k (see Figure 3) :(5.5) A k = { x + ix ; x + √ x = a k } , where a k is chosen so that A k passes through(5.6) α k = α + k (1 + ω ) ∈ ( ∂ Ω) T . The vertices on A k ∩ Ω are written as(5.7) α k,(cid:96) = α k + (cid:96) (1 + ω ) , (cid:96) = 0 , , , · · · . Figure 3.
Line A k Lemma 5.2.
Let A k ∩ ∂ Ω = { α k, , α k,m } . Then, there exists a unique solution (cid:98) u to the equation (5.8) (cid:0) − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ (cid:1)(cid:98) u = 0 in ◦ Ω , with partial Dirichlet data (cid:98) f such that (5.9) (cid:40) (cid:98) f ( α k, ) = 1 , (cid:98) f ( z ) = 0 for z ∈ ∂ Ω \ (cid:0) ( ∂ Ω) R ∪ α k, ∪ α k,m (cid:1) and partial Neumann data (cid:98) g = 0 on ( ∂ Ω) L . It satisfies (5.10) (cid:98) u ( x + ix ) = 0 if x + √ x < a k . An important feature is that (cid:98) u vanishes below the line A k . By using this property,we reconstructed the vertex potentials and defectes of the hexagonal lattice in [4].We make use of the same idea. NVERSE SCATTERING ON THE QUANTUM GRAPH FOR GRAPHENE 19
Let (cid:98) u be a solution of the equation(5.11) ( − (cid:98) ∆ V ,λ + (cid:98) Q V ,λ ) (cid:98) u = 0 , in ◦ Ω , which vanishes in the region x + √ x < a k . Let a, b, b (cid:48) , c ∈ V and e , e (cid:48) ∈ E be asin Figure 4. Figure 4. (cid:98) u ( b ) and (cid:98) u ( b (cid:48) )Then, evaluating the equation (5.11) at v = a and using (2.7), (2.8), we obtain(5.12) 1 ψ ba (1 , λ ) (cid:98) u ( b ) + 1 ψ b (cid:48) a (1 , λ ) (cid:98) u ( b (cid:48) ) = 0 . Here, for any edge e ∈ E , we associate an edge [ e ] without orientation and a function φ [ e ] ( z, λ ) satisfying (cid:18) − d dz + q e ( z ) − λ (cid:19) φ [ e ] ( z, λ ) = 0 , for 0 < z < ,φ [ e ] (0 , λ ) = 0 , φ (cid:48) [ e ] (0 , λ ) = 1 . By the assumption (Q-3), φ [ e ] ( z, λ ) is determined by e and independent of theorientation of e . Then, the equation (5.12) is rewritten as(5.13) (cid:98) u ( b ) = − φ [ e ] (1 , λ ) φ [ e (cid:48) ] (1 , λ ) (cid:98) u ( b (cid:48) ) . Let e k, , e (cid:48) k, , e k, , e (cid:48) k, , · · · be the series of edges just below A k starting from thevertex α k , and put(5.14) f k,m ( λ ) = − φ [ e k,m ] (1 , λ ) φ [ e (cid:48) k,m ] (1 , λ ) . Then, we obtain the following lemma.
Lemma 5.3.
The solution (cid:98) u in Lemma 5.2 satisfies (cid:98) u ( α k,(cid:96) ) = f k, ( λ ) · · · f k,(cid:96) ( λ ) . Reconstruction procedure.
We now prove Theorem 1.1 by showing thereconstruction algorithm of the potential q e ( z ). . We first take a sufficiently large hexagonal parallelogram Ω as in Figure 2which contains all the supports of the potential q e ( z ). . For an arbitrary k , draw a line A k as in Figure 3 and take the boundarydata (cid:98) f having the properties in Lemma 5.2. . Compute the values of the associated solution (cid:98) u to the boundary valueproblem in Lemma 5.2 at the points α k,(cid:96) , (cid:96) = 0 , , , · · · . . Look at Figure 2. Two edges e and e (cid:48) between A k and A (cid:48) k are said tobe A (cid:48) k -adjacent if they have a vertex in common on A (cid:48) k (see Figure 4). Take two A (cid:48) k -adjacent edges e and e (cid:48) between A k and A (cid:48) k , and use the formula (5.14) tocompute the ratio of φ [ e ] (1 , λ ) and φ [ e (cid:48) ] (1 , λ ). . Rotate the whole system by the angle π and take a hexagonal parallelo-gram congruent to the previous one. Then, the roles of A k and A (cid:48) k are exchanged.One can then compute the ratio of φ [ e ] (1 , λ ) and φ [ e (cid:48) ] (1 , λ ) for A (cid:48) k -adjacent pairs inthe sense after the rotation, which are A k -adjacent before the rotation.After the 4th and 5th steps, for all pairs e and e (cid:48) which are either A k -adjacentor A (cid:48) k -adjacent, one has computed the ratio of φ [ e ] (1 , λ ) and φ [ e (cid:48) ] (1 , λ ). . Take a zigzag line on the hexagonal lattice (see Figure 5), and take anytwo edges e and e (cid:48) on it. They are between A k and A (cid:48) k for some k . Then, usingthe 4th and 5th steps, one can compute the ratio of φ [ e ] (1 , λ ) and φ [ e (cid:48) ] (1 , λ ) bycomputing the ratio for two successive edges between e and e (cid:48) . Figure 5.
Zigzag line in the hexagonal lattice . For a sufficiently remote edge e (cid:48) , one knows φ [ e (cid:48) ] (1 , λ ) since q e (cid:48) ( z ) = 0on e (cid:48) . One can thus compute φ [ e ] (1 , λ ) for any edge e . Then, by the analyticcontinuation, one can compute the zeros of φ [ e ] (1 , λ ) for any edge e . . Note that the zeros of φ [ e ] (1 , λ ) are the Dirichlet eigenvalues for theoperator − ( d/dz ) + q e ( z ) on (0 , q e ( z ).We have now completed the proof of Theorem 1.1.Note that for the 1st step, we need a-priori knowledge of the size of the supportof the potential q E ( z ). The knowledge of the D-N map is used in the 2nd step (inthe proof of Lemma 5.1). In the 3rd step, one uses the equation (5.8) and the factthat (cid:98) u = 0 below A k .The proof of Theorem 1.2 requires no essential change. Instead of sin √ λz √ λ and sin √ λ (1 − z ) √ λ , we have only to use the corresponding solutions to the Schr¨odingerequation (cid:0) − ( d/dz ) + q ( z ) − λ ) ϕ = 0. NVERSE SCATTERING ON THE QUANTUM GRAPH FOR GRAPHENE 21
Acknowledgement
K. A. is supported by Grant-in-Aid for Scientific Reserach (C)17K05303, Japan Society for the Promotion of Science (JSPS). H. I. is supportedby Grant-in-Aid for Scientific Research (C) 20K03667, JSPS. E. K. is supportedby the RFBR grant No. 19-01-00094. H. M. is supported by Grant-in-Aid forYoung Scientists (B) 16K17630, JSPS. The authors express their gratitude to thesesupports.
References [1] S. Agmon and L. H¨ormander,
Asymptotic properties of solutions of differential equationswith simple characteristics , J. d’Anal. Math., (1976), 1-38.[2] K. Ando, Inverse scattering theory for discrete Schr¨odinger operators on the hexagonal lat-tice , Ann. Henri Poincar´e (2013), 347-383.[3] K. Ando, H. Isozaki and H. Morioka, Spectral properties of Schr¨odinger operators on perturbedlattices , Ann. Henri Poincar´e (2016), 2103-2171.[4] K. Ando, H. Isozaki and H. Morioka, Inverse scattering for Schr¨odinger operators on per-turbed lattices , Ann. Henri Poincar´e (2018), 3397-3455.[5] K. Ando, H. Isozaki and H. Morioka, Correction to : Inverse scattering for Schr¨odingeroperators on perturbed lattices , Ann. Henri Poincar´e (2019), 337-338.[6] K. Ando, H. Isozaki, E. Korotyaev and H. Morioka, Inverse scattering on the quantum graph— Edge model for graphene , arXiv:1911.05233.[7] S. Avdonin, B. P. Belinskiy, and J. V. Matthews,
Dynamical inverse problem on a metrictree , Inverse Porblems (2011), 075011.[8] M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BCmethod , Inverse Problems (2004), 647-672.[9] J. von Below, A characteristsic equation associated to an eigenvalue problem on c -networks ,Linear Algebra Appl. (1985), 309-325.[10] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs , Mathematical Surveysand Mnonographs , AMS (2013).[11] N. Bondarenko and C. T. Shieh,
Partial inverse problem for Sturm-Liouville operators ontrees , Proceedings of the Royal Society of Edingburgh (2017), 917-933.[12] N. Bondarenko,
Spectral data characterization for the Sturm-Liouville operator on the star-shaped graph , arXiv:2009.02522v1[13] G. Borg,
Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Dif-ferentialgleichung durch die Eigenwerte , Acta Math. (1946), 1-96.[14] B. M. Brown and R. Weikard, A Borg-Levinson theorem for trees , Proc. Royal Soc. Lond.Ser. A Math. Phys. Eng. Sci.
Spectra of self-adjoint extensions and applicationsto solvable Schr¨odinger operators , Rev. Math. Phys. (2008), 1-70.[16] C. Cattaneo, The spectrum of the continuous Laplacian on a graph , Monatsh. Math. (1997), 215-235.[17] T. Chen, P. Exner and O. Turek,
Inverse scattering for quantum graph vertices , Phys. Rev.A (2011), 86:062715.[18] F. Chung,
Spectral Graph Theory , AMS. Providence, Rhodse Island (1997).[19] Y. Colin de Verdi`ere,
Spectre de graphes , Cours sp´ecialis´es , S. M. F., Paris, (1998).[20] E. B. Curtis and J. A. Morrow, Inverse Problems for Electrical Networks , On Applied Math-ematics, World Scientific, (2000).[21] D. Cvetkovic, M. Doob, I. Gutman and A. Torgasev,
A recent result in the theory of graphspectra , Annals of Discrete Mathematics , North-Holland Publishing Co., Amsterdam(1988).[22] D. Cvetkovic, M. Doob and H. Saks, Spectra of graphs, Theory and applications , 3rd edition,Johann Ambrosius Barth, Heidelberg (1995).[23] P. Exner, A. Kostenko, M. Malamud and H. Neidhardt,
Spectral theory for infinite quantumgraph , Ann. Henri Poincar´e (2018), 3457-3510.[24] M. S. Eskina, The direct and the inverse scattering problem for a partial difference equation ,Soviet Math. Doklady, (1966), 193-197. [25] B. Gutkin and U. Smilansky, Can one hear the shape of a graph?
J. Phys. A (2001),6061-6068.[26] H. Isozaki and E. Korotyaev, Inverse problems, trace formulae for discrete Schr¨odinger op-erators , Ann. Henri Poincar´e, (2012), 751-788.[27] H. Isozaki and H. Morioka, Inverse scattering at a fixed energy for discrete Schr¨odingeroperators on the square lattice , Ann. l’Inst. Fourier (2015), 1153-1200.[28] E. Korotyaev and I. Lobanov, Schr¨odinger operators on zigzag nanotubes , Ann. HenriPoincar´e (2007), 1151-1076.[29] E. Korotyaev and N. Saburova, Schr¨odinger operators on periodic discrete graphs , J. Math.Anal. Appl. (2014), 576-611.[30] E. Korotyaev and N. Saburova,
Spectral band localization for Schr¨odinger operators on peri-odic graphs , Proc. Amer. Math. Soc. (2015), 3951-3967.[31] E. Korotyaev and N. Saburova,
Scattering on metric graphs , arXiv:1507.06441v1 [math.SP]23 Jul 2015.[32] E. Korotyaev and N. Saburova,
Estimates of bands for Laplacians on periodic equilateralmetric graphs , Proc. Amer. Math. Soc. (2016), 1605-1617.[33] E. Korotyaev and N. Saburova,
Effective masses for Laplacians on periodic graphs , J. Math.Anal. Appl. (2016), 104-130.[34] V. Kostrykin and R. Schrader,
Kirchhoff’s rule for quantum wires , J. Phys. A (1999),595-630.[35] P. Kuchment, Quantum graph spectra of a graphyne structure , NanoNMTA, (2013), 107-123.[36] P. Kuchment and O. Post, On the spectra of carbon nano-structures , Commun. Math. Phys. (2007), 805-826.[37] P. Kuchment and B. Vainberg,
On absence of embedded eigenvalues for Schr¨odinger operatorswith perturbed periodic potentials , Comm. PDE, (2000), 1809-1826.[38] P. Kurasov, Schr¨odinger operators on graphs and geometry. I. Essentially bounded potentials ,J. Funct. Anal. (2008), 934-953.[39] N. Levinson,
The inverse Sturm-Liouville problem , Mat. Tidsskr. B. (1949), 25-30.[40] K. Mochizuki and I. Yu. Trooshin,
On the scattering on a loop-shaped graph , Progress ofMath. (2012), 227-245.[41] S. Nakamura,
Modified wave operators for discrete Scr¨odinger operators with long-rangeperturbations , J. Math. Phys. (2014), 112101.[42] H. Niikuni, Spectral band structure of periodic Schr¨odinger operators with two potentials onthe degenerate zigzag nanotube , J. Appl. Math. Comput. (2016) 50:453-482.[43] K. Pankraskin,
Spactra of Schr¨odinger operators on equilateral quantum graphs , Lett. Math.Phys. (2006), 139-154.[44] D. Parra and S. Richard, Spectral and scattering theory for Schr¨odinger operators on per-turbed topological crystals , Rev. Math. Phys. (2018), Article No. 1850009, pp 1-39.[45] V. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph , SIAMJ. Math. Anal. (2000), 801-819.[46] O. Post, Spectal Analysis on Graph-like Spaces , Lecture Notes in Mathematics ,Springer, Heidelberg (2012).[47] J. P¨oschel and E. Trubowitz,
Inverse Spectral Theory , Academic Press, Boston, (1987).[48] W. Shaban and B. Vainberg,
Radiation conditions for the difference Schr¨odinger operators ,Applicable Analysis, (2001), 525-556.[49] Y. Tadano, Long-range scattering for discrete Schr¨odinger operators , Ann. Henri Poincar´e (2019), 1439-1469.[50] F. Visco-Comandini, M. Mirrahimi, and M. Sorine, Some inverse scattering problems onstar-shaped graphs , J. Math. Anal. Appl. (2011), 343-358.[51] X. C. Xu and C. F. Yang,
Determination of the self-adjoint matrix Schr¨odinger operatorswithout the bound state data , Inverse Problems (2018), 065002 (20pp).[52] V. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs , Inverse Prob-lems (2005), 1075-1086.[53] V. Yurko, Inverse spectral problems for differential operators on spatial networks , Russ. Math.Surveys , No 3 (2016), 539-584. NVERSE SCATTERING ON THE QUANTUM GRAPH FOR GRAPHENE 23 (K. Ando)
Department of Electrical and Electronic Engineering and Computer Sci-ence, Ehime University, Matsuyama, 790-8577, Japan
Email address : [email protected] (H. Isozaki) Graduate School of Pure and Applied Sciences, Professor Emeritus, Uni-versity of Tsukuba, Tsukuba, 305-8571, Japan
Email address : [email protected] (E. Korotyaev) Department of Math. Analysis, Saint-Petersburg State University,Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia, National Research Univer-sity Higher School of Economics, St. Petersburg, Russia
Email address : [email protected] (H. Morioka) Department of Electrical and Electronic Engineering and ComputerScience, Ehime University, Matsuyama, 790-8577, Japan
Email address ::