Inverse problems for quantum graph associated with square and hexagonal lattices
K. Ando E. Blåsten P. Exner H. Isozaki E. Korotyaev M. Lassas J. Lu H. Morioka

14pt

A Software Visualization Approach for 

Multiple Visual Output Devices

    Malte Hansen
Department of Computer Science 

Kiel University

Kiel, Germany 

malte.hansen@email.uni-kiel.de
Heiko Bielfeldt
Department of Computer Science 

Kiel University

Kiel, Germany 

heikobielfeldt@gmail.com
Armin Bernstetter

GEOMAR Helmholtz Centre for Ocean Research Kiel

Kiel, Germany 

abernstetter@geomar.de
Tom Kwasnitschka

GEOMAR Helmholtz Centre for Ocean Research Kiel

Kiel, Germany 

tkwasnitschka@geomar.de
Wilhelm Hasselbring
Department of Computer Science 

Kiel University

Kiel, Germany 

        hasselbring@email.uni-kiel.de

    September 9, 2024
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

§ ABSTRACT 
We solve  inverse  problems from the D-N map for the quantum graph on a finite domain in a square lattice and that on a hexagonal lattice, as well as inverse scattering problems from the S-matrix for a  locally perturbed square lattice and  a hexagonal lattice.





§ INTRODUCTION



 §.§ Gel'fand problem


In the International Congress of Mathematics at Amsterdam in 1954, I.M. Gel'fand raised the following problem (an extended form): Let (M,g) be a compact Riemannian manifold with boundary. Let λ_0 < λ_1 ≤⋯ be the eigenvalues and φ_0(x), φ_1(x), …  the associated orthnormal eigenvectors for the operator H = - c(x)^2Δ_g + V(x) on M with the Dirichlet boundary condition on ∂ M.
Then, from the knowledge of the boundary spectral data (BSD)  {(λ_n,∂/∂νφ_n|_∂ M)  ;   n = 0, 1, 2, …}, where ν is the unit normal at the boundary, can one determine the manifold M and the operator H? 


For the case of Riemannian manifold, this problem was solved by Belishev-Kurylev <cit.> using the boundary control method developped  by Belishev <cit.>.  The solution is unique up to a diffeomorphsim leaving ∂ M invariant, and this is the only obstruction for this problem.



 §.§ Quantum graph Hamiltonian and discrete operator



We are interested in the analogue of Gel'fand's problem on graphs. There are two models in which one can do that. The first one concerns graphs Γ = {𝒱,ℰ} understood as a vertex set 𝒱 and an edge set ℰ determined by an adjacency matrix; we are then interested in the discrete operator defined on the vertex set,
 

    Ĥû(x) = 1/μ_x∑_y ∼ x g_xy(û(y) - û(x)) +  q(x)û(x),    x ∈𝒱,

 
where y ∼ x means that x and y are the endpoints of a same edge e = e_xy∈ℰ, μ : 𝒱→ℝ_+ = (0,∞) is a weight on 𝒱, g : ℰ→ℝ_+ is a weight on ℰ and q : 𝒱→ℝ is a scalar potential. The other one concerns the so-called quantum graphs <cit.> in which the edges are identified with line segments and the Hamiltonian is a collection of one-dimensional Schrödinger operators defined on them,
 

    H_ℰ = {h_e =  - d^2/dz^2 + V_e(z),   z ∈ [0,ℓ_e] }_e ∈ℰ,

 
with a real-valued potential V_e(z). To make such an operator self-adjoint, one has to match the function properly at the vertices; we choose the simplest possibility assumig the δ-coupling condition, see (<ref>) below. 

 It was proved in <cit.>  that in the case of the discrete graph,  the graph structure and the coefficients of (<ref>) are determined by BSD  under the Two-Point Condition introduced there. We emphasize here that the inverse problem for the discrete graph cannot be solved without such conditions, as documented by a counter example given in <cit.>. Recall that the knowledge of BSD is equivalemt to the knowledge of the associated D-N map for all energies. The indicated result has various applications, in particular, those concerning the following three  issues:
 


  * Inverse boundary value problems for random walks.

  * Inverse scattering problems for locally perturbed discrete periodic graphs.

  * Inverse boundary value problems and inverse scattering problems for quantum (metric) graph.

 
The problem (1) has been discussed in <cit.>, where it was shown that the graph structure and the transition matrix of the random walk can be uniquely recovered from the distribution of the first passing time on the boundary, or from the observation on the boundary of one realization of the random walk.  

The problem (2) is  considered in <cit.>.
For a locally perturbed discrete periodic graph, the structure of the perturbed subgraph, along with the edge weight g and the potential q can be recovered from the scattering matrix at all energies provided that the Two-Points Condition is preserved under the perturbations.

For a  fixed energy λ, the S-matrix S(λ) of the whole system and the D-N map Λ(λ) determine each other, and thus the problem is reduced to that of the bounded domain, to which we can apply the result of <cit.>. In particular, if two locally perturbed periodic lattices have the same S-matrix for all energies, we can conclude:
 (i) If μ = μ', then g = g' and q = q'. (ii) If q = q', then μ = μ' and g = g'. (iii) In particular, if μ(v) =  deg (v), μ'(v') =  deg (v'), then g = g' and q=q'.

The well known duality between the discrete and metric graphs <cit.> allows to determine the spectrum of an equilateral `continuous' graph from that of its discrete counterpart and vice versa, and therefore the discrete Gel'fand problem is expected to play a role in inverse problems for quantum graphs. In <cit.>, we have considered such equilateral graphs, i.e. those
in which all the edges have the same length and the one-dimensional Hamiltonian has the same potential on them.
Letting C_v be the δ-coupling constant (see (<ref>)  below), and d_v the degree of v ∈𝒱, it was assumed that C_v/d_v is a constant independent of v ∈𝒱. We then proved that if two such quantum graphs 𝔾_Γ and 𝔾_Γ' have the same D-N map (or the same S-matrix) for all energies, then there exists a bijection 𝔾_Γ→𝔾_Γ' preserving the edge relation. Moreover,  we have d_v = d_v' and C_v = C_v' ∀ v ∈𝒱.
Therefore, the S-matrix of an equilateral quantum graph determines its graph structure. 

In this paper, we continue the investigation of <cit.>,  and consider the problem of determining local perturbation of C_v and V_e(z) for lattices such that ℓ_e = 1 holds for all e ∈ℰ and the degree d_v is the same for all v ∈𝒱.
Our method is different from that of <cit.> relying strongly on the explicit form of the lattice. Therefore, although we are convinced that the result will be true for a larger class of lattices, we restrict ourselves to the proof  for  the square and hexagonal situations.




 §.§ Edge and vertex  Schrödinger operators in the quantum graph


Let us  recall basic facts about quantum graphs. Let  Γ = {𝒱, ℰ} be a quantum (or metric) graph with a vertex set 𝒱 and edge set ℰ. We assume that  each edge e has unit length and identify it with the interval [0,1]. Consider a quantum-graph  Schrödinger operator, called the edge Schrödinger operator in this paper,
 

    H_ℰ = {h_e =  - d^2/dz^2 + V_e(z),   z ∈ [0,1] }_e ∈ℰ,

 
with a real-valued potential satisfying the symmetry condition
 

    V_e(z) ∈ L^2((0,1)),    V_e(z) = V_e(1 -z),   ∀ e ∈ℰ.

 
The generalized Kirchhoff condition, or the δ-coupling condition, is imposed[Here,  v ∈ e means that v is an end point of an edge e and the derivative is conventionally taken in the outward direction.]:
 

    ∑_v ∈ eu'_e(v) = C_vu_e(v),   ∀ v ∈𝒱^o := 𝒱∖∂𝒱,

 
C_v being a real constant. Here, ∂𝒱 is the boundary of 𝒱, which will be chosen suitably later. Let ϕ_e(z,λ) be the solution of
 

    {   (- d^2/dz^2 + V_e(z))ϕ_e(z,λ) = λϕ_e(z,λ),   
    z ∈ [0,1], 
       ϕ_e(0,λ) = 0,   ϕ'_e(0,λ) = 1,     ^'= d/dz.
    .

 
We put
 

    ϕ_e0(z,λ) = ϕ_e(z,λ),   ϕ_e1(z,λ) = ϕ_e(1 - z,λ).

 
Each edge e is parametrized as e(z), 0 ≤ z ≤ 1. Letting e(0) = v, e(1) = w, any solution u = {u_e(z,λ)}_e ∈ℰ of the equation (h_e - λ)u = 0 can be written as
 

    u_e(z,λ) = u(v,λ)ϕ_e0(z,λ)/ϕ_e0(1,λ) +  u(w,λ)ϕ_e1(z,λ)/ϕ_e1(1,λ).

 
The edge Schrödinger operator H_ℰ is related to the vertex  Schrödinger operator in the following way: we define the operators Δ_𝒱,λ and Q_𝒱,λ on 𝒱 by
 

    (Δ_𝒱,λu)(v) = 1/d_v∑_w ∼ v1/ϕ_e0(w,λ)u(w,λ),

 

    Q_𝒱,λ = 1/d_v∑_v ∈ eϕ'_e0(1,λ)/ϕ_e0(1,λ) + C_v/d_v,

 
where d_v is the degree of v ∈𝒱. Then, the δ-coupling condition (<ref>) is rewritten in the form of vertex Schrödinger equation
 

    (- Δ_𝒱,λ + Q_𝒱,λ)u(v) = 0,   ∀ v ∈𝒱^0.

 

 

In our previous work <cit.>, we studied spectral properties of quantum graphs on a class of  locally (i.e. on a bounded part) perturbed periodic lattices including the square and hexagonal ones which are the object of interest in this paper. We have found, in particular,
 
 
that the N-D map (and the D-N map as well) for the vertex Schrödinger operator on the interior domain and the S-matrix  for the edge Schrödinger operator on the whole system determine each other, cf. Corollary 6.16 in <cit.>. Therefore, one can  reduce the inverse boundary value problem for these lattices to that on a domain of the shape to be described below, namely the rectangular domain for the square lattice and the hexagonal parallelogram for the hexagonal lattice.




 §.§ Inverse boundary value problem for square and hexagonal lattices


Let Γ_0 = {𝒱_0, ℰ_0} be the square or the hexagonal lattice in ℝ^2 with vertex set 𝒱_0 and edge set ℰ_0. Assume that we are given a bounded domain Ω in ℝ^2 and a subgraph Γ = {𝒱, ℰ}, where ℰ = ℰ_0 ∩Ω, 𝒱 = 𝒱_0 ∩Ω. We define ∂ℰ to be the set of v ∈𝒱 such that v is an end point of some edge in ℰ, and deg_Γ(v) = 1, where deg_Γ(v) is the degree of v in the graph Γ. We further assume that Γ has the following properties[These assumptions were used in <cit.>; we state them again to make the paper self-contained.]:

(i)  ∂ℰ = ∂𝒱. 

(ii) d_v = 1  for v ∈∂𝒱.

(iii) The unique  continuation  property  holds  for  the vertex Schrödinger  operator  in  the  exterior  domain 𝒱_ext := 𝒱_0 ∖𝒱  in the following sense: if u satisfies the equation (- Δ_𝒱,λ + Q_𝒱,λ)u = 0 in 𝒱_ext and u = 0  in {v ∈𝒱_ext  ;   |v| > R} for some R > 0,  then u = 0 on 𝒱_ext.


The D-N map for the edge  Schrödinger operator is defined by
 

    Λ_ℰ(λ) : f→u_e'(v),   
    e(0) = v ∈∂𝒱,

 
where u_e is the solution to the equation
 

    {    (h_e - λ)u_e = 0,    on   ℰ,
       u_e = f,    on   ∂ℰ = ∂𝒱, 
       δ-coupling  condition.
    .

 
The D-N map for the vertex  Schrödinger operator is defined by
 

    Λ_𝒱(λ) : f(v) →1/ϕ_e0(1,λ)u(w),  v = e(0) ∈∂𝒱,  w = e(1) ∈𝒱^o.

 
where u is the solution to the equation
 

    {   (- Δ_𝒱,λ + Q_𝒱,λ)u(v) = 0,    v ∈𝒱^0, 
       u(v) = f(v),    v ∈∂𝒱.
    .

 
Then, by Lemma 3.1 in <cit.>,  Λ_ℰ(λ) and Λ_𝒱(λ) determine each other under the assumptions (i), (ii), (iii).



Our first main result is as follows:
 


Let Ω be  a bounded domain in the 2-dimensional square or hexagonal lattice having the properties (i), (ii), (iii), and consider the edge  Schrödinger operator H_ℰ = {- d^2/dz^2 + V_e(z)}_e ∈ℰ assuming the δ-coupling condition and the Dirichlet boundary condition on ∂ℰ.  Then one can uniquely reconstruct V_e(z) and C_v for all e ∈ℰ and v ∈𝒱
from the D-N map of H_ℰ- λ for all values of the energy λ, provided we know V_e(z) for all the edges e adjacent to the boundary of 𝒱.

 



 §.§ Inverse scattering problem


In <cit.>, we have discussed the spectral and scattering theory for Schrödinger operators on a class of locally perturbed periodic quantum graphs. Here we will use these results for perturbations of square and hexagonal lattices quantum graphs preserving the lattice structure. Assume that the length of  each edge is one. The assumptions on V_e(z) and C_v are as follows:

(iv)  There exists a constant C_0 ∈ℝ such that C_v = C_0 except for a finite number of vertices v ∈𝒱.

(v)  There exists V_0 ∈ L^2((0,1)) such that V_e(z)  = V_0(z) except for a finite number of edges e ∈ℰ.

One can then define the S-matrix S(λ) for the Hamiltonian of the quantum graph built on Γ. Consider a bounded domain  Ω in ℰ
which contains all the indicated perturbations, in particular, assume that V_e(z) = V_0(z) holds on any edge e adjacent to ∂𝒱.
In Corollary 6.16 in <cit.>, we have proven that the S-matrix S(λ) and the D-N map Λ_ℰ(λ) for 𝒱 determine each other.
Applying Theorem <ref>, we obtain our second main result:

 


Consider the Schrödinger operator H_ℰ = {- d^2/dz^2 + V_e(z)}_e ∈ℰ on the 2-dimensional square or hexagonal lattice Γ satisfying the conditions (i)-(v). Then one can uniquely reconstruct V_e(z) and C_v for all e ∈ℰ and v ∈𝒱 from the knowledge of the S-matrix S(λ) of H_ℰ for all energies λ.

 

Let us remark that by assumption we know V_0(z) and C_0 a priori.
 






 §.§ Related works

The study of spectral and scattering theory on quantum graphs is now making a rapid progress. Although  the topics of this paper are restricted to inverse problems on quantum graphs for square and hexagonal lattices,
there are plenty of articles devoted to this subject. Our previous work <cit.>, on which the present paper is based, deals with the forward problem and some types of inverse problems. We add here a brief look at the related works.

A general survey of discrete graph and quantum graph properties can be found in the monographs <cit.>, see also the paper <cit.>.
For dicussion of the δ-coupling and related topics we refer to <cit.>. The relation between edge Schrödinger operators and vertex Schrödinger operators was studied in <cit.>. Spectral properties of quantum graphs are discussed in <cit.>. The wave operators for discrete Schrödinger operators are investigated in <cit.>. Various inverse problems for the quantum graphs are studied in <cit.>, see also <cit.>. Finally, for earlier results on the inverse scattering for discrete Schrödinger operators on locally perturbed periodic graphs see <cit.>.




 §.§ Acknowledgement

The authors express their gratitude for the funding obtained. P.E. was supported  by the EU under the Marie Skłodowska-Curie Grant No 873071. H.I. was supported by Grant-in-Aid for Scientific Research (C) 20K03667 and (C) 24K06768 Japan Society for the Promotion of Science. H.M. was supported by Grant-in-aid for young scientists 20K14327 Japan Society for the Promotion of Science. The work of E.B. was supported by the Research Council of Finland
through the Flagship of Advanced Mathematics for Sensing, Imaging and
Modelling (decision number 359183).




§ SQUARE LATTICE


As we have proven in Theorem 5.7 of <cit.>, the S-matrix and the D-N map determine each other, if the unique continuation theorem holds in the exterior domain. Then, we can change the domain Ω as long as the conditions (i), (ii), (iii) hold. Therefore, to prove  Theorem <ref>, we have only to consider the case in which Ω is a rectangular domain as below.

Given a square lattice Γ_0 = {𝒱_0, ℰ_0} in ℝ^2, let Ω be its rectangular domain as sketched in Figure <ref>, and Γ = {𝒱, ℰ}, where 𝒱 = 𝒱_0∩Ω, ℰ = ℰ_0 ∩Ω. The black dots  there denote the boundary points satisfying d_v = 1 for v ∈∂𝒱, while d_v = 4 for v ∈𝒱^o. The  boundary ∂𝒱 consists of four parts (∂𝒱)_T, (∂𝒱)_B, (∂𝒱)_L, (∂𝒱)_R, where the top (∂𝒱)_T and the left side (∂𝒱)_L are given by
 

    (∂𝒱)_T = {a_1, a_2, …,a_m},   (∂𝒱)_L = {b_1, b_2,…, b_n},

 
and the bottom (∂𝒱)_B and the right side (∂𝒱)_R are defined similarly.
 

 

Let  - Δ_𝒱, λ + Q_𝒱,λ be the vertex Hamiltonian introduced in Subsection <ref>.
By a cross with center v_0 we mean the graph shown in Figure <ref>.
 

 
Denoting by e_i the edge with the endpoints v_0 and v_i, we can rewrite the equation (- Δ_𝒱,λ + Q_𝒱,λ)u = 0 as
 

    ∑_i=1^41/ϕ_e_i(1,λ)u(v_i) =
    (∑_i=1^4ϕ'_e_i(1,λ)/ϕ_e_i(1,λ) + C_v_0)u(v_0).

 
The key to the inverse procedure is the following partial data problem <cit.>. Denoting 𝒱^o = 𝒱∖∂𝒱, we define Neumann derivative[Note that this definition of Neumann derivative differs from that in <cit.>; here we adopt the one employed in <cit.>.] on the boundary for the vertex Hamiltonian by
 

    (∂_νû)(v) = -  1/d_v∑_w ∼ v, w ∈𝒱^o1/ϕ_e0(w,λ)u(w),    v ∈∂𝒱,

 
where ϕ_e0(w,λ) is given in (<ref>).



 

(1) Given partial Dirichlet data f on ∂𝒱∖(∂𝒱)_R, and partial Neumann data g on (∂𝒱)_L, there is a unique solution u on 𝒱 to the equation
 

    {    (- Δ_𝒱,λ + Q_𝒱,λ)u = 0    in 𝒱^o,
       u =f   on ∂𝒱∖(∂𝒱)_R, 
       ∂_νu = g   on  (∂𝒱)_L.
    .

 
(2) Given the D-N map Λ_𝒱(λ), partial Dirichlet data f_2 on ∂𝒱∖(∂𝒱)_R and partial Neumann data g on (∂𝒱)_L, there exists a unique f on ∂𝒱 such that f = f_2 on ∂𝒱∖(∂𝒱)_R and Λ_𝒱(λ)f = g on (∂𝒱)_L. Moreover, f is uniquely determined by the D-N map.

 

Let A_k be the line with the slope -1 passing through a_k as sketched in Figure <ref>. Denote the vertices on A_k ∩𝒱 by
 

    a_k = a_k,0,   a_k,1,  …  ,   a_k^∗,

 
successively. Then, A_k ∩∂𝒱 = {a_k, a_k^∗}.
 

 



(1) There exists a unique solution u to the equation
 

    (- Δ_𝒱,λ + Q_𝒱,λ)u = 0    in 𝒱^o,

 
with the partial Dirichlet data  f  on ∂𝒱∖(∂𝒱)_R 
Is this correction OK? such that
 

    {   f(a_k) = 1, 
       f(v) = 0    for 
    v ∈∂𝒱∖((∂𝒱)_R
    ∪{a_k
    })
    .

 a_k^∗ is removed.
 
and the partial Neumann data g = 0 on (∂𝒱)_L. 

(2) This solution satisfies
 

    u(v) = 0    in vertices below (not including) the line A_k.

(3)
Moreover, the values of f on (∂𝒱)_R and

a_k^∗ are uniquely determined by the D-N map.

 

The argument is the same as in Lemma 6.2 of <cit.>, which is in turn based on Lemma 6.1 of the same paper. The assertion (3) follows from
claims (2), (3) of the indicated Lemma 6.1.


We determine V_e(z) and C_v inductively by sliding down the line A_k. Assuming that we know V_e(z) and C_v for all e and v above A_k, we determine V_e(z) between A_k and A_k-1,  C_v for v ∈ A_k. Since the D-N map is given, by the induction hypothesis,
we can then compute u(v,λ) in Lemma <ref> for all v above A_k, including this line as well, as a meromorphic function of λ by using the equation (<ref>)[More detailed explanation is given in the proof of Lemma <ref> below.].

Let us compute the values of u(v,λ) more carefully. Observing Figure <ref>,
 

 
we use the equation (<ref>) for the cross with center r_1. As u = 0 at r_1, r_2, s_1, s_2, we have
 

    1/ϕ_e_0(1,λ)u(p_0,λ) +
    1/ϕ_e_1(1,λ)u(p_1,λ) = 0.

 
Observing next the cross with center p_1, we obtain
 

    1/ϕ_e_3(1,λ)u(p_3,λ) +
    1/ϕ_e_4(1,λ)u(p_4,λ) =
    (
    ∑_i=1^4ϕ'_e_i(1,λ)/ϕ_e_i(1,λ) + C_p_1)u(p_1,λ).

 


(1) Assume that we know u(v,λ) for v = p_0, p_1, and V_e(z) for e = e_0. Then, we can determine V_e(z) for e = e_1.

(2) Assume that we know u(v,λ) for v = p_3, p_4, p_1. Assume also that we know V_e(z) for e = e_3, e_4, e_1.
Then, we can determine V_e(z) for e = e_2, and C_v for v = p_1.

 

As
ϕ_e_1(1,λ) = - u(p_1,λ)/u(p_0,λ)ϕ_e_0(1,λ), one can compute the zeros of ϕ_e_1(1,λ) from the given data. This means that the Dirichlet eigenvalues referring to the potential  V_e_1(z) are determined by these data. Indeed, by Borg's theorem (see e.g. <cit.>, p. 55 and <cit.>, p. 27), a symmetric potential is determined by its Dirichlet eigenvalues. This allows to construct V_e_1(z) uniquely proving thus the first claim.

Rewrite next relation (<ref>) as
 

    ϕ'_e_2(1,λ)/ϕ_e_2(1,λ) + C_p_1  = 1/ϕ_e_3(1,λ)u(p_3,λ)/u(p_1,λ) + 1/ϕ_e_4(1,λ)u(p_4,λ)/u(p_1,λ)
    - ∑_i=1,3,4ϕ'_e_i(1,λ)/ϕ_e_i(1,λ).

 
Both sides are meromorphic with respect to λ∈ C and the singular points of the right-hand side are determined by the given data only. Noting that ϕ'_e(1,λ) ≠ 0 if ϕ_e(1,λ) = 0 (recall that ϕ_e(1,λ) = ϕ'_e(1,λ) = 0 implies ϕ_e(z,λ) = 0 for all z), we see that the zeros of ϕ_e_2(1,λ) are determined exclusively by the given data and that the value of C_p_1 plays no role in fixing the zeros of ϕ_e_2(1,λ). Since these zeros are the Dirichlet eigenvalues referring to the potential  V_e_2(z), Borg's theorem implies that V_e_2(z) is determined by the data uniquely. The equation (<ref>) then gives C_p_1 which proves the second claim.


As in Figure <ref>, we draw the line, denoted there as c, passing through the upper-left corner of the lattice 𝒱^o, and call it A_0, and later also B_n+1.

We need one more preparatory result before stating the following lemma. Comparing Figures <ref> and <ref>, we rewrite the equation (<ref>) above for the cross with the center a_k-1,i+1 in the form
 

    u(a_k,i+1,λ) = - ϕ_e'_k,i(1,λ)/ϕ_e_k,i(1,λ)u(a_k,i,λ),

 
and the equation (<ref>) for the cross with center a_k,i+1 as
 

    1/ϕ_e_k+1,i(1,λ)u(a_k+1,i,λ) +
    1/ϕ_e'_k+1,i+1(1,λ)u(a_k+1,i+1,λ)
    
       =   (
    ϕ'_e'_k,i(1,λ)/ϕ_e'_k,i(1,λ) +
    ϕ'_e_k,i+1(1,λ)/ϕ_e_k,i+1(1,λ) +
    ϕ'_e_k+1,i(1,λ)/ϕ_e_k+1,i(1,λ) +
    ϕ'_e'_k+1,i+1(1,λ)/ϕ_e'_k+1,i+1(1,λ) + C_a_k,i+1)u(a_k,i+1,λ).

 

 


Let 1 ≤ k ≤ m-1, and assume that we know V_e(z) and C_v for all e and v above the line A_k (not including the points v∈ A_k). Then
one is able to determine V_e(z) and C_v for all e between A_k and A_k-1 and C_v for all v ∈ A_k.

 

Observe Figure <ref> and put i = 0. Considering the cross with the center a_k-1,1, one can employ the first claim of Lemma <ref> to determine V_e(z) for e = e'_k,0. Using further the second claim, one can determine V_e(z) for e = e_k,1 and C_v for v = a_k,1. This is the first step; proceeding then inductively with i = 1, 2, … we prove obtain the lemma.


We have thus determined all the V_e(z) and C_v for the edges e and vertices v above the line A_0 (excluding v∈ A_0). Modifying the used argument, we can deal with the part below the line A_0 = B_n+1.

 


Let 2 ≤ℓ≤ n+1, and assume that we know V_e(z) and C_v for all the edges e and vertices v above B_ℓ, where v ∉B_ℓ. Then we can determine V_e(z) and C_v for all the e between B_ℓ and B_ℓ-1 and C_v for all v ∈ B_ℓ.

 

Consider the case ℓ = n+1 and denote the points in 𝒱∩ B_n+1  by
 

    c,  c_1,  …,  c^∗.

 
We use Lemma <ref> taking the partial  Dirichlet data as

    {   f(b_n) = 1, 
       f(v) = 0,    v ∈∂𝒱∖ ((∂𝒱)_R ∪{c, c^∗}),
     .

  
and the partial Neumann data as
 
  
 
    {   g(b_n) = 0, 
       g(v) = 0,    v ∈∂𝒱∖ ((∂𝒱)_L ∪{b_n}),
     .

  
to which the solution u constructed there corresponds. Then u = 0 holds below B_n+1 except at b_n, in fact, this is true for the vertices adjacent to (∂𝒱)_L in view of the Neumann data. Inspecting the equation, the same is true on the next line. Arguing then as in Lemma <ref> we obtain the claim.


We have thus proven the following theorem.
  


Let Ω be a rectangular domain as in Figure <ref>. From the knowledge of the D-N map of - Δ_𝒱,λ + Q_𝒱,λ for any energy λ, one can then determine all the V_e(z) and C_v, provided we know V_e(z) for all the edges e adjacent to the boundary of 𝒱.

  

Theorems <ref> and <ref> for the square lattice then follow from Theorem <ref>.




§ HEXAGONAL LATTICE





 §.§ Hexagonal parallelogram

Let us next consider the hexagonal lattice. The first three steps of the construction are parallel to the square case discussed above.
We identify ℝ^2 with ℂ, and put
 

    ω = e^π i/3,    v_1 = 1 + ω,    v_2 = √(3) i,

 

    p_1 = ω^-1 = ω^5,    p_2 = 1,

 
Given n = n_1 + i n_2 ∈ℤ[i]= ℤ + iℤ, we denote
 

    ℒ_0 = { v(n)  ;   n ∈ℤ[i]},    v(n) = n_1 v_1 + n_2 v_2,

 
and define the vertex set 𝒱_0 by
 

    𝒱_0 = 𝒱_01∪𝒱_02,   𝒱_0i = p_i + ℒ_0.

 
Let 𝒟_0 be the Wigner-Seitz cell of 𝒱_0. It is a hexagon the six vertices of which are at the points ω^k, 0 ≤ k ≤ 5, and the center at the origin. Take the set D_N := {n ∈ℤ[i]  ;   0 ≤ n_1 ≤ N,   0 ≤ n_2 ≤ N}, and put
 
 
    𝒟_N = ⋃_n ∈ D_N( 𝒟_0 +  v(n));

 
the number N we have to choose large enough. This is a parallelogram in the hexagonal lattice as sketched in Figure <ref>.
 

 
The piecewise linear loop which is the perimeter of 𝒟_N has interior angles switching between the values 2π/3 and 4π/3. Let 𝒜 be the set of vertices with the former angle, and to each z ∈𝒜 we attach a new edge e_z,ζ with the new vertex ζ = t(e_z,ζ) at its terminal point, naturally laying outside of 𝒟_N. Let
 

    Ω = {v ∈𝒱_0  ;   v ∈𝒟_N}

 
be the set of vertices in the interior of the resulting discrete graph; its boundary consists of the added vertices, ∂𝒱 = {t(e_z,ζ)  ;   z ∈𝒜}, and one is able to divide it into four parts, naturally labeled as top, bottom, right, and left:
 

    (∂𝒱)_T =   {α_0 , ⋯ , α_N }, 
    
     (∂𝒱)_B =    {2ω^5 + k(1 + ω)   ;   0 ≤ k ≤ N}, 
    
     (∂𝒱)_R =    { 2+ N(1+ω ) + k√(3) i   ;   1 ≤ k ≤ N }∪{ 2+N(1+ω ) +N√(3) i + 2 ω ^2  } , 
    
     (∂𝒱)_L =   {2ω^4}∪{β_0,⋯,β_N} ,

 
where α_k = β _N + 2ω + k (1+ω ) and β _k = -2 + k√(3) i for 0≤ k ≤ N. The perturbations we are going to consider are again supposed to be of compact support, hence taking N large enough one can ensure that they are supported within 𝒟_N.

For 0 ≤ k ≤ N, consider the line A_k as sketched in Figures <ref> and <ref>:
 

 

    A_k = {x_1 + ix_2  ;   x_1 + √(3)x_2 = a_k},

 
where a_k is chosen so that A_k passes through
 

    α_k =  α_0 + k (1+ ω )  ∈
    (∂𝒱)_T.

 
The vertices belonging to A_k∩Ω are written as
 

    α_k,ℓ = α_k + ℓ (1 + ω^5 ),   ℓ = 0, 1, 2, ⋯.

 




 §.§ Special solutions to the vertex Schrödinger equation

We consider the following Dirichlet problem for the discrete Schrödinger equation
 

    {    ( - Δ_𝒱,λ + Q_𝒱,λ)u = 0    in  𝒱^o, 
       u = f   on  ∂𝒱.
    .

 
As in the case of the square lattice, the following lemmata hold.




(1) Given partial Dirichlet data f on ∂𝒱∖(∂𝒱)_R, and partial Neumann data g on (∂𝒱)_L, there is a unique solution u on 𝒱 to the equation
 

    {    (- Δ_𝒱,λ + Q_𝒱,λ)u = 0,    in  𝒱^o,
       u =f,    on  ∂𝒱∖(∂𝒱)_R, 
       ∂_νu = g,    on   (∂𝒱)_L.
    .

 
(2) Given the D-N map Λ_𝒱(λ), partial Dirichlet data f_2 on ∂𝒱∖(∂𝒱)_R and partial Neumann data g on (∂𝒱)_L, there exists a unique f on ∂𝒱 such that f = f_2 on ∂𝒱∖(∂𝒱)_R and Λ_𝒱(λ)f = g on (∂𝒱)_L. Moreover, f is uniquely determined by the D-N map.



 


Let A_k ∩∂𝒱 = {α_k,0,α_k,m}. 

(1) There exists a unique solution u to the equation
 

    (- Δ_𝒱,λ + Q_𝒱,λ)u = 0    in  𝒱^o,

 
with the partial Dirichlet data f such that
 

    {   f(α_k,0) = 1, 
       f(z) = 0    for  
    z ∈∂𝒱∖((∂𝒱)_R
    ∪α_k,0∪
    )
    .

As in Lemma 2.2, α_k,m is removed.
 
and the partial Neumann data g = 0 on (∂𝒱)_L. 

(2) The said solution satisfies
 

    u(x_1 + ix_2) = 0    if  
    x_1 + √(3) x_2 < a_k.

(3)
Moreover, the values of f on (∂𝒱)_R and α_k,m are uniquely determined by the D-N map.

 


 


The values of u on A_k can be computed by only using the D-N map, the edge potentials V_e(z) for the edges e in the halfplane x_1 + √(3)x_2 ≥ a_k, and the vertex potentials C_v for vertices v in the halfplane x_1 + √(3)x_2 > a_k.

 

The claim is obtained using equation (<ref>) and the D-N map. Specifically, we regard the said equation as the Cauchy problem with initial data on the top and left sides of 𝒱. More precisely, we argue in the same way as in <cit.>, Lemma 6.1. Let us inspect Figure <ref>. Given the D-N map, one can compute the values of u(v) at the vertices v on the line x_1 = -1.  They are zero in the halfplane x_1 + √(3)x_2 < a_k, since the same is true for relevant boundary data. Next we observe the values of u(v) on the adjacent line, x_1 = - 1/2. We start from ω^4 and go up. Then, by using the D-N map for ω^4 and then the equation (<ref>) for the vertices above ω^4, we see that u(v) = 0 holds for v in the halfplane x_1 + √(3)x_2 < a_k when x_1 = -1/2. Repeating this procedure, we conclude that u(v) = 0 holds in the lower halfplane x_1 + √(3)x_2 < a_k. To compute the values of u(v) in the upper halfplane, x_1 + √(3)x_2 ≥ a_k, we observe the top
 boundary (∂𝒱)_T. We first compute u(v) for vertices v adjacent to (∂𝒱)_T. They are determined by the D-N map. Next we observe u(v) on the neighboring line, the second adjacent to the boundary. This time, we start from the rightmost vertex (for which, by virtue of claim (3) from Lemma <ref>, the value u(v) is determined by the D-N map) and go down to the left. With the aid of the equation, we can then determine u(v) by using the values of V_e(z) and C_v in the halfplane x_1 + √(3)x_k > a_k, which are known by assumption. Finally, using the equation, the values of u(v) on the line x_1 + √(3)x_2 = a_k are computed from those in the halfplane x_1 + √(3)x_2 > a_k and V_e(z), C_v specified in the lemma.



Now we pass to the reconstruction procedure.

Let u be the solution to the equation (<ref>) specified in Lemma <ref>. If a_k is large enough, all the perturbations lie below the line A_k, hence the support of u is disjoint with them. Let us slide A_k downwards, and observe the equation when we first touch the perturbation. Let a, b, b', c ∈𝒱 and e, e' ∈ℰ be as in  Figure <ref>.
 

 
Then, evaluating the equation (<ref>) at v = a, we obtain
 

    u(b) = - ϕ_e0(1,λ)/ϕ_e'0(1,λ) u(b').

 
Let e_k,1, e'_k,1, e_k,2, e'_k,2, ⋯ be the series of edges just below A_k starting from the vertex α_k, and put
 

    f_k,m(λ) = - ϕ_e_k,m0(1,λ)/ϕ_e'_k,m0(1,λ).

 
Then we see that the  solution u in  Lemma <ref> satisfies
 

    u(α_k,ℓ) = f_k,1(λ)⋯ f_k,ℓ(λ) ;

 
note that the right-hand side does not depend on C_v.
 

 

To proceed, observe the graph detail sketched in Figure <ref>.
 


As in Lemma <ref>, assume that we know V_e(z) and C_v for all edges e and vertices v above A_k. Then, assuming that we know V_e(z) for e = e_k,1, one can determine V_e(z) for e = e'_k,1 and e = e_k,2, and also C_v for v = α_k,2.

 

In view of (<ref>), since we know u(α_k,ℓ) for all ℓ, we also know the corresponding f_k,ℓ(λ). Furthermore, since we know ϕ_e_k,1(1,λ), we get from (<ref>) ϕ_e'_k,10(1,λ) for all values of the energy λ. The zeros of ϕ_e'_k,10(1,λ) are the Dirichlet eigenvalues of - (d/dz)^2 + V_e(z), e = e'_k,1. By  Borg's theorem, these eigenvalues determine the potential V_e(z) at the edge e = e'_k,1.

Making the equation (<ref>) explicit at the vertex v = α_k,2 and noting that u = 0 holds below A_k, we have
 

    ϕ'_e_k,20(1,λ)/ϕ_e_k,20(1,λ)  = 1/ϕ_e”_k,20(1,λ)u(v”)/u(v)  - ϕ'_e”_k,20(1,λ)/ϕ_e”_k,20(1,λ) - ϕ'_e'_k,10(1,λ)/ϕ_e'_k,10(1,λ) - C_v,

 
where v = α_k,2 =  e”_k,2(0), v” = e”_k,2(1). The both sides of (<ref>) are meromorphic function in the complex plane, and we note that u is also meromorphic with respect to λ by construction. The singularities of the left-hand side are thus determined by those of the right-hand side. Furthermore, the singularities of the right-hand side can be computed without using C_v. Therefore, the zeros of ϕ_e_k,20(1,λ) are  determined by V_e(z) for edges e in the halfplane x_1 + √(3)x_2 ≥ a_k and e = e_k,1,  e'_k,1 as well as C_w for w in the open halfplane x_1 + √(3)x_2 > a_k. As before, Borg's theorem determines V_e_k,2(z), which in turn determines ϕ_e_k,20(1,λ) and ϕ'_e_k,20(1,λ). The value of C_v is then computed from the formula (<ref>) for regular points λ.


Lemma <ref> implies the following result.



Let u be as in Lemma <ref>, and assume that we know V_e(z) for all the edges e above A_k, and C_v for all the v above A_k but not on A_k. Then, one can determine all the V_e(z) for e between A_k and A'_k, and C_v for all the v on A_k.

 

 

Let  A_k,B_l be as in Figure <ref>.  By Lemma <ref>, we can compute (from the knowledge of the D-N map) the edge potentials between A_k,A_k' and between B_l,B_l', and also compute C_v for the vertices on A_k,B_l. We call this the initial procedure.

Next we make a step inside 𝒟_N. For the notational reason, we determine the edge potentials below B'_ℓ. To be able to repeat the described procedure after passing to the neighboring “lower” line B_l-1, it suffices to know the edge potential for the edges below B'_ℓ and A'_k with the endpoints on B'_ℓ, as well as C_v for such endpoints on B'_ℓ in Figure <ref>.



The following “local” version of Lemma <ref> holds.
 

In Figure <ref>, if one knows the edge potentials for the edges ab and ac,  one can compute the edge potential of the edge ac' and the vertex potential C_v for v=a.

 

 

As in Lemma 3.5, consider the equation for A_k-1, as shown in Figure <ref>; for the sake of clarity and brevity, we denote it by u:=u_A_k-1. Using the edge identification e=ab, we also use the symbol ϕ_ab(1,λ) for ϕ_e0(1,λ).

The procedure to “to compute ϕ” means to express it using the known quantities in the successive steps:

(i) Evaluating the equation (<ref>) at b_2, we can compute u_A_k-1(b_1).


(ii) Evaluating the equation at b_4, we can compute u_A_k-1(b_5). 

(iii) Evaluating the equation at b_5, we can compute u_A_k-1(b). 

(iv) The potential at the edges bb_1,ba and the vertex potential C_v at b are found during the initial procedure for the lines B.
Hence solving the equation at b, we can compute u_A_k-1(a). 

(v) The potentials at the edges ab,ac are found during the initial procedure for the lines B. Since u_A_k-1(c)=u_A_k-1(c')=0 and u_A_k-1 at a,b are known, one can determine the potential at ac' and C_v at v=a by the same argument as in Lemma 3.7.

Let us explain the step (v) in detail. We consider the equation for u at vertex a:
 

    (-Δ_𝒱,λ+Q_a,λ)u(a)=0,

 
where
 

    (Δ_𝒱,λ u)(a)=1/3∑_w∼ a1/ϕ_aw(1,λ)u(w)=1/31/ϕ_ab(1,λ)u(b),

 
since u(c)=u(c')=0. By (<ref>),
 

    Q_a,λ   =   1/3∑_w∼ aϕ_aw'(1,λ)/ϕ_aw(1,λ)+C_a/3
       =   1/3(ϕ_ab'(1,λ)/ϕ_ab(1,λ)+ϕ_ac'(1,λ)/ϕ_ac(1,λ)+ϕ_ac''(1,λ)/ϕ_ac'(1,λ)+C_a).

 
This equation yields
 

    ϕ_ac''(1,λ)/ϕ_ac'(1,λ)= 1/ϕ_ab(1,λ)u(b)/u(a)-ϕ_ab'(1,λ)/ϕ_ab(1,λ)-ϕ_ac'(1,λ)/ϕ_ac(1,λ)-C_a.

 
By the above construction, u(a) is a meromorhic function of λ. All the quantities on the right-hand side of (<ref>) except C_a are known. This means the singularities of the right-hand side of (<ref>) are known. Hence we can determine the singularities of the left-hand side of
(<ref>) without knowing C_a. The singularities of the left-hand side are the Dirichlet eigenvalues for the edge ac', which determine the edge potential of ac' by Borg's theorem. This shows ϕ_ac'(z,λ), z∈ [0,1] can be computed. Inserting ϕ_ac'(1,λ),ϕ'_ac'(1,λ) back into (<ref>) gives C_a.



Note that in the above proof, we use only the knowledge of edges between B_ℓ and B_ℓ'. We emphasize that the knowledge of edges below both of A_k and B_ℓ, which is still unknown, is not used. Then we repeat Lemma <ref> by taking the function u for A_k-2,A_k-3,⋯, and thus we recover the edge potential for all the  edges just below B'_ℓ and left to A_k-1,

and C_v for all  vertices between B_ℓ and B_ℓ' in Figure <ref>. This shows that we can push the lines B_l as “low” as needed.

By a symmetric reasoning, choosing the function u in the different direction (i.e., choosing u_B_j for the proper j), one can show that the lines A_k can also be pushed “down” as low as needed. But in fact, this is not needed, since pushing the lines B_l “down” alone can already recover the whole perturbed region.


We arrive thus at the following conclusions:
 


Let Ω be a hexagonal paralellogram as in Figure <ref>. Then from the D-N map of - Δ_,λ + Q_𝒱,λ, one can determine all the V_e(z) and C_v, provided we know V_e(z) for all the edge e adjacent to the boundary of 𝒱.



Theorems <ref> and <ref> for the hexagonal lattice then follow from Theorem <ref>.


 99




 A1 K. Ando, Inverse scattering theory for discrete Schrödinger operators on the hexagonal lattice. Ann. Henri Poincare 14 (2013), 347–383.

AIM16 K. Ando, H. Isozaki, H. Morioka, Spectral properties for Schrödinger operators on perturbed lattices. Ann. Henri Poincare 17 (2016), 2103–2171.

AIM18
K. Ando, H. Isozaki and H. Morioka,  Inverse scattering for Schrödinger operators on perturbed periodic lattices, Ann. Henri Poincaré 19 (2018), 3397-3455.

AvdBelMat11
S. Avdonin, B. P. Belinskiy, and J. V. Matthews, Dynamical inverse problem on a metric tree, Inverse Porblems 27 (2011),  075011.

Bel87
M. Belishev, An approach to multidimensional inverse poblems for the wave equation. (Russian) Dokl. Akad. Nauk SSSR. 297 (1987), 524-527.

Bel04
M. I. Belishev, Boundary spectral inverse  problem on a class of graphs (trees) by the BC method, Inverse Problems 20 (2004), 647-672.

BelKur92
M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. PDE. 17 (1992), 767-804.

Below85
J. von Below, A characteristsic equation associated to an eigenvalue problem on c^2-networks, Linear Algebra Appl. 71 (1985), 309-325.

BerkolaikoKuchment2013
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Amer. Math. Soc., Providence, R.I. (2013).

BILL2021
E. Blåsten, H. Isozaki, M. Lassas and J. Lu, The Gel'fand's inverse problem for the graph Laplacian, J. Spectral Theory 13 (2023), 1-45.

BILL2
E. Blåsten, H. Isozaki, M. Lassas and J. Lu, Inverse problems for discrete heat equations and random walks for a class of discrete graphs, SIAM Discrete Math. 37 (2023), 831-863.

BEILL
E. Blåsten, P. Exner, H. Isozaki, M. Lassas and J. Lu, Inverse problems for locally perturbed lattices - Discrete Hamiltonian and quantum graph, to appear in Ann. Henri Lebesgue.

BondaShieh17
N. Bondarenko and C. T. Shieh, Partial inverse problem for Sturm-Liouville operators on trees, Proceedings of the Royal Society of Edingburgh 147A (2017), 917-933.




Bonda20
N. Bondarenko, Spectral data characterization for the Sturm-Liouville operator on the star-shaped graph, Anal. Math. Phys. 10 (2020), 83

BroWei09
B. M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. Royal Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 2062  (2005), 3231-3243.

BPG08
J. Brüning, V. Geyler and K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schrödinger operators, Rev. Math. Phys. 20 (2008), 1-70.




Cattaneo97
C. Cattaneo,  The spectrum of the continuous Laplacian on a graph, Monatsh. Math. 124 (1997), 215–235.

ChExTu10
T. Cheon, P. Exner and O. Turek, Approximation of a general singular vertex coupling in quantum graphs, Ann. Phys. 325 (2010), 548-578.

Exner96
P. Exner, Weakly coupled states on branching graphs, Lett. Math. Phys. 38 (1996), 313–320.

Exner97
P. Exner,
A duality between Schrödinger operators on graphs and certain Jacobi matrices, Ann. Inst. Henri Poincaré 66 (1997), 359–371.

EKMN17
P. Exner, A. Kostenko, M. Malamud and H. Neidhardt, Spectral theory for infinite quantum graph, Ann. Henri Poincaré 19 (2018), 3457-3510.

ExnerKovarik2015
P. Exner and H. Kovařík, Quantum Waveguides, Springer, Cham Heidelberg New York Dordrecht London (2015).

ExnerPost09
P. Exner and O. Post, Approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds, J. Phys. A: Math. Theor. 42 (2009), 415305.

Es
M.S. Eskina, The direct and the inverse scattering problem for a partial difference equation, Soviet Math. Doklady, 7 (1966), 193-197.

GutSmil01
B. Gutkin and U. Smilansky, Can one hear the shape of a graph? J. Phys. A 34 (2001), 6061-6068.

FrYu01
G. Freiling and V. Yurko, Inverse Sturm-Liouville Problems and their Applications, Nova Science Publishers, Hauppauge (2001).

IK H. Isozaki, E. Korotyaev, Inverse problems, trace formulae for discrete Schrödinger operators. Ann. Henri Poincare 13 (2012), 751–788.




IsoMo15
H. Isozaki and H. Morioka, Inverse scattering at a fixed energy for discrete Schrödinger operators on the square lattice. Ann. l'Inst. Fourier 65 (2015), 1153–1200.

KoLo07
E. Korotyaev and I. Lobanov, Schrödinger operators on zigzag nanotubes, Ann. Henri Poincaré 8 (2007), 1151-1076.

KoSa14
E. Korotyaev and N. Saburova, Schrödinger operators on periodic discrete graphs, J. Math. Anal. Appl. 420 (2014), 576-611.





KoSa15b
E. Korotyaev and N. Saburova, Estimates of bands for Laplacians on periodic equilateral metric graphs,  Proc. Amer. Math. Soc. 114 (2016), 1605-1617.


KoSa15
E. Korotyaev and N. Saburova, Scattering on periodic metric graphs, Rev. Math. Phys. 32 (2020), 2050024 (51 p.).








KN22
A. Kostenko, N. Nicolussi, Laplacians on Infinite Graphs, Mem. EMS, Berlin 2022.

KostrykinSchrader1999
V. Kostrykin, R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A: Math. Gen. 32 (1999),  595-630.

Ku24
P. Kurasov, Spectral Geometry of Graphs, Birkhäuser, Berlin 2024.

MochiTroosh12
K. Mochizuki and I.Yu. Trooshin:  On the scattering on a loop-shaped graph, Progress of Math. 301 (2012), 227-245.

Nakamura14
S. Nakamura, Modified wave operators for discrete Schrödinger operators with long-range perturbations, J. Math. Phys. 55 (2014), 112101.

Niikuni16
H. Niikuni, Spectral band structure of periodic Schrödinger operators with two potentials on the degenerate zigzag nanotube, J. Appl. Math. Comput. (2016) 50:453-482.

Pank06
K. Pankrashkin, Spactra of Schrödinger operators on equilateral quantum graphs, Lett. Math. Phys. 77 (2006), 139-154.

Pankrashkin13
K. Pankrashkin: An example of unitary equivalence between self-adjoint extensions and their parameters, J. Funct. Anal. 265 (2013), 2910–2936.

ParRich18
D. Parra and S. Richard, Spectral and scattering theory for Schrödinger operators on perturbed topological crystals,
Rev. Math. Phys. 30 (2018), Article No. 1850009, pp 1-39.

Pivo00
V. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal. 32 (2000), 801-819.

Post12
O. Post, Spectal Analysis on Graph-like Spaces, Lecture Notes in Mathematics 2039, Springer, Heidelberg (2012).

PoTru
J. Pöschel and E. Trubowitz, Inverse Spectral Theory,
Academic Press, Boston, (1987).

Tadano19
Y. Tadano, Long-range scattering for discrete Schrödinger operators, Ann. Henri Poincaré 20 (2019), 1439-1469.

VisComMirSor11
F. Visco-Comandini, M. Mirrahimi, and M. Sorine,  Some inverse scattering problems on star-shaped graphs, J. Math. Anal. Appl. 387  (2011), 343-358.

YangXu18
X. C. Xu and C. F. Yang, Determination of the self-adjoint matrix Schrödinger operators without the bound state data,
Inverse Problems 34 (2018), 065002 (20pp).

Yurk05
V. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems 21 (2005), 1075-1086.

Yurko16(1)
V. Yurko, Inverse spectral problems for differential operators  on spatial networks, Russ. Math. Surveys 71, No 3 (2016), 539-584.













 

 