35

数理解析研究所講究録
第2074巻 2018年 35-46

INVERSE SCATTERING ON GRAPHEN— VERTEX MODEL AND EDGE MODELHIROSHI ISOZAKI

1. INTRODUCTION

1.1. Brief summary. The problem we addrees here is the spectral theory andinverse scattering associated with Schrödinger operators on perturbed periodic lat‐tices. Among many periodic structures appearing in material science, we are in‐terested in the hexagonal lattice because of its generality in carbonic materials inphysical side and also in the geometry of lattice problem in mathematical side. \mathrm{A}well‐known example is the graphen, whose simplest model is the hexagonal latticewith Hamiltonian defined by a difference operator between vertices of the nearest
neighbors. We call it a vertex model. As perturbations, one can consider poten‐
tials or defects, which means to deform the lattice structure by removing or addingedges. The scattering theory developed for the continuous model for Schrödinger
operators is also extended to this discrete model. Our main concern is the inverse

problem. Our results are:\bulletA compactly supported scalar potential is uniquely reconstructed from theS‐matrix of one fixed energy.

The S‐matrix determines the structure of the planar graph in a finite part.
\bulletThe convex hull of the defects of the form of convex polygon is determinedby the S‐matrix of one fixed energy.
\bulletThere is another model for graphen, in which we take into account of effects ofedges. We consider -(d/dz)^{2}+q_{\mathrm{e}}(z) on each edge \mathrm{e} with Kirchhoff conditions onvertices. We call it an edge model, although it is usually called a metric graph.
The study of spectral properties for the edge model is reduced to that of the vertex
model. We can then obtain the following result.
\bulletAssume that q_{\mathrm{e} (z) \in L^{2}(0,1) and real‐valued, symmetric, i.e. q_{\mathrm{e} (z)q_{\mathrm{e}}(1-z) , moreover q_{\mathrm{e}}(z)=0 except for a finite number of edges \mathrm{e} . Then,the q_{\mathrm{e}}(z)s are determined by the S‐matrix for all energies.

=

Our theory, in particular the forward problem of scattering, can be extended
to more general lattices. However, for the simplicity of explanation, we restrict
ourselves here to the hexagonal lattice.
Date: October 7, 2017.

2000 Mathematics Subject Classification. Primary 81\mathrm{U}40 , Secondary 47\mathrm{A}40.
Key words and phrases. Schrödinger operator, lattice, inverse scattering.

36
HIROSHI ISOZAKIThis is a joint work with K. Ando, E. Korotyaev and H. Morioka.
2. HEXAGONAL LATTICE AND ITS HAMILTONIAN

Figure 1 represents a hexagonal lattice.

We can find two lattices : the one

consisting of white dots, and the other black dots. Letting\mathcal{L}_{0}=\{\mathrm{v}(n);n=(n_{1}, n_{2})\in \mathrm{Z}^{2}\},\mathrm{v}(n)=n_{1}\mathrm{v}_{1}+n_{2}\mathrm{v}_{2},\mathrm{v}_{1}=1+ $\omega$, \mathrm{v}_{2}= $\omega$(1+ $\omega$) , $\omega$=e^{ $\pi$ i/3},
we define the vertex set \mathcal{L}_{0} by\mathcal{V}_{0}= (p_{1}+\mathcal{L}_{0})\cup(p_{2}+\mathcal{L}_{0}).

Consequently, the wave function has two components in \ell^{2}(\mathrm{Z}) . Denoting it by
û(n) (ûl (n) , û2 (n) ), we define the vertex Laplacian by=( $\Delta$\hat{}\mathcal{V} û)(n)=\displayst le\frac{1}3\left(bgin{ary}l \sum_{| athrm{v}(n')-\mathr{v}(n)|=1\hat{u}_2(n')\ sum_{|\athrm{v}(n)-\mathr{v}(n)|=\mathr{l}\atu_{1}(n,)\ed{ary}\ight)

FIGURE 1. Hexagonal lattice

To study the wave propagation on the lattice, let us first recall the case ofcontinuous model, in which the free Schrödinger equation is(-\triangle- $\lambda$)u=0.

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INVERSE SCATTERING ON GRAPHEN

Passing to the Fourier transform, it becomes (| $\xi$|^{2}- $\lambda$)\overline{u}( $\xi$)

=

0

1

. The physicalsolution is the L^{2} ‐density on the sphere, hence has the asymptotic expansionu(x)=C($\lambda$)\displaystyle\int_{S^{n-1} e^{i\sqrt{$\lambda$}$\omega$\cdotx}\overline{u}(\sqrt{$\lambda$}$\omega$)d$\omega$\displaystyle\simeqC_{+}\frac{e^{i\sqrt{$\lambda$}r {r^(n-1)/2} (\sqrt{}$\lambda\theta$) \displaystle\frac{e^-i\sqrt{$\lambda$}r {^(n-1)/2} ( \sqrt{}$\lambda\theta$)
û

+C‐

û ‐

with $\theta$=x/r, r= |x| . This also holds for the perturbed equation (-\triangle+V(x)-$\lambda$)u=0 , where V(x) is compactly supported, i.e.

u(x)\displaystyle \simeq C_{+}\frac{e^{i\sqrt{$\lambda$}r }{r^{(n-1)/2} $\varphi$_{out}( $\theta$)+C_{-}\frac{e^{-i\sqrt{$\lambda$}r }{r^{(n-1)/2} $\varphi$_{in}( $\theta$) .

(2.1)
The operator

S( $\lambda$):L^{2}(S^{n-1})\ni$\varphi$_{in}\rightarrow$\varphi$_{out}\in L^{2}(S^{n-1})
is the \mathrm{S} ‐matrix. Note that the sphere \sqrt{ $\lambda$}S^{n-1} is the characteristic surface of theoperator -\triangle- $\lambda$.

The free Schrödinger equation on the hexagonal lattice is(-\triangle_{\mathcal{V} - $\lambda$)\hat{u}=0\wedge.
Passing to the Fourier series, we have

(H_{0}(x)- $\lambda$)u(x)=0 ,on\mathrm{T}^{2}=(\mathrm{R}/2 $\pi$ \mathrm{Z})^{2},

H_{0}(x)=-\displaystyle \frac{1}{3}\left(\begin{ar ay}{l l } & 0 & & 1+ & e^{-ix_{1} +e^{-ix_{2} \ 1+ & e^{ix_{1} & +e^{ix_{2} & & 0 \end{ar ay}\right).

The physical solution is an L^{2} ‐density supported in a submanifold on the torus:

M_{ $\lambda$}=\{x\in \mathrm{T}^{2};\det(H_{0}(x)- $\lambda$)=0\},\displaystyle \det(H_{0}(x)- $\lambda$)=$\lambda$^{2}-\frac{1}{9}\{3+2(\cos x_{1}+\cos x_{2}+\cos(x_{1}-x_{2})\}.

This is the characteristic surface of the difference operator -\triangle_{\mathcal{V} \wedge- $\lambda$ , and is called
the Fermi surface. The solution

u(x) is then written asu(x)=\displaystyle \frac{ $\varphi$(x)}{p(x, $\lambda$)-i0}-\frac{ $\varphi$(x)}{p(x, $\lambda$)+i0},$\varphi$(x)\in L^{2}(M_{ $\lambda$}) , p(x, $\lambda$)=\det(H_{0}(x)- $\lambda$).

This also holds asymptotically (in the sense of singularity) for the perturbed lattice:(2.2)
and the operator

u(x)\displaystyle \simeq\frac{$\varphi$_{out}(x)}{p(x, $\lambda$)-i0}-\frac{$\varphi$_{in}(x)}{p(x, $\lambda$)+i0},
S( $\lambda$):L^{2}(M_{ $\lambda$})\ni$\varphi$_{in}\rightarrow$\varphi$_{out}\in L^{2}(M_{ $\lambda$})is the \mathrm{S} ‐matrix.

The two terms appearing in the right‐hand side of (2.1) are called outgoing andincoming waves. As is seen here, they are distinguished by their spatial behaviors at
1_{\mathrm{I}\mathrm{n} this note, the Fourier trasnsform of a distribution u(x) on \mathrm{R}^{d} is denoted by \overline{u}( $\xi$) , while
the Fourier coefficient of a distribution u(x) on the torus \mathrm{T}^{d} is denoted by û(n)38
HIROSHI ISOZAKI

infimity. For the lattice, it does not work well, since the Fermi surface is in general
not strictly convex and we encounter difficulties in applying the stationary phasemethod to the integral on it. Therefore, we pass to the Fourier series and observe
the singularities of wave functions on the torus. This leads us to the formulation(2.2).
As a perturbation of the vertex model, we add a scalar potential, or (and) replace
its finite part by a general graph. The associated Hamiltonian is denoted by \hat{H}_{\mathcal{V} .
If these perturbations are assumed to be bounded self‐adjoint and confined in acompact part, the spectral theory can be developped easily.

We turn to the edge model, which is usually called the metric graph. We consider
a hexagonal lattice with vertex set \mathcal{V} and edge set \mathcal{E} . Each edge \mathrm{e}\in \mathcal{E} is identified
with the interval [0 , 1 ] . Functios on the edge set \mathcal{E} are denoted by \^{u}=\^{u} \mathcal{E}\{\hat{u}_{\mathrm{e} (z);\mathrm{e}\in \mathcal{E}\} . The Hamitonian \hat{H}_{\mathcal{E} is defined by=(\displaystyle \hat{H}_{\mathcal{E} \hat{u}_{\mathcal{E} )_{\mathrm{e} (z)= (-\frac{d^{2} {dz^{2} +q_{\mathrm{e} (z) \hat{u}_{\mathrm{e} (z) , \mathrm{e}\in \mathcal{E},assuming the Kirchhoff condition on û \mathcal{E} . For \mathrm{e} \in \mathcal{E} , denote its end points by\mathrm{e}(0)=0, \mathrm{e}(1)=1 . Then, the Kirchhoff condition is(K‐1) û \mathcal{E} (z) is continuous on \mathcal{E}.

(K‐2) For each \mathrm{e}\in \mathcal{E} , ûe \in C1 ([0,1and\displaystyle\sum_{\mathrm{e}(0)=v}\hat{u}_{\mathrm{e} '(0)=\sum_{\mathrm{e}(1)=v}\hat{u}_{\mathrm{e} '(1),\foral v\in\mathcal{V}.
The assumption on the potentials are as follows:(E‐1) q_{\mathrm{e} (z) is real‐valued, and q_{\mathrm{e}}(z)\in L^{2}(0,1) .
(E‐2) q_{\mathrm{e}}(z)=0 except for a finite number of edges.
(E‐3) q_{\mathrm{e}}(z)=q_{\mathrm{e}}(1-z) .
This makes \hat{H}_{\mathcal{E} self‐adjoint on L^{2}(\mathcal{E}) .

There is a simple relation between the vertex Laplacian and the edge Laplacian.
On each edge, the solution of the Schrödinger equation(-(d/dz)^{2}+q_{\mathrm{e} (z)- $\lambda$)\hat{u}_{\mathrm{e} =\hat{f_{\mathrm{e} }

,

on

(0,1)

is written as

c_{\mathrm{e}(1, $\lambda$)\displayst le\frac{$\phi$_{\mathrm{e}0(z,$\lambda$)}{$\phi$_{\mathrm{e}0(1,$\lambda$)}+c_{\mathrm{e}(0, $\lambda$)\frac{$\phi$_{\mathrm{e}1(z,$\lambda$)}{$\phi$_{\mathrm{e}1(0,$\lambda$)}+r_{\mathrm{e}($\lambda$)\hat{f_\mathrm{e} ,r_{\mathrm{e} ( $\lambda$)= (-(d/dz)_{D}^{2}+q_{\mathrm{e} (z)- $\lambda$)^{-1}(d/dz)_{D}^{2} is the Dirichlet Laplacian on (0,1) , and $\phi$_{\mathrm{e}i}(z, $\lambda$) is the solution towhere
the equation(-(d/dz)^{2}+q_{\mathrm{e} (z)- $\lambda$)$\phi$_{\mathrm{e}i}(z, $\lambda$)=0,$\phi$_{\mathrm{e}0}(0, $\lambda$)=0, $\phi$_{\mathrm{e}0}'(0, $\lambda$)=1,

on(0,1) ,

39
INVERSE SCATTERING ON GRAPHEN$\phi$_{\mathrm{e}\mathrm{i} (1, $\lambda$)=0, $\phi$_{\mathrm{e}1}'(1, $\lambda$)=-1.
The coefficients c_{\mathrm{e} (1, $\lambda$) , c_{\mathrm{e} (0, $\lambda$) are determined by the Kirchhoff condition. The
point is that

Kirchhoff condition=

Vertex equation

Namely, the following equation holds:\displayst le\sum_{\mathrm{e}(0)=v}\frac{1}$\phi$_{\mathrm{e}0(1,$\lambda$)}c_{\mathrm{e}(1, $\lambda$)+\sum_{\mathrm{e}(1)=v}\frac{1}$\phi$_{\mathrm{e}1(0,$\lambda$)}c_{\mathrm{e}(0, $\lambda$)+\displaystle\sum_{\mathrm{e}(0)=v}\frac{$\phi$_{\mathrm{e}1'(0,$\lambda$)}{ \phi$_{\mathrm{e}1(0,$\lambda$)}c_{\mathrm{e}(0, $\lambda$)-\sum_{\mathrm{e}(1)=v}\frac{$\phi$_{\mathrm{e}0'(1,$\lambda$)}{ \phi$_{\mathrm{e}0(1,$\lambda$)}c_{\mathrm{e}(1, $\lambda$)=\displaystyle\sum_{\mathrm{e}(1)=v}\frac{d} z}r_{\mathrm{e}($\lambda$)\hat{f_\mathrm{e} |_{z=1}-\sum_{\mathrm{e}(0)=v}\frac{d} z}r_{\mathrm{e}($\lambda$)\hat{f_\mathrm{e} |_{z=0}.
Therefore, when q_{\mathrm{e}}(z)=0 , we have the following representation:( \displaystyle\hat{H}_{\mathcal{E}^{(0)}-$\lambda$)^{-1}\hat{f})|_{\mathrm{e}=c_{\mathrm{e}(1, $\lambda$)\frac{\sin\sqrt{$\lambda$}z{\sqrt{$\lambda$}+c_{\mathrm{e}(0, $\lambda$)\frac{\sin\sqrt{$\lambda$}(1-z)}{\sqrt{$\lambda$}+r_{\mathrm{e}^{(0)}($\lambda$)\hat{f_{\mathrm{e} ,on each edge \mathrm{e} , andc_{\mathrm{e} (1, $\lambda$)=c_{\mathrm{e}' (0, $\lambda$)= (\hat{R}_{\mathcal{V} ^{(0)}( $\lambda$)\hat{F}_{\mathcal{E} ^{(0)}( $\lambda$)\hat{f})(v),

for \mathrm{e}(1)=\mathrm{e}'(0)=v , where\displaystyle\hat{R}_{\mathcal{V} ^{(0)}($\lambda$)=\frac{\sin\sqrt{$\lambda$} {\sqrt{$\lambda$} (-\trianglev\wedge+\cos\ qrt{$\lambda$})^{-1}(\hat{F}_{\mathcal{E} ^{(0)}( $\lambda$)\hat{f}) (v)=-\displaystyle \frac{1}{3}(\sum_{\mathrm{e}(1)=v}\frac{d}{dz}r_{\mathrm{e} ^{(0)}( $\lambda$)\hat{f_ \mathrm{e} |_{z=1}-\sum_{\mathrm{e}(0)=v}\frac{d}{dz}r_{\mathrm{e} ^{(0)}( $\lambda$)\hat{f_ \mathrm{e} |_{z=0}).

This formula suggests that the properties of the continuous spectrum of the edge
Schrödinger operator are inherited from those of the vertex Schrödinger operator.
The spectra of the vertex and edge Schrödinger operators are as follows.

Lemma 2.1. (1) $\sigma$_{e}(\hat{H}_{\mathcal{V} )=[-1, 1].
(2) $\sigma$(\hat{H}_{\mathcal{E} )=[0, \infty)\cup$\sigma$_{D} , where $\sigma$_{D}=\displaystyle \bigcup_{\mathrm{e}\in \mathcal{E} $\sigma$_{p}(-(d/dz)_{D}^{2}+q_{\mathrm{e} (z) .
3. RELLICH TYPE THEOREM

To study the continuous model for the Schrödinger operator on \mathrm{R}^{n} , appropriate

function spaces are the Besov type spaces B^{*}(\mathrm{R}^{n}) and B_{0}^{*}(\mathrm{R}^{n}) defined byB^{*}(\displaystyle \mathrm{R}^{n})\ni f\Leftrightar ow\sup_{R>1}\frac{1}{R}\int_{|x|<R}|f(x)|^{2}dx<\infty,
B_{0}^{*}(\displaystyle \mathrm{R}^{n})\ni f\Leftrightar ow\lim_{R\rightar ow\infty}\frac{1}{R}\int_{|x|<R}|f(x)|^{2}dx=0.

The following theorem, proven by Rellich and Bekua, is fundamental in studying
the continuous spectrum.

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HIROSHI ISOZAKI

Theorem 3.1. Let $\lambda$ > 0 . If u(x) \in B_{0}^{*}(\mathrm{R}^{n}) satisfies the Helmholtz equation(-\triangle- $\lambda$)u=0 on \{|x| >R\} for a constant R>0 , then u(x)=0 on \{|x| >R\}.
This theorem implies the non‐existence of eigenvalues embedded in the continu‐
ous spectrum. It also palys an important role in the inverse scattering theory.

We consider the counter part of this theorem on the lattice. For the periodic
lattice \mathcal{V} , we define the Besov type speces by

B^{*}(\displaystyle \mathcal{V})\ni\hat{v}\Leftrightar ow\sup_{R>1}\frac{1}{R}\sum_{|n<R}|\hat{v}(n)|^{2}<\infty,
B_{0}^{*}(\displaystyle \mathcal{V})\ni\hat{v}\Leftrightar ow\lim_{R\rightar ow\infty}\frac{1}{R}\sum_{|n<R}|\hat{v}(n)|^{2}=0.

Passing to the Fourier series, we defineB^{*}(\mathrm{T}^{d})\ni v\Leftrightarrow\hat{v}\in B^{*}(\mathcal{V})

B_{0}^{*}(\mathrm{T}^{d})\ni v\Leftrightarrow\hat{v}\in B_{0}^{*}(\mathcal{V}),.

Sometimes, it is more convenient to use the Fourier transform. Multiplying a cut‐

off function (a partition of unity) and passing to the Fourier transform \overline{v}( $\xi$) , we canalso defineB^{*}(\displaystyle \mathrm{T}^{d})\ni v\Leftrightar ow\sup_{R>1}\frac{1}{R}\int_{|x<R}|\overline{v}( $\xi$)|^{2}d $\xi$<\infty,
B_{0}^{*}(\displaystyle \mathrm{T}^{d})\ni v\Leftrightar ow\lim_{R\rightar ow\infty}\frac{1}{R}\int_{|x<R}|\overline{v}( $\xi$)|^{2}d $\xi$=0.

For the lattice space, the Rellich type theorem should be rephrased as follows.

Theorem 3.2. Suppose \^{u}\in B_{0}^{*}(\mathcal{V}) satisfies (-\hat{ $\Delta$}_{\mathcal{V} - $\lambda$)\hat{v}=0 near infinity of thelattice space. Then, \hat{v}=0 near infinity.

Extending û to be 0 in the finite part and passing to the Fourier series, we obtain
the equation(H_{0}(x)- $\lambda$)u(x)=f(x),where f(x) is a trigonometric polynomial, since its Fourier coefficients are compactly
supported. Then, the Rellich type theorem on the lattice is formulated and provedin the following form.

Theorem 3.3. Let $\lambda$\in(-1,1)\backslash \{0, \pm 1/3, \pm 1\} . Let u(x) \in B_{0}^{*}(\mathrm{T}^{2}) satisfy (H_{0}(x)$\lambda$)u(x)f(x) , where f(x) is a trigonometric polynomial. Then, u(x) is also a=

trigonometric polynomial.

The proof relies on the HilbertNullStellenSatz ([18], [2]). Similar theorem also
holds for the edge Hamiltonian. In particular, the non‐existnece of embedded eigen‐values follows. Put$\sigma$_{\mathcal{V}}=(-1,1)\backslash \{0, \pm 1/3, \pm 1\},$\sigma$_{\mathcal{E} =(0, \infty)\backslash ($\sigma$_{D}\cup\{ $\lambda$;-\cos\sqrt{ $\lambda$}=0, \pm 1/3, \pm 1\}).

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INVERSE SCATTERING ON GRAPHEN

Corollary 3.4. (1) $\sigma$_{p}(\hat{H}_{\mathcal{V} )\cap$\sigma$_{\mathcal{E} =\emptyset.
(2) $\sigma$_{p}(\hat{H}_{\mathcal{E} )\cap$\sigma$_{\mathcal{E} =\emptyset.
For the preceeding results about the spectra of vertex models and edge models,see e.g. [6], [16], [14], [5], [6], [13].
4. FORWARD PROBLEM

For the vertex model, the space \hat{B}(\mathcal{V}) is defined by\displaystyle \hat{B}(\mathcal{V})\ni\hat{f}\Leftrightar ow\sum_{j=0}^{\infty}r_{j}^{1/2}(\sum_{r_{j-1}\leq|n<r_{j} |\hat{f}(n)|^{2})^{1/2}<\infty,
where r_{-1}
space

=

0, r_{j}=2^{j}(j \geq 0) . Then, the spaces \hat{B}(\mathcal{V}) , \hat{B}^{*}(\mathcal{V}) rig the Hilbert\ell^{2}(\mathcal{V}) :\hat{B}(\mathcal{V}) \subset\ell^{2}(\mathcal{V})\subset\hat{B}^{*}(\mathcal{V})For the edge model, the Besov type spaces \hat{B}(\mathcal{E}) , \hat{B}^{*}(\mathcal{E}) , \hat{B}_{0}^{*}(\mathcal{E}) are defined similarly,.

\mathrm{e}.\mathrm{g}.

\displaystyle\hat{B}^{*}(\mathcal{E})\ni\hat{f}\Leftrightar ow\sup_{R>1}\frac{1}{R}\sum_{\mathrm{e}\subsetB_{R}\Vert\hat{f_{\mathrm{e} \Vert_{L^{2}(0,1)}^{2}<\infty,

where B_{R}=\{x\in \mathrm{R}^{2}; |x| <R\} . Their counter parts on the torus are defined by

B(\mathrm{T}^{2}\times I_{\mathcal{E} )\ni f\Leftrightar ow\hat{f}\in\hat{B}(\mathcal{E})B^{*}(\mathrm{T}^{2}\times I_{\mathcal{E} )\ni f\Leftrightar ow\hat{f}\in\hat{B}^{*}(\mathcal{E})
B_{0}^{*}(\mathrm{T}^{2}\times I_{\mathcal{E} )\ni f\Leftrightar ow\hat{f}\in\hat{B}_{0}^{*}(\mathcal{E}),,,where I_{\mathcal{E} =(0,1) , and \hat{f} denotes the Fourier coefficient of f(x, z) with respect to x

(with a suitable cut‐off function).
(\hat{H}_{\mathcal{V} -z)^{-1} and \hat{R}_{\mathcal{E} (z)Let \hat{R}_{\mathcal{V} (z)=*=(\hat{H}_{\mathcal{E} -z)^{-1} . Then, the following weak‐limits exist.

Theorem 4.1. For $\lambda$\in$\sigma$_{\mathcal{V} ,\hat{R}_{\mathcal{V} ( $\lambda$\pm i0)\in \mathrm{B}(\hat{B}(\mathcal{V});\hat{B}^{*}(\mathcal{V}) .

Theorem 4.2. For $\lambda$\in$\sigma$_{\mathcal{E} ,\hat{R}_{\mathcal{E} ( $\lambda$\pm i0) \in \mathrm{B}(\hat{B}(\mathcal{E});\hat{B}^{*}(\mathcal{E}) .

Once we have proven this limiting absorption principle, one can follow the sta‐tionary theory of scattering developed by Kato‐Kuroda [12], Ikebe [10], Agmon[1] without any difficulty, namely, existence and completeness of time‐dependent
wave operators, eigenfunction expansion theory, unitarity of the \mathrm{S} ‐matrix and its
representation by generalized eigenfunctions.

In some energy region, one can define the \mathrm{S} ‐matrix by using the asymptotic
behavior of wave functions at the infinity of the lattice space. However, for both ofthe vertex model and the edge model, it is more convenient to obeserve the behaviorof singularities of wave functions on the torus, since there is no restricton of energy.
For the vertex model, it is stated as follows. Recall that H_{0}(x) has two eigenvalues$\lambda$_{j}(x) and the eigenprojections P_{j}(x) , j=1 , 2. Define

M_{ $\lambda$,j}=\{x\in \mathrm{T}^{2};$\lambda$_{j}(x)= $\lambda$\},42
HIROSHI ISOZAKI

M_{ $\lambda$}=M_{ $\lambda$,1}\cup M_{ $\lambda$,2}.
Theorem 4.3. Let $\lambda$ \in $\sigma$_{\mathcal{V} . For any $\varphi$^{in} \in L^{2}(M_{ $\lambda$}) , there exist a unique $\varphi$^{out} \inL^{2}(M_{ $\lambda$}) and û \in B ( \mathcal{V} ) such that*(\hat{H}_{\mathcal{V} - $\lambda$)\hat{u}=0,

u\displaystyle \simeq\frac{1}{2 $\pi$ i}\sum_{j=1,2}\frac{1}{$\lambda$_{j}(x)- $\lambda$\mp i0}\otimes P_{j}(x)$\varphi$^{out}-\displaystyle \frac{1}{2 $\pi$ i}\sum_{j=1,2}\frac{1}{$\lambda$_{j}(x)+ $\lambda$\mp i0}\otimes P_{j}(x)$\varphi$^{in},where f\simeq g means f-g\in B_{0}^{*}(\mathrm{T}^{2}) . The unitary operator

S( $\lambda$):L^{2}(M_{ $\lambda$})\ni$\varphi$^{in}\rightarrow$\varphi$^{out}\in L^{2}(M_{ $\lambda$})is the S‐matrix.

For the edge model, the above theorem is stated as follows.

Theorem 4.4. Let $\lambda$\in$\sigma$_{\mathcal{E} . For any incoimg data $\phi$^{in}\in L^{2}(M_{-\cos\sqrt{ $\lambda$}}) , there exist\hat{}*

a unique solution û \in B ( \mathcal{E} ) of the equation(\hat{H}_{\mathcal{E} - $\lambda$)\hat{u}=0,
and an outgoing data $\varphi$^{out}\in L^{2}(M_{-\cos\sqrt{ $\lambda$}}) satisfyingu\displaystyle \simeq B( $\lambda$)\sum_{j=1,2}\frac{1}{$\lambda$_{j}(x)+\cos\sqrt{ $\lambda$}-i $\epsilon$( $\lambda$)}\otimes P_{j}(x)$\varphi$^{out}-B( $\lambda$)\displaystyle \sum_{j=1,2}\frac{1}{$\lambda$_{j}(x)+\cos\sqrt{ $\lambda$}+i $\epsilon$( $\lambda$)}\otimes P_{j}(x)$\varphi$^{in},where f\simeq g means f-g\in B_{0}^{*} (\mathrm{T}^{2} \times I_{\mathcal{E} ) . The unitary operator

S( $\lambda$):L^{2}(M_{-\cos\sqrt{ $\lambda$}}) \in$\varphi$^{in}\rightarrow$\varphi$^{out}\in L^{2}(M_{-\cos\sqrt{ $\lambda$}})is the S‐matrix.

Here B( $\lambda$) \in \mathrm{B}(B^{*}(\mathrm{T}^{2}\times I_{\mathcal{E} );B^{*}(\mathrm{T}^{2}\times I_{\mathcal{E} ) and $\epsilon$( $\lambda$)=\pm 1 , however we omit the
precise definition.
5. FROM \mathrm{S} ‐MATRIX TO D‐N MAP

As in the case of continuous model, the inverse scattering on the whole spaceis reduced to an inverse boundary value problem in a bounded domain. Take asufficiently large bounded set containing all perturbations. Then, the equation in
the whole space is split into three parts: the exterior boundary value problem, theinterior boundary value problem and the (integral) equation on the boundary.
We follow the same approach for the vertex model and edge model. Take abounded domain $\Omega$_{int} enclosing all perturbations, and let $\Omega$_{ext} be the exterior do‐main so that$\Omega$_{ext}\cup$\Omega$_{int}=\mathcal{V}, \partial$\Omega$_{\mathrm{e}xt}=\partial$\Omega$_{int}.

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INVERSE SCATTERING ON GRAPHEN

Consider the Dirichlet problem\left{\begin{ar y}{l (-\hat{$\Delta$}_{\mathcl{V}-$\lambda$)\hat{u}=0,\mathrm{i}\mathrm{n}$\Omega$_{int},\ ^{u}=f\hat{},\mathrm{o}\mathrm{n}\partil$\Omega$_{int}. \end{ar y}\right.
The Neumann derivative is defined by\partial_{\mathrm{v} \hat{u}=-\hat{$\Delta$}_{\mathcal{V} \hat{u}|_{\partial$\Omega$_{int}Then, the D‐N map, Dirichlet‐to‐Neumam map, is defined by$\Lambda$_{int}( $\lambda$):\hat{f}\rightar ow\partial_{ $\nu$}\hat{u}.
The \mathrm{S} ‐matrix is defined in the whole space and in the exterior domain as well. We

denote them S( $\lambda$) and S_{ext}( $\lambda$) , and define the scattering amplitudes

S( $\lambda$)=I-2 $\pi$ iA( $\lambda$) , S_{ext}( $\lambda$)=I-2 $\pi$ iA_{ext}( $\lambda$).

Then, the following formula holds:(5.1)

A_{ext}( $\lambda$) — A ( $\lambda$ )=î( + ) ( $\lambda$)B_{ $\Sigma$}( $\lambda$)^{-1}(\hat{I}^{(-)}( $\lambda$))^{*},

B_{ $\Sigma$}( $\lambda$)=$\Lambda$_{int}( $\lambda$)-$\Lambda$_{ext}( $\lambda$)- $\lambda \chi \Sigma$.

Here, $\Sigma$=\partial$\Omega$_{int}=\partial$\Omega$_{ext} , î( \pm ) ( $\lambda$ ) : L^{2}(M_{ $\lambda$})\rightarrow l^{2}( $\Sigma$) are some injective operators,$\Lambda$_{ext}( $\lambda$) is the D‐N map in the exterior domain, and x $\Sigma$ is the characterisitic functionof $\Sigma$ . Since all perturbations are confined to $\Omega$_{int} , we know î (\pm)( $\lambda$) and $\Lambda$_{ext}( $\lambda$) .
Therefore, (5.1) means that S( $\lambda$) and $\Lambda$_{int}( $\lambda$) determine each other.
The edge model can be dealt with similarly.

6. INVERSE SCATTERING

6.1. Vertex model‐ reconstruction of the potential. In the inverse boundary
value problem for the continuous case, the exponentially growing solution for the
Schrödinger equation plays an important role. It solves the Schrödinger equation(-\triangle+V(x)- $\lambda$)u =0 in \mathrm{R}^{n} , and is exponentially growing in a half‐space, andexponentially decaying in the opposite half‐space. There is a counter part of thissolution in the discrete model. In [11], we used a solution to the discrete Schrödinger
equation on the square lattice, which vanishes in a half‐space and non‐zero in the
opposite half‐space, to recontsruct the scalar potential from the \mathrm{S} ‐matrix. A similar
idea works well for the hexaogonal lattice.

Theorem 6.1. Consider the vertex Schrödinger operator on the hexagonal latticewith a compactly supported scalar potential. Then, the potential is uniquely recon‐
structed from the S‐matrix S( $\lambda$) for an arbitrarily given fixed energy $\lambda$\in$\sigma$_{\mathcal{V} .

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HIROSHI ISOZAKI

6.2. Vertex model‐recovery as a planar graph. Another central issue in the
inverse boundary value problem for the continuous model is the reconstruction of

the Riemannian metric from the D‐N map with 0 energy. This is an interpretationof the inverse problem of electrical impedance tomography. We expect that the
metric is determined by the D‐N map up to a diffeomorphism which leaves theboundary of the domain invariant. It is proved in 2‐dimensions in the general
setting, and for the real analytic case with dimension n\geq 3.
The counter part of the Riemannian manifold in the discrete model is the planargraph. By the works of Colin de Verdière [7], [8] and Curtis‐Morrow [9], it is known
that the D‐N map determines the planar graph up to elementary transformations.
In the case of hexagonal lattice, the D‐N map for the planar graph corresponds to
the \mathrm{S} ‐matrix with energy at the bottom of the continuous spectrum. Therefore, we
obtain the following theorem.

Theorem 6.2. For a sequence of energies $\lambda$_{n} \in $\sigma$_{\mathcal{V} such that $\lambda$_{n} \rightarrow -1 , the S‐

matrices S($\lambda$_{n}) determine the perturbed finite part of the hexagonal lattice up toelementary transformations.

The equivalence of the \mathrm{S} ‐matrix and the D‐N map can be proven for a wide classof lattices, e.g. those introduced in [2]. Therefore, Theorem 6.2 holds for such classof vertex models.

6.3. Vertex model ‐ proving defects. A physically important example of theperturbation as a graph is the defect of the lattice. We consider the case where thedefects are bounded. By virtue of Theorem 6.2, one can reconstruct the perturbed
graph topologically. However, it does not give the information of the location of
defects. To study it, we utilize the above mentioned analogue of exponentiallygrowing solution to the Schrödinger equation on the hexagonal lattice. Take astraight line L in the lattice and construct the solution to the free Schrödinger
equation which vanishes below L but does not vanish above L . Suppose that the
all defects are lying below L . If we take it as an input of the D‐N map, the output isthe same as that of the unperturbed case. Let us move the line L downwards. Then,it will touch the defects. At this moment, the output of the D‐N map changes, andwe conclude that there is a defect on the line L . Therefore, we can enclose thedefects from above. Changing the dierction of the line L , we get the following
theorem.

Theorem 6.3. Assume that the defects consist of finite convex polygons. Then,from the S‐matrix S( $\lambda$) with an arbitrarily fixed energy $\lambda$\in$\sigma$_{\mathcal{V} , one can determine
the convex hull of the defects.

Theorems 6.1 ,6.2, 6.3 will appear in [3].
6.4. Edge model‐ reconstuction of the potential. One can use the same idea

for the edge model. By the above procedure, one can compute the coefficients for the45
INVERSE SCATTERING ON GRAPHEN

perturbed vertex Laplacian from the D‐N map. It consists of $\phi$_{\mathrm{e}0}(0, $\lambda$) , where $\phi$_{\mathrm{e}0}(z)is the solution to the Schrödinger equation (-(d/dz)^{2}+q_{\mathrm{e}}(z)- $\lambda$)$\phi$_{\mathrm{e}0}(z, $\lambda$)=0 withinitial data $\phi$_{\mathrm{e}0}(0, $\lambda$)=0 . Then, the zeros of $\phi$_{\mathrm{e}(0)}(0, $\lambda$) are the Dirichlet eigenvaluesof the operator -(d/dz)^{2}+q_{\mathrm{e}}(z)- $\lambda$ . Recall Borg’s theorem [4], which is the starting
point of the inverse spectral theory: A symmetric potential on (0,1) is determinedby its Dirichlet eigenvalues. We have thus arrived at the following theorem.
Theorem 6.4. Let I be an open interval in $\sigma$_{\mathcal{E} . Then, the potential is determinedfrom the S‐matrix S( $\lambda$) for all energies $\lambda$\in I.

One can also deal with non‐zero back ground potentials.

Theorem 6.5. Let q_{0}(z) \in L^{2}(0,1) be real and symmetric. Suppose that the edge
potentials q_{\mathrm{e} (z) coincide with q_{0}(z) except for a finite number of edges. Let I be an
open interval in $\sigma$_{\mathcal{E} . Then, the potential q_{\mathcal{E} is determined from the S‐matrix S( $\lambda$)
for all energies $\lambda$\in I.
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HIROSHI ISOZAKIPROFESSOR EMERITUS, UNIVERSITY OF TSUKUBA, TSUKUBA, 305‐8571, JAPAN