INVERSESCATTERING ON

GRAPHEN- VERTEX MODEL

AND EDGE MODEL

Hiroshi Isozaki

University of Tsukuba, Professor Emeritus

Tosio Kato CentennialConference

September 8, 2017, TokyoUniversity

Joint work with

Kazunori Ando Ehime University

Hisashi Morioka Doshisha University

Evgeny Korotyaev St. Petersburg

State University

Based on the works

• Spectral properties of Schroedinger operators on perturbed lattices,

K. Ando, H. Isozaki and H. Morioka,

Ann. Henri Poincare, 17 (2016), 2103-2171

• Inverse scattering for Schroedinger operators on perturbed lattices,

K. Ando, H. Isozaki and H. Morioka, in preparation

• The work in progress,

K. Ando, H. Isozaki, E. Korotyaev and H. Morioka,

Graphen as a material

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プレゼンテーションのノート

These are the papers of a team of engineering department of Tsukuba University

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It is an array of carbon atoms. Computing the current, they conclude that there is a hole inside the lattice.

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Another experiment. One atom of carbon is replaced by that of nitrogen. I will talk about two simple mathematical models describing the wave motion on the periodic lattice. But, I am sure that the real physics is much more complicated.

The plan of the talk

Vertex Laplacian and

edge Laplacian

Helmholtz equations

Inverse scattering

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You find two lattices in this figure. One is made of black dots, the other white dots. This multiple lattice structure is a feature of periodic system.

KAGOME LATTICE

GRAPHITE

DIAMOND LATTICE

A possible extension

• Asymptotically cylindrical domain = carbon nanotube

Multiple Lattice Structure

• Basic lattice

• Vertex set

Wave functions and Laplacians on the vertex model• The wave function has two components

each of which corresponds to respective lattices.

• The Laplacian takes the mean

over nearest neighbors

Scattering in the continuous model

• For the case of continuous model, the free Schroedinger equation is

• Passing to the Fourier transform, it becomes

• We consider the solution living on the sphere

• Which has the asymptotic expansion

Incoming and outgoing states

• It also holds for the perturbed equation :

• The operator is the S-matrix.

• Remember that the sphere is the characteristic surface of the Laplacian.

Fermi surface (Characteristic surface)• The free Schroedinger equation on the hexagonal lattice is

• Passing to the Fourier series, it becomes

• So, the solution lives on the submanifold in the 2-dim. torus

Scattering on the lattice

• The physical solution is written as

• This holds also asymptotically (in the sense of singularities) for the perturbed lattice.

• The operator is the S-matrix.

Radiation on the lattice

• The outgoing radiation and incoming radiation are distinguished by their spatial behavior at infinity.

• For the lattice, this does not work well, since the Fermi surface is complicated and we have difficulties in applying the stationary phase method to the integral on it.

• We pass to the Fourier series and observe the singularities of wave functions on the torus.

What is the perturbed lattice?

Given a hexagonal lattice, perturb its finite part by

adding a scalar potential, or (and)

replacing a finite part by a general graph.

Edge model (metric graph)

• Hexagonal lattice with vertex set and edge set . Each edge

is oriented and identified with

• The Laplacian is defined to be

assuming the Kirchhoff condition

Kirchhoff condition

• For each , denote its end points by

• is continuous on

• For each , , and at each vertex

Assumptions on the edge Laplacian

• is real-valued, and

• except for a finite number of edges

• On each edge

Relation between vertex Laplacian and edge Laplacian

Trick• On each edge, the solution of the Schroedinger equation

is written as

• The coefficients are determined by the Kirchhoff condition

Kirchhoff condition = Vertex equation• The Kirchhoff condition is rewritten as

Which is the Schroedinger equation on the vertex set.

The symmetric assumption of the potential is used here.

Rellich type theorem ーContinuous case• Define the Besov type space by

Fourier Analysis on the lattice

• Define the Besov type space by

• Passing to the torus by the Fourier series, we have

Sometimes, it is more convenient to pass to the Fourier transform• Multiply a cut off function (a partition of unity), and pass to the Fourier

transform

Rellich type theorem on the lattice

• Suppose satisfies the Helmholtz equation near infinity of the lattice

• Extending to be zero in the finite part, and passing to the Fourier series

where is a trigonometric polynomial

Rellich type theorem

• The HilbertNullStellenSatz from Algebraic Geometry (in its simplest form) is used in the proof.

• This theorem implies the non-existence of embedded eigenvalues.

• It also plays an important role in the inverse problem.

• The same theorem also holds for the edge model.

The tools we are using

• Sometimes, the discrete model is harder than the continuous model. Because, we do not have so many tools available in the former.

We are using

• Real Fourier analysis (Ps. D. OP.’s)

• Complex analysis (Simplest cases in the theory of several complex variables and algebraic geometry)

Forward problem ---- stationary scattering theory

Resolvent estimates --- Limiting

absorption principle

Spectral representation S-matrix Helmholtz equation

Forward problem

• Limiting absorption principle

• Spectral representation

• S-matrix

Expansion in the sense of singularities

• Instead of observing the spatial behavior of the resolvent at infinity, we observe its singularities by passing it on the torus by using the Fourier series.

• Then, the most singular part of the resolvent has the form

which gives us the spectral representation and the S-matrix.

Helmholtz equation and S-matrix

• has eigenvalues , and eigenprojections

The same theorem holds for the edge model

It is hard to tell the details of the formulas

• Sorry. I can only give the flavor

Interior boundary value problem

• Neumann derivative

• DN map

The whole space problem and the interior problem

• Divide the lattice into two parts : exterior domain and interior domain

• Then, the Schroedinger equation on the whole space is divided into

the exterior problem + the interior problem + the integral equation on the boundary

S-matrix at a fixed energy determines the D-N map

The same is true for the edge model

• The S-matrix for the whole space problem determines the D-N map for the interior boundary value problem and vice versa.

• Moreover, the S-matrix for the edge model is computed from that of the vertex model.

The inverse scattering problem is reduced to the inverse boundary value problem• In the case of the planar graph, the inverse problem from the D-N map has is

solved by

Colin de Verdiere (1994, 1996)

Curtis-Morrow (1994)

Similarity between the inverse problemsfor the continuous case and discrete case

• In 2-dim, the metric is determined by the D-N map up to a diffeomorphism leaving the boundary invariant

• The same is considered to be true (and is proved for the analytic case) for higher dimensions.

• The planar graph is determined from the diffeomorphism up to elementary transformations.

ELEMENTARY TRANSFORMATIONS

So, in general, local perturbations in a periodic graph is determined, as a planar graph up to elementary transformations, from the S-matrix of all energies.

• This is true for a wider class of periodic graph.

• It determines the “toplogy” and the “conductivity” of the planar graph.

• However, it does not determine the location of defects.

Conitnuous case－exponentially growingsolutions to the SchroEdinger equation

• So, it is exponentially growing in a half-space and exponentially decaying in the opposite half-space

• Faddeev (1966), Calderon (1980), Sylvester-Uhlmann (1987)

Discrete analogue of the exponentially growing solution

• It is non-zero above , and zero below

Potential is computed from the S-matrix.

• Adopting this solution as a boundary data, one can compute the potential by using the solution of the interior Dirichlet problem.

• Recall that the S-matrix and the D-N map are mutually related:

Reconstruction of the potential

This idea can be also used to detect the defects of the lattice

• Consider the solution for the free equation which vanishes below the line .

• If is very high, it satisfies the Schroedinger equation on the lattice with defects.

• Let the line move downward. Then the solution touches the defect, and has a different behavior.

• It will be observed by the D-N map.

Proving for the defects of the lattice

The same idea works well for the edge model

Reduction to 1-dimensional inverse spectral problem• By observing the waves on the lattice, one can determine the coefficients of the

vertex Laplacian.

• They are the values at the end point of solutions to the equation

• They give the Dirichlet eigenvalues.

Borg’s classical result (1946)

• The symmetric potential is determined by the Dirichlet eigenvalues.

Problems onthe tree

The inverse problem on the tree is hard to solve when it contains a loop.

The edge model has loops, however, our method works well.

I t is because we are using the special structure of the periodic lattice, in particular, the analogue of exponentially growing solution for the continuous model.

NON-ZERO BACK GROUND POTENTIALS

Thank you for your attention!