Unified transform method for the Schr¨odinger equation on a simple metric graph Текст научной статьи по специальности «Математика»

УДК 517.953

Unified Transform method for the Schrodinger Equation on a Simple Metric Graph

Gulmirza Khudayberganov* Zarifboy A. Sobirov^ Mardonbek R. Eshimbetov*

National University of Uzbekistan Universitet Str. 4, Tashkent, 100174 Uzbekistan

Received 16.11.2018, received in revised form 06.02.2019, accepted 05.04.2019 Integral-representation of solutions of the initial-boundary value problems for the Schrodinger equation on simple metric graphs was obtained with the use of the Fokas method. This method uses special generalization of the Fourier transform that is referred to as the unified transform. Obtained representation of solutions of the problem for open and closed simple star graphs allows one to identify transmitted, reflected and trapped waves at the graph branching point.

Keywords: ¡Schrodinger equation, metric graph, branched structure, unified transform, initial problem, Fourier transform, boundary value problem. DOI: 10.17516/1997-1397-2019-12-4-412-420.

Introduction

It is known that branched structures and networks are used to model many complex systems from physics, biology, ecology, sociology, economy and finance [1,2]. In this paper we present the Fokas method for the solution of time dependent Schrodinger equation on simple star graphs. Metric graphs with Schrodinger equation accompanied together with Kirchhoff conditions at the vertex are referred to as quantum graphs [3-5]. Usually static Schrodinger equation is used to study physical properties of quantum graphs (see [3-7] and references therein). Non linear Schrodinger equation on two dimensional thin tabular branched domain was studied [8]. It was proved that the problem on metric graphs for one dimensional non linear Schrodinger equation on metric graph with gluing (Kirchhoff) conditions at the vertex can be obtained when width of the branches tends to zero. Similar convergence result in the case of linear Schrodinger equation with different approaches was obtained [6,7].

Schrodinger equation can be also treated as the equation with imaginary time. The heat equation on branched structures was firstly used in the 50's of the nineteenth century. Thomson (Lord Kelvin) used the heat equation (Thomson's cable equation) as mathematical model of signal decay in submarine (underwater) telegraphic cables (Ch.IV in [9]). Later this method was widely used in neuroscience to analyse data collected from intracellular microelectrode recordings and to analyse the electrical properties of neuronal dendrites (see [10,11]). Initial and boundary value problems for some other types of PDE on metric graphs and their possible applications can be found in [12-14].

In this paper we consider time-dependent Schrodinger equation. To construct solution of Schrodinger equation on simple star graphs with bounded and semi-infinite bonds we use the

* gkhudaiberg@mail.ru tsobirovzar@gmail.com tmr.eshimbetov92@mail.ru © Siberian Federal University. All rights reserved

Fokas unified transform method [15-19]. Our solution gives more detailed information on scattering at the branching point of the graph and dynamics near vertex.

1. Simple star graph with semi-infinite bonds

Let us consider simple star graph r with three semi-infinite bonds connected at the point O. The point O is the vertex of the graph. We label bonds of the graph as Bj, j = 1,2,3 (Fig. 1). Let us define coordinate xj on the bond Bj, j = 1, 2, 3, and xj e (0, to). We denote the graph by At each bond the coordinate of the vertex point O is equal to zero. Further we will use x instead of x~.

Fig. 1

In each bond of the graph we consider Schrodinger equation

iqj)(x,t) = aq{xjX>(x,t), x G Bj, t> 0, j = 1, 2, 3, (1)

-h

where a = -—. Initial conditions are 2m

q(j)(x, 0) = q(j)(x), x G Bj, j = 1, 2, 3, (2)

and asymptotic conditions are

lim q(j)(x,t)=0, t > 0, j = 1, 2, 3. (3)

At the vertex point the solution satisfies the following gluing (Kirchhoff) conditions

q(1)(+0,t) = q(2)(+0,t) = q(3)(+0,t), t > 0, (4)

q«(+0,t) + ¿2qX2>(+0,t) + ¿2qX3)(+0,t)=0, t > 0. (5)

Now we consider standard steps in the application of the Fokas method. We begin with the so-called "local relations" (see [17]):

(e-ikx+wtq(j)(x,t))^ = (ae-ikx+wt (kq(j)(x,t) - iqXj)(x,t))) , j = 1, 2, 3, (6)

where w(k) = -iak2.

Integrating over domain rœ x (0, t) and applying Green's theorem, we find (Fig. 2)

' kx+wt

0 = / e-ikx qj (x)dx - e-ikx+wt q(j) (x, t)dx-JO Jo

- J aews (kq(j)(0,s) - iq{xj)(0,s)) ds, j = 1, 2, 3, (7)

,s) - iqx (^ s

O

where k G C.

Fig. 2

One can notice that this procedure is equivalent to application of the Fourier transform in (1) with respect to time and coordinate. But in this Green's theorem we use special dispersion relation w = -iak2 mentioned above. So, in this Fourier transform parameters w and k are not independent. Relation between these parameters (or new variables) should be chosen from the following equations

q(j)(k,t) = e-ikxq(j)(x,t)dx, j = 1, 2, 3, J 0

qj\k)= l e-ikxq0> (x)dx, j = 1, 2, 3.

0

ikx (j) i

Using these definitions, the global relation (7) is rewritten as

ewtqfj)(k,t) = q0j)(k) — kag0(w,t) + iagj(w,t), j = 1, 2, 3,

(8)

where {k € C : Imk > 0}, j = 1, 2, 3.

Since the dispersion relation w = -iak2 is invariant with respect to transform k ^ —k then functions g0(w,t), gj(w,t), j = 1, 2, 3 are also invariant with respect to this transform. Thus we can substitute -k instead of k into (8) and obtain

ewtqfjj)(—k, t) = q0j)(—k)+ kag0(w,t) + iagj(w,t), j = 1, 2, 3,

where {k € C : Imk < 0}, j = 1, 2, 3.

Inverting the Fourier transforms in (8), we have

1

(9)

1

q(j)(x,t) = — eikx-wtq0j) (k)dk+

2n J-oo

ikx wt

( — kag0(w, t) + iagj(w, t))dk, j = 1, 2, 3.

(10)

The region D is shown in Fig. 3. The integrand of the second integral in (10) is entire and decays as k ^ <x for k € {Imk > 0} \D(2). Using the analyticity of the integrand and applying Jordan's lemma, we can replace the contour of integration of the second integral (see [17-19]):

q(j)(x,t) = — eikx-wtq0j) (k)dk+

2n J

+ 1 i eikx-wt(—kag0(w,t) + iagj(w,t))dk, j = 1, 2, 3, 2n JdD(2)

where D = {k € C : Re(—ik2) < 0} = D(2) |J D(4).

Fig. 3

We need to find unknowns gj(w,t), g0(w,t), j = 1,2,3 in this representation of the solution. Solving (8) for iagj(w,t), we find

iagj(w,t) = ewtcfj)—k,t) - q(j)—k) - kag0(w,t), Imk > 0.

(12)

So, we have

1 f+,x / q(j)(x,t) = 2n I eikx-wtqj> (k)dk-

- — i eikxq(j)(-k,t)dk +— i eikx-wtq0j)(-k)dk-

2n JdD(2) 2n JdD(2)

dD(2) ikx-wt

n JdD(2)

elkx-wtkag0(w, t)dk, j = 1, 2, 3. (13)

Taking to account that functions elkx and q(j)(—k,t) are holomorphic and bounded in {k e C : Imk > 0}, we conclude that the following integral (see [17-19])

eikxqij)( -k,t)dk, 0 < x < x, t > 0

2n JdD(2)

' dD(2)

tends to zero at k ^ x.

The term eikxq(j)(-k,t) gives rise to the term

1

_L f eikxq(j)(-k,t)dk, 0 <x< x, t> 0 2n JdD(2)

which vanishes because both eikx and q(j)(-k,t) are bounded and analytic in the upper half of the complex k plane, and furthermore q(j)(k,t) is of O (k) as k ^ to:

qfj)(k,t)= I eikxq(j) (x, t)dx -

k) as q(j) (0, t)

kxqij)(

k

k.

Thus, Cauchy's theorem supplemented with Jordan's lemma in the domain D(2) implies the desired result. Hence

q(j (x, t) = — eikx-wtc0) (k)dk +— eikx-wtqj] (-k)dk-

2n J2n JdD(2)

-i

n JdD(2)

eikx-wtkago(w,t)dk, j = 1, 2, 3. (14)

1

o

(15)

Using vertex conditions, we obtain

ewtq(1)(-k, t) = ^(-k) + kag0(w, t) + iag1(w, t) ewtq(2)(-k, t) = q02)(-k) + kag0(w, t) + iag2(w, t) ewtf3)(-k, t) = q03)(-k) + kag0(w, t) + iag3(w, t) S21qi1)(+0, t) + S2q¥\+0, t) + S2q£\+0, t) = 0

where {k G C : Imk < 0}. Solving this equations for go, we have k ( 52q01)(-k) + S2q02)(-k) + SlqOV—k)

kago(w,t) =--WtW+M-+

+ ewt (S^—k, t) + SK^ (-k, t) + S23q(3)(-k, t)) + S2 + S2 + S2 . ( )

Substituting (16) into (14) and taking to account that some integrals vanish, we obtain the solution of the problem

i r

11 ikx-

1

qtj)(x,t) = — eikx-wtq0j) (k)dk+

2n J-oo

+ 2nJBDW e s2 + ¿2 + ¿290 ( k)dk+

+ 1 i eikx-wt S^)(-k) + S3f(-k) dk, j = h 2, 3. (17)

nJdD2 ¿2 + ¿¿2 + s2 ' J ' ' v ;

First term in this solution represents free wave propagation. Second and third terms represent reflection and transmission to the other bonds. In conclusion we have

Theorem 1. If initial data q0\x) are analitical functions in Bj for j = 1, 2, 3, then the solution of problem (1)-(6) is given by (17).

2. Simple star graph with bounded bonds

Here we consider simple graph r2 with three bounded bonds connected at the point O. The point O is the vertex of the graph. Let us denote bonds of the graph by Bj, j = 1, 2,3. We define coordinate x1 G (0,L1) on the bond B1. We also define coordinates x2 G (0,L2 ) and x3 G (0,L3) on the bonds B2 and B3, respectively. At each bond the coordinate of the vertex point O is equal to zero. We denote this graph by r. We define Schrodinger equations on each bond

iq(j)(x,t) = aqXjX(x,t), x G Bj, t> 0, j = 1, 2, 3 (18)

with initial conditions

qtj)(x, 0) = q(j)(x), x G Bj, j = 1, 2, 3 (19)

and boundary conditions

qtj)(Lj,t) = h{0j)(t), t > 0, j = 1, 2, 3. (20)

We suppose that solution satisfies the Kirchhoff conditions (4) and (5) at the vertex point.

Global relations for this case have the form

ewtq(jj)(k, t) - cf0j)(k) = e-ikLja(kh0j)(w,t) - ih[j)(w,t)) - akg0(w,t) + iagj(w,t), (21) where {k G C : Imk > 0}, j = 1, 2, 3,

nLj nLj

qj)(k,t)= e-ikxq(j)(x,t)dx; q0i](k)= e-ikxq0j) (x)dx,

OO

(' t C t

h{j)(w,t) = ewsqxj)(Lj,s)ds; h0j](w,t)= ewsq(j)(Lj, s)ds,

OO r t {■ t

gj (w,t)= ewsqxj) (0, s)ds; g0(w,t) = ews q(j) (0, s)ds, j = 1, 2, 3.

qj (0,s)ds; g0(w,t)= ewsq(j)( 10 Jo

Since the dispersion relation w = -iak2 is invariant with respect to transform k ^ —k then

functions g0(w,t), gj(w,t), h(j)(w, t), h0j)(w,t), j = 1, 2, 3 are also invariant with respect to this transform. Thus we can substitute -k instead of k into (21) and obtain

ewtq°j)(-k,t) - cf0j)(-k) = eikLja(-kh{0j)(w,t) - ih(j)(w,t)) + akgo(w,t) + iagj(w,t), (22)

where {k G C : Imk < 0}, j = 1, 2, 3.

From global relations (21), (22) and the vertex condition

ô( gi(w,t)+ S^g2 (w, t) + S2g3(w,t)=0

we obtain

kago(w,t) = S2B1A2A3 + S2B2A1A3 + SBA1A2 X

x ( S2 A2A3 (eikLl q0o](k) - e-ikLl ^(-k) + 2kah(1 (w,t)-

-ewt (eikLi tj°1)(k, t) - e-ikLl q(1)(-k,t)^ + +S2A1A3 (eikL2q(2) (k) - e-ikL2q(2) (-k) + 2kah%)(w,t)- (23)

-ewt (eikL2q°2)(k,t) - e-ikLq(2)(-k,t)^ + +S&1A2 (eikL3q0(3 (k) - e-ikL3q(3) (-k) + 2kah0g)(w,t)--ewt (eikL3 q(3)(k,t) - e-ikL q(3)(-k

where Aj = eikLj - e-ikLj, Bj = eikLj + e-ikLj, j = 1, 2, 3.

Introducing G(j) (k, t) = q(j)(k) - kag0(w, t) + e-ikLj ■ kah(0')(w, t), j = 1, 2, 3, we can rewrite

ewtq°j)(k,t) = G(j)(k,t)+ iagj(w,t) - ie-ikLjah[j)(w,t), ewtq(j)(-k, t) = G(j)(-k, t) + iagj (w,t) - ieikL ahj (w,t).

(24)

So, we have

iagj (w,t) = - A- (eikLi G(j)(k,t) - e-ikL> G(j)(-k,t)) +

+ — (ewt (eikLj q(j)(k,t) - e-ikLq(j)(-k,t)

iah[j)(w,t) = -j (GU)(k,t) - Gj)(-k,t)) +

+ -j (ewt (q{j)(k,t) - $j)(-k,t))) , j = 1, 2, 3.

(26)

As in the case of the first problem we can state Theorem 2. The solution of problem (18)-(20), (4), (5) is given by

1 f-N

q(j)(x,t) = — eikx-wtq0j>(k)dk-

2n

1 f ikx-ikjwt q0oj)(k) - 9oj)(-k) - 2kago(w,t) + 2eikL kah0j)(w,t)

e

dk+ (27)

2n JdD(4) Aj

+ i eikx-wt e-ikLjqjj)(-k) - eikLj 9oj)(k) - 2kah<j)(w,t) + 2e-ikLj kago(w,t) ^

2n J8D2 Aj '

where j = 1, 2, 3, g0 (w,t) is defined by (23).

3. The general star graph

Here we generalize the obtained above results to the case of more general star graph which has number of finite and semi-infinite bonds. We consider metric graph r3 which obtained by connecting n finite B1,B2,... ,Bn and m semi infinite Bn+1, Bn+2,..., Bn+m bonds at one point that is the vertex of the graph. As in previous cases, bonds Bj, (j = 1, n) correspond to intervals (0, Lj) and bonds Br, (r = n +1,n + m) correspond to intervals (0, to). It defines coordinates in each bond.

Theorem 3. The solution of the IBVP problem on r3 has the form q(j)(x,t) = — eikx-wtqj)(k)dk-

2n J

1 r eikx-ikL3-wt 9oj)(k) - 9oj)(-k) - 2kago(w,t) + 2eikL kahQj)(w,t) dk+ (2g)

2n JdD(4) Aj

+ f eikx-wt e-ikLj 9oj)(-k) - eikLj 9oj)(k) - 2kahQj)(w,t) + 2e-ikLj kago (w,t) ^

1 r

1 -i f

q(r) (x, t) =— eikx-wtq0r) (k)dk +— eikx-wtq0r)(-k)dk-2n J2n J9D(2)

1

(29)

eikx-wt kg0(w, t)dk, (j = 1,n, r = n +1,n + m), n JdD(2)

where

kag0(w,t) = -s-:-x

2-^j = 1 sj Aj + ¿^ r=n+1 ¿r

n e2 n+m

Ef [eikLq0\k) - e-ikL9oj)(-k)+2kah0j)(w,t)]+ £ ¿l^(k)

j=l j r=n+l

A: = eikLi - e-ikLi, B: = eikLi + e-ikL=, (j = 1~n,).

The proof of the theorem is similar to given above theorems and can be obtained by combining

previous results.

Authors are grateful to the reviewer for helpful comments and references which certainly helped

to improve the paper.

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Метод унифицированного преобразования (Фокса) для уравнения Шредингера на простом метрическом графе

Гулмирза Худайберганов Зарифбой А. Собиров Мардонбек Р. Эшимбетов

Национальный университет Узбекистана им. М. Улугбека

ВУЗ городок, Ташкент, 100174 Узбекистан

В настоящей работе мы получили интегральное представление решения начально-краевых .задач для уравнения Шредингера в простых звездообразных графах методом Фокаса. Этот метод использует унифицированное преобразование Фурье. Полученные нами решения дают более точную информацию о динамике рассеяния в точке разветвления графа. В частности, можно увидеть отдельные слагаемые интегрального представления, соответствующие отражению и прохождению волны через вершину графа.

Ключевые слова: уравнение Шредингера, метрический граф, разветвленные структуры, унифицированное преобразование, начально-краевая задача, преобразование Фурье, краевая задача.