CC BY
12
Inverse dynamic problem for the wave equation with periodic
boundary conditions
A. S. Mikhaylov1'2, V. S. Mikhaylov1'2
1 Saint Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, 7, Fontanka, Saint Petersburg, 191023 Russia
2 Saint Petersburg State University, 7/9 Universitetskaya nab., Saint Petersburg, 199034 Russia
mikhaylov@pdmi.ras.ru, vsmikhaylov@pdmi.ras.ru
We consider the inverse dynamic problem for the wave equation with a potential on an interval (0, 2n) with periodic boundary conditions. We use a boundary triplet to set up the initial-boundary value problem. As inverse data we use a response operator (dynamic Dirichlet-to-Neumann map). Using the auxiliary problem on the whole line, we derive equations of the inverse problem. We also establish the relationships between dynamic and spectral inverse data.
Keywords: inverse problem, Boundary Control method, Schrodinger operator.
Received: 10 January 2019 Revised: 24 January 2019
1. Introduction
Inverse problems for one-dimensional continuous and discrete systems plays an important role for the creation of new nano-devices, to mention just [1,2] and references therein. In the present paper, we set up and study the inverse dynamic problem for a wave equation with a potential on an interval with periodic boundary conditions. The control problems for dynamical systems for wave equation with periodic boundary conditions (the density allows certain dependence on time) were considered in [3,4]. The spectral problem for a Schrodinger operator on an interval with periodic and anti-periodic boundary conditions are used for treating the spectral problem for a Schrodinger operator with periodic potential on R, see [5]. The inverse spectral problem with periodic boundary conditions for Schrodinger operator plays an important role for studying inverse problems on graphs with cycles [6].
In the previous papers by the authors, the "dynamic" approach to inverse spectral problems based on ideas of the Boundary Control method [7,8] was developed in the cases of Schrodinger operator on a half-line [9-12] and finite and semi-infinite Jacobi matrices [13,14]. We believe that our "dynamic" methods will help us to establish new relationships and develop new tools for studying the inverse problems with periodic potential, and will also stimulate studying inverse problems on graphs with cycles [6, 15].
For a potential q e C2(0, 2n) we consider an operator H in L2(0,2n) given by:
DOI 10.17586/2220-8054-2019-10-2-115-123
(Hf )(x) = -f ''(x) + q(x)f (x), x g (0, 2n), domH = {f g H2(0, 2n) | f (0) = f '(0) = f (2n) = f '(2n) = 0} .
Then
(H*f )(x) = -f''(x) + q(x)f (x), x g (0, 2n), dom H * = {f g H 2(0, 2n)} .
For a continuous function g we introduce the notations:
go := lim g(0 + e), g2n := lim g(2n - e).
Let B := R2. The boundary operators r0j1 : dom H* ^ B are introduced by the rules:
Integration by parts for u, v g dom H* shows that the abstract second Green identity holds:
(H V v)L2(0'2n) - H*v)L2(0'2n) = r0v)B - (F0^ F1v)B .
The mapping
r :=
To Ti
: dom H * h^ B x B
is surjective. Then a triplet {B, r0, ri} is a boundary triplet for H* (see [16]).
Let T > 0 be fixed. We use the triplet {B, r0, r1} to set up the following initial-boundary value problem:
( utt + H*u = 0, t > 0,
(r°u)(t) = /2(t)J, t>0,
u(^, 0) = ut(•, 0) = 0.
(1)
Here the vector function F
fi /2 € ¿2(0, T), is interpreted as a boundary control. The solution to (1) is
denoted by uF . The response operator is introduced by the rule
(RTF) (t) := (riuF) (t), t > 0.
The speed of the wave propagation in the system (1) equal to one, which is why the natural set up of the dynamic inverse problem (IP) is to find a potential q(x), x € (0, 2n) from the knowledge of a response operator R2n (see also [7,8,17,18]).
In the second section, we derive the representation formula for the solution uF, introduce the auxiliary dynamical system on the real line (see also [19]), and use the finiteness of the speed of wave propagation to establish relationships between the problem with periodic boundary conditions and problem on R. In the third section, on the basis of this relationship, we obtain the suitable version of Krein and Gelfand-Levitan equations of the dynamic inverse problem. In the last section we derive the spectral representation of the response operator and dynamic representation of a Weyl function associated with {B, r0, r1}.
2. Forward problem, auxiliary dynamical system
We introduce the outer space of the system (1), the space of controls as ft := L2(0,T; R2), F € ft,
F = (/\. By q we also denote the same potential, periodically continued to the whole real line: q(x+2n) = q(x), /2
x € R.
Theorem 1. The solution to (1) with a control F € ft n C0°(R+), admits the following representation: 1) For 0 < t < 2n
uF (x,t)= uF+ (x,t) + uF (x, t) (2)
11 1
= 2/i(t - x) - 2/2(t - x) + y w°(x, s)/i(t - s) + w°(x,s)/2(t - s) ds
11 ft
-o/i(t + x - 2n) --/2(t + x - 2n)+ / w2n (x, s)/i(t - s)+ w2n (x,s)/2(t - s) ds.
2 2 J2n-x
where kernels (x,t) satisfy the following Goursat problems:
w°tt(x, t) - w°xx(x,t) + q(x)w°(x, t) =0, 0 < x < t,
d 0 q(x)
—wi(x,x)=--, x> 0,
dx 4
w2ntt(x,t) - w2nxx(x,t) + q(x)w2n(x,t) =0, 0 < 2n - x < t,
d n„.. „ q(x)
—(x, 2n - x) = , x > 0,
dx 1 4
w0(0, s) = (2n,s), wQx(0,s) = w?nx(2n,s).
w0tt(x,t) — w0xx(x,t) + q(x)wQ(x, t) =0, 0 < x < t, d Q/ \ q(x)
dxwQ(x,x) = 4 ,
(3)
—w,' (x, 2n — x) = — dx2
w0(0,s) = (2n,s),
x > 0,
0 + q q(x)
w2nii(x,t) - w,nxx(x,t) + q(x)w,n(x, t) =0, 0 < 2n - x < t,
(4)
4
x > 0,
2) 0« 0 < t < 4n
U (x,t)= U- + (x,t)+ u1 (x,t)+ U- + (x,t)+ U- (x,t)
1 ^
2"
t
11 r
= 2fi(t - x) - 2f,(t - x)^y w0(x, s)/i(t - s) + wQ(x,s)/,(t - s) ds
11 I"t
-1 /i(t + x - 2n) - 1 /,(t + x - 2n)+ / (x, s)/i(t - s) + (x,s)/,(t - s) ds
2 2 J2n-x
+ 2/i(t - 2n - x) - 1 /,(t - 2n - x)
t-2n
+ / wQ(x, s)/1(t - 2n - s) + w0(x, s)/2(t - 2n - s) ds
x
- 2/i(t + x - 4n) - 2/2 (t + x - 4n)
p t-2n
+ / w2n(x, s)/i(t - 2n - s) + w|n(x, s)/2(t - 2n - s) ds.
J 2n —x
where the integral kernels w0^, w^^ satisfy certain Goursat problems and the following compatibility conditions:
w
,.q
0,2(0, s) = w2n2(2n, s), wQ,2x(0, s) = w2n2x(2n, s), 0 < s < 4n,
2x 1,2x
w0,2(2n, s) = W0,2(0, s - 2n), w0 2x(2n, s) = W02x(0, s - 2n), 0 < s < 4n, „ ï -27,2x(0, s) = W2X,
w2n2(0, s) = w2n2(2n, s - 2n), w2n2 (0, s) = Win2 (2n, s - 2n), 0 < s < 4n.
3) On 0 < t < 2nn, n > 1 :
2
U2 (x, t) = uf+ (x, t) + U2 (x, t) + ... + un+ (x, t) + U-2 (x, t),
(5)
where
1
1
+
r+2(fc-i)n
i+ (x, t) = 2/i(t - x - 2(k - 1)n) - 2/2(t - x - 2(k - 1)n) wi(x + 2(k - 1)n, s)/i(t - s) + w2(x + 2(k - 1)n, s)/2(t - s) ds
1
1
U-2 (x, t) = -^/i(t + x - 2kn) - ^/2(t + x - 2kn) + / wi(x - 2kn, s)/i(t - s) + W2(x - 2kn, s)/2(t - s) ds
—x
t
and kernels wi,2 satisfy the following Goursat problem:
witt(x, t) — t) + q(x)wi(x, t), 0 < |x| < t < 2nn,
x > 0, x < 0,
d q(x)
dxwi (x, x) =--— ,
d , q(x)
(6)
d-wi(x, —x) = —- 4
W2tt(x, t) — W2xx(x, t) + q(x)w2(x, t), 0 < |x| < t < 2nn, x > 0, x < 0.
d q(x)
dxw2(x,x) = —,
d , \ q(x) dxw2(x —x) = —;
(7)
Several remarks have to be made.
Remark 1. The proof of the representation (2) is straightforward and similar to one in [19]. If F g FT, the function uF defined by (2) is a generalized solution to (1) for t g (0,2n).
Remark 2. The compatibility conditions in (3), (4) is used in the next subsection to relate the solution of the problem with periodic boundary conditions with one of the problem on the whole line.
Since we consider the periodic boundary conditions, sometimes it would be convenient for us to interpret the interval as a ring:
Remark 3. The compatibility conditions in 2) allows one to construct the "general" Goursat problems in 3). The physical meaning of the representation (5) is clear: the members of the sum indexed with "+" corresponds to waves that move clockwise, ones indexed with "— " correspond to waves moving counterclockwise.
The response operator RT : ft ^ ft with the domain DR = {ft n C0°(O, T; R2)} is defined by the rule
(RTF)(t) := (riuF) (t), 0 <t<T.
Representation (2) implies the following
Corollary 1. The response operator has a form: 1) on an interval (0,2n):
(rt f > <«=—1 —+ r • f:
(8)
where
R(t) :=
rii(t) ri2(t) ,r21(t) r22(t)
5(0,t)
c(0,t)
(0,t)
—w0(0,t) —w0(0,t)
w2nx(0,t) w2n3 —w2n (0,t) —w2n (0,t)
is a response matrix, 2) on an interval (0,2nn) :
(-RT f) (t)
r 2 U o2kn) — ** (f2,
(9)
where the integral kernel R is expressed in terms of solutions to Goursat problems (6), (7).
Remark 4. Due to the finite speed of wave propagation in system (1), the natural set up of IP is to recover the potential on (0, 2n) from R2n, that is why, for solving IP we can consider the system for times less or equal 2n.
2.1. Auxiliary problem on R
We introduce the the potential if by the rule
'q(x), 0 < x < 2n,
0, x > 2n,
q(x + 2n), —2n < x < 0, ^0, x < —2n,
q(x)
(10)
u
u
w
w
i
2
(Hff)(x) = — f "(x) + i(x)f (x), x g :
For this potential, we consider an operator H in L2 (R) given by:
'(x
domH = {f g H2(R) | f (0) = f '(0) = 0} .
Then:
(H*f )(x) = —f''(x)+ i(x)f (x), x g R, dom HP = {f g L2(R) | f g H2(—to, 0), f g H2(—to, 0)} . For a continuous function g we denote:
g± := lim g(0 ± e). The boundary operators r0j1 : dom H* ^ B are introduced by the rules
iow := — W-\ , fiw :=i f + W- ) .
\ w+ — wl 2 \ —w+ — w_ I
We consider the initial boundary value problem for an auxiliary dynamical system on R:
'vtt + vxx + fv = 0, x g R, 0 < t < 2n,
(rov)(t)= f (t^ , 0 <t< 2n, (11)
0) = v^, 0) = 0.
In [19] the dynamic IP for (11) was studied, where as a inverse data the authors used the response operator, introduced by the rule:
(RTF) (t) := (f ivF) (t), t > 0.
On comparing the representation (2) with one obtained in [19] in Theorem 1, one deduce that for 0 < t < 2n the following equality holds:
VF (xt)=JUF+ (x,t), 0 <x< 2n, (12)
V ' 7 \uF- (x + 2n,t), —2n<x< 0.
Moreover, one has that:
R2nF = r1uF = f 1VF = R2nF, 0 < t < 2n. (13)
Thus we reduced our initial IP to the IP for dynamical system (11) of recovering the potential i(x), on the interval —n < x < n from R2n.
3. Equations of IP
In this section, we briefly outline the results of [19] in applying to our situation. Fix a parameter 0 < T < n and introduce the inner space, the space of states of the system (11) as ht := L2(—T, T). The representation (12) and Theorem 1 imply that vF(•, T) g ht.
A control operator WT : ft ^ ht is defined by the formula WTF := vF(-,T). The reachable set is defined by the rule:
UT := WTft = {vF(•, T) | F gft} .
It will be convenient for us to associate the outer space ht = L2(—T,T) with a vector space L2(0,T;R2) by setting for a g L2(—T, T) (we keep the same notation for a function)
a = ( ai(x) | g L2(0,T;R2), a1(x) := a(x), a2(x) := a(—x), x g (0,T). ya2(x) J
Theorem 2. The control operator is a boundedly invertible isomorphism between ft and ht, and UT = ht. The connecting operator CT : ft ^ ft is introduced via the quadratic form:
(CTF1,F2)ft = (vFl(•,T),vF2(•,T))ht .
The crucial fact in the Boundary Control method is that the connecting operator is expressed in terms of inverse dynamic data:
Theorem 3. The connecting operator CT admits the following representation:
1 /i(tA , rT ^ ^ /i(s)
(ct F ) «>=2 1; C (t.s) p d„
where
Ci,i(t, s)= Pi(2T - t - s) - pi(|t - s|), pi(s) = / rii(a) da,
J °
/ ri2(a) da, s > 0,
J°
Ci,2(t,s)= pi(2T - t - s) - pi(t - s), pi(s) = ^ „_s
/ ri2(a) da, s < 0,
C2,i(t, s) = -f2i(t - s) - r2i(2T - t - s), f2i(s) = j r2l(S)), s> 0 n
[ -r2i(-s), s < 0,
C2,2(t, s) = -r22(|t - s|) - r22(2T - t - s).
3.1. Krein equations
Let y(x) be a solution to the following Cauchy problem:
i -y" + §y = 0, x € (-T, T), 1 y(0) = 0, y'(0) = 1.
(14)
We set up the special control problem: to find F € ft such that WTF = y in ht. By the Theorem 2, such a control F exists, but we can say even more:
Theorem 4. The solution to a special control problem is a unique solution to the following Krein equation:
(CTF) (t) = (T - t)Q , t € (0, T). (15)
Representation formulas (2) and (12) imply that that the solution F to a special control problem satisfies relations:
y(T)= (T,T) = 1 /i(0) - 1 /2(0), y(-T)= (-T, T) = -1 /i(0) - 1 /2(0).
Thus solving (15) for all T € (0, n), we recover the solution y(x) to (14) on the interval (-n,n). Then the
y" (x)
potential q(x), x € (-n, n) can be recovered as q(x) = —-—, x € (-n, n), and consequently
y(x)
/ \ I 5(x), 0 < x < n, q(x) =
I </(x - 2n), n < x < 2n.
3.2. Gelfand-Levitan equations
We introduce the notations:
CT = 2(1 + C), (C/)(t) = 2 £ Cfts)/^ ds,
JT : ft ^ ft, (JTF) (t) = F(T -1),
(5 = JTCJT, (cf) (t) = ^ C(t,s)F(s) ds. (16)
Let m(x,t) € C ((0, n)2, R2x2) denotes a matrix-valued function such that m(x,t) = 0 when x > t. In [13] it was proved the following
Theorem 5. The unique solution to the Gelfand-Levitan equation
~ rn „
m(x, s) + C(x, s) + / C(x, a)m(a, s) da = 0, 0 < x < s < n.
o
where the kernel C is defined by (16), determines the potential by the formula:
{2— (mn(x,x) — m12(x, x)), x g (0, n), d
—2— (mn(2n — x, 2n — x) + m12(2n — x, 2n — x)), x g (n, 2n). dx
4. Relationship between dynamic and spectral inverse data
The problem of finding relationships between different types of inverse data is very important in inverse problems theory. We can mention [9,10,13,20-22] on some recent results in this direction. Below we show the relationships between the dynamic response function, matrix spectral measure and Weyl matrix.
4.1. Response function and spectral measure
Consider two solutions to the equation:
—¿'' + q(x)^ = A^, 0 < x < 2n, (17)
satisfying the Cauchy data:
y(0, A) = 0, y>'(0, A) = 1, 0(0, A) = 1, 0'(0, A) = 0. The eigenvalues and normalized eigenfunctions of (17) with periodic boundary conditions:
¿(0) = ¿(2n), ¿'(0)= ¿'(2n). (18)
are denoted by {An, Let y„ g R be such that:
y„(x) = y#n¥>(x, An) — 7„0(x, An),
we point out that there can be eigenvalues of multiplicity two. We evaluate:
yn(0) = —Yn, yn(2n) = An^(2n, A„) + Y„0(2n, A„), yn(0) = An, yn(2n) = An^'(2n, An) + 7n0'(2n, A„).
Then:
riyn = H *<J> + yn<22') = . (19)
2 \-yn(0) - yn(2n) = / \ Yn ,
Let F g ft n (0, T; R2), and uF be a solution to (1). On multiplying (1) by yn and integrating by parts, we get the following relation:
/*2n /*2n /*2n /*2n
0 = / wiiyn dx — / MFœyn dx + / q(x)wFyn dx = / wFyn dx ./o ./o ./o ./o
+ (wF , Hyn) + (ri«F, roy^B — (TowF,
i 2^ /Y
J wFyn dx + An (mf , yn) — II
.Mm /A
/2 (t)/ \Yn
Looking for the solution to (1) in the form:
uF
= ^3 cfc (t)yfc (x), (20)
k=1
we plug (20) into (1) and multiply by yn and integrate over (0,2n) to get:
'—' /»2n '—^
£4'(*)y*(x)yn(x) dx + / (t)yfc(x)Anyn(x) dx =
o
7i(t)\ /A /2(t)/ Ay
B
B
Thus we obtain that cn(t), n > 1, satisfies the following Cauchy problem:
/7i(tA (fnx
cn (t) + A„c„(t) =
cn(0) = 0, cn(0) = 0.
the solution of which is given by the formula:
/2(t)/' Wn
Cn(t)= /
Jq
Then for uf (20) we have the expansion:
sin yA"(t - s)
V^n
(/i(s)An + /2(s)Yn) ds.
>,t) = Y
k = i '
sin^t - s)
%/An
(/i(s)An + /2(s)Yn) ds (^„^(x,A„) - Y„0(x, A„))
œ />t sin^t - s)
E
k=i '
VAn
A
k7n
n \ ^ / A
0
n
Yn ,
/i(s) /2 (s),
y(x, An) -0(x, An)
fœ f si^v/A(t - s)
-œ ./0
VA
d£(A)
/i (s)\ Mx,A) /2 (sW A -0(x,A)y
Where d£(A) is a matrix measure (see [5]) introduced by the rule:
e w°(f
{k | Ak<A} V'V V7ny Thus, the response operator RT is given by:
(RF)(t) = rivF = Y cfc(t)riyfc = Y ck(t)
k=i
Y
k = i '
sin a/AI(t - s)
'Ak
VYk )
, ^ (/i(s)Ak + /2(s)Yk) dslAk)
0 V Ak ^Yk y
fœ /•t sin vA(t - s) /7i(sA , „
-œi, —tHdS(A) /wds- 0 <(.
4.2. Weyl function and response function
Let Na := ker (H* - AI), we observe that any ^(x, A) G Na is given by:
-0(x, A) = ciy(x, A) + C2^(x, A).
We evaluate:
^q = c2, = ciy(2n) + C2^(2n),
^0 = ci, ^2 n = ci^'(2n) + c20'(2n).
Thus the following relations hold:
-y(2n) 1 - 0(2n)\ /cA
1 - y'(2n) -0'(2n) / \c2
r^ = -
1/1 + y'(2n) 0'(2n) \/ci
2 I -y(2n) - (1 + 0(2n)W Vo2 / '
The Weyl matrix is given by (see [16]):
so we have:
M (A) = r (r, | )-
1
-0'(2n) - (1 - y'(2n))
M (A) = If1 + y'(2n) 0,(2^) _
() 2 I -y(2n) - (1 + 0(2n)W detr, I- (1 - 0(2n)) -y(2n)
(21)
(22)
(23)
(24)
B
t
u
q
0
i
Evaluating the last expression we get the following formula for the Weyl matrix:
where
F(x, A) = y/(x, A) + 0(x, A)
is a Lyapunov function.
In [9] the authors established the relationship between the Weyl function and the kernel of dynamic response operator (see also [10,13,22]). Note that one needs to know the response for all t > 0. Then, cf. (9):
where this equality is understood in a weak sense. Acknowledgements
The research of Victor Mikhaylov was supported by RFBR 17-01-00529. Alexandr Mikhaylov was supported by RFBR 17-01-00099; A. S. Mikhaylov and V. S. Mikhaylov were partly supported by RFBR 18-01-00269 and by the Ministry of Education and Science of Republic of Kazakhstan under grant AP05136197.
[1] Rietman E.A. Molecular Engineering of Nanosystems. Berlin, Springer Science & Business Media, 2013.
[2] Voda A. Micro, Nanosystems and Systems on Chips: Modeling, Control, and Estimation. New York, John Wiley & Sons, 2013.
[3] Avdonin S.A., Belinskiy B.P. and Ivanov S.A. On controllability of an elastic ring. Appl. Math. Optim., 2009, 60(1), P. 71103.
[4] Avdonin S.A., Belinskiy B.P. and Pandolfi L. Controllability of a nonhomogeneous string and ring under time dependent tension. Math. Model. Nat. Phenom., 2010, 5(4), P. 4-31.
[5] Levitan B.M. Inverse Sturm-Liouville problems. VNU Science Press, Utreht, the Netherlands, 1987.
[6] Kurasov P.B. Inverse problems for Aharonov-Bohm rings. Math. Proc. Cambridge Philos. Soc., 2010, 148(2), P. 331-362.
[7] Belishev M.I. Recent progress in the boundary control method, Inverse Problems, 2007, 23(5), P. R1-R67.
[8] Belishev M.I. Boundary control and tomography of Riemannian manifolds (the BC-method). Uspekhi Matem. Nauk, 2017, 72(4), P. 3-66 (in Russian).
[9] Avdonin S.A., Mikhaylov V.S., Rybkin A.V. The boundary control approach to the Titchmarsh-Weyl m—function. Comm. Math. Phys., 2007, 275(3), P. 791-803.
[10] Mikhaylov A.S., Mikhaylov V.S. Relationship between different types of inverse data for the one-dimensional Schrödinger operator on the half-line. J. Math. Sci. (N.Y.), 2017, 226(6), P. 779-794.
[11] Mikhaylov A.S., Mikhaylov V.S. Inverse dynamic problems for canonical systems and de Branges spaces. Nanosystems: Physics, Chemistry, Mathematics, 2018, 9(2), P. 215-224.
[12] Mikhaylov A.S., Mikhaylov V.S. Boundary Control method and de Branges spaces. Schrödinger operator, Dirac system, discrete Schrodinger operator. Journal of Mathematical Analysis and Applications, 2018, 460(2), P. 927-953.
[13] Mikhaylov A.S., Mikhaylov V.S., Simonov S.A. On the relationship between Weyl functions of Jacobi matrices and response vectors for special dynamical systems with discrete time. Mathematical Methods in the Applied Sciences, 2018, 41(16), P. 6401-6408.
[14] Mikhaylov A.S., Mikhaylov V.S. Dynamic inverse problem for Jacobi matrices, 2019, (to appear in Inverse Problems and Imaging).
[15] Belishev M.I., Wada N. On revealing graph cycles via boundary measurements. Inverse Problems, 2009, 25(10), P. 105011-21.
[16] Behrndt J., Malamud M.M., Neidhart H. Scattering matrices and Weyl functions. Proc. London Math. Soc., 2008, 97, P. 568-598.
[17] Avdonin S.A., Mikhaylov V.S. The boundary control approach to inverse spectral theory. Inverse Problems, 2010, 26(4), P. 045009-19.
[18] Belishev M.I., Mikhaylov V.S. Unified approach to classical equations of inverse problem theory. Journal of Inverse and Ill-posed Problems, 2012, 20(4), P. 461-488.
[19] Mikhaylov A.S., Mikhaylov V.S. On an inverse dynamic problem for the wave equation with a potential on a real line. Zapiski Seminarov POMI, 2017, 461, P. 212-231.
[20] Belishev M.I. On relation between spectral and dynamical inverse data. J. Inv. Ill-posed problems, 2001, 9(6), P. 647-665.
[21] Belishev M.I. On a relation between data of dynamic and spectral inverse problems. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 2003, 297, P. 30-48, translation in J. Math. Sci. (N.Y.), 2005, 127(6), P 2353-2363.
[22] Mikhaylov A.S., Mikhaylov V.S. Quantum and acoustic scattering on R+ and a representation of the scattering matrix. Proceedings Days on Diffraction 2017, IEEE, 2017, P. 237-240.
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