Scattering Scheme with Many Parameters and Translational Models of Commutative Operator Systems

The scattering scheme with many parameters for a commutative system of linear bounded operators {T1; T2}, when T1 is a contraction, is built. Using this construction of the scattering scheme, the translation model of the semigroup with two parameters T(n) = T1^n1 T2^n2 , n = (n1; n2) belongs Z+^2 is...

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Дата:	2007
Автор:	Zolotarev, V.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2007
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/7617
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Scattering scheme with many parameters and translational models of commutative operator systems / V.A. Zolotarev // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 4. — С. 424-447. — Бібліогр.: 8 назв. — англ.

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spelling	irk-123456789-76172010-04-07T12:01:02Z Scattering Scheme with Many Parameters and Translational Models of Commutative Operator Systems Zolotarev, V.A. The scattering scheme with many parameters for a commutative system of linear bounded operators {T1; T2}, when T1 is a contraction, is built. Using this construction of the scattering scheme, the translation model of the semigroup with two parameters T(n) = T1^n1 T2^n2 , n = (n1; n2) belongs Z+^2 is obtained. Description of characteristic properties of the dilation U of the contraction T1, that follows from the commutative property of the operators T1 and T2, in terms of external parameters lies in the basis of the method of the construction of the translational models for T(n). 2007 Article Scattering scheme with many parameters and translational models of commutative operator systems / V.A. Zolotarev // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 4. — С. 424-447. — Бібліогр.: 8 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/7617 en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The scattering scheme with many parameters for a commutative system of linear bounded operators {T1; T2}, when T1 is a contraction, is built. Using this construction of the scattering scheme, the translation model of the semigroup with two parameters T(n) = T1^n1 T2^n2 , n = (n1; n2) belongs Z+^2 is obtained. Description of characteristic properties of the dilation U of the contraction T1, that follows from the commutative property of the operators T1 and T2, in terms of external parameters lies in the basis of the method of the construction of the translational models for T(n).
format	Article
author	Zolotarev, V.A.
spellingShingle	Zolotarev, V.A. Scattering Scheme with Many Parameters and Translational Models of Commutative Operator Systems
author_facet	Zolotarev, V.A.
author_sort	Zolotarev, V.A.
title	Scattering Scheme with Many Parameters and Translational Models of Commutative Operator Systems
title_short	Scattering Scheme with Many Parameters and Translational Models of Commutative Operator Systems
title_full	Scattering Scheme with Many Parameters and Translational Models of Commutative Operator Systems
title_fullStr	Scattering Scheme with Many Parameters and Translational Models of Commutative Operator Systems
title_full_unstemmed	Scattering Scheme with Many Parameters and Translational Models of Commutative Operator Systems
title_sort	scattering scheme with many parameters and translational models of commutative operator systems
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2007
url	http://dspace.nbuv.gov.ua/handle/123456789/7617
citation_txt	Scattering scheme with many parameters and translational models of commutative operator systems / V.A. Zolotarev // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 4. — С. 424-447. — Бібліогр.: 8 назв. — англ.
work_keys_str_mv	AT zolotarevva scatteringschemewithmanyparametersandtranslationalmodelsofcommutativeoperatorsystems
first_indexed	2025-07-02T10:25:53Z
last_indexed	2025-07-02T10:25:53Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2007, vol. 3, No. 4, pp. 424�447 Scattering Scheme with Many Parameters and Translational Models of Commutative Operator Systems V.A. Zolotarev Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University 4 Svobody Sq., Kharkiv, 61077, Ukraine E-mail:Vladimir.A.Zolotarev@univer.kharkov.ua Received February 27, 2006 The scattering scheme with many parameters for a commutative system of linear bounded operators fT1; T2g, when T1 is a contraction, is built. Using this construction of the scattering scheme, the translation model of the semigroup with two parameters T (n) = T n1 1 T n2 2 , n = (n1; n2) 2 Z 2 + is obtained. Description of characteristic properties of the dilation U of the contraction T1, that follows from the commutative property of the operators T1 and T2, in terms of external parameters lies in the basis of the method of the construction of the translational models for T (n). Key words: scattering scheme with many parameters, translational model, commutative operator system. Mathematics Subject Classi�cation 2000: 47A45. The construction of the functional and translational models for a contracting operator T (kTk � 1) and its unitary dilation U [3, 6] is based on the study of the basic properties of the wave operators W� and scattering operator S [6]. Immediate generalization of these constructions for the case of the commutative operator system fT1; T2g, [T1; T2] = 0 is not trivial and not always possible. In this paper, a new method generalizing the scattering scheme on the case of many parameters by P. Lax and R. Fillips is presented. This method uses the isometric expansion � Vs; + V s �2 1 [7] based on isometric dilation for the commutative operator system fT1; T2g of the class C (T1) [7, 8] constructed in [8]. Unlike in a one-variable situation, two scattering operators S(p; k) and ~S(p; k), p, k 2 Z+ appear here. These operators have the property S�(0; 0) = ~S(0; 0). Using the method presented in [6], the generalization of construction of translational mo- dels for one operator T1 for the commutative operator system fT1; T2g, when one of the operators, e. g., T1, is a contraction, is presented in this paper. c V.A. Zolotarev, 2007 Scattering Scheme with Many Parameters 1. Isometric Dilations of Commutative Operator System I. Consider the commutative system of linear bounded operators fT1; T2g, [T1; T2] = T1T2 � T2T1 = 0 in the separable Hilbert space H. Hereinafter, we will suppose that one of the operators of the system fT1; T2g, e.g., T1, is a contraction, kT1k � 1. Following [4, 7, 8], de�ne the commutative unitary expansion for the system fT1; T2g. De�nition 1. Let the commutative system of the linear bounded operators fT1; T2g be given in the Hilbert space H, where T1 is a contraction, kT1k � 1. The set of mappings V1 = � T1 � K � ; V2 = � T2 �N K � : H �E ! H � ~E; + V 1= � T �1 � �� K� � ; + V 2= � T �2 � ~N� �� K� � : H � ~E ! H �E; (1:1) where E and ~E are Hilbert spaces, is said to be the commutative unitary expansion of the commutative system of operators T1, T2 in H, [T1; T2] = 0, if there are such operators �, � , N , � and ~�, ~� , ~N , ~� in the Hilbert spaces E and ~E, where �, � , ~�, ~� are selfadjoint, that the following relations take place: 1) + V 1 V1 = � I 0 0 I � ; V1 + V 1= � I 0 0 I � ; 2) V �2 � I 0 0 ~� � V2 = � I 0 0 � � ; + V �2 � I 0 0 � � + V 2= � I 0 0 ~� � ; 3) T2�� T1�N = ��; T2 � ~N T1 = ~� ; 4) ~N �� N = K�� ~�K; 5) ~NK = KN: (1:2) Consider the following class of commutative systems of linear operators fT1; T2g. De�nition 2. The commutative system of operators T1, T2 belongs to the class C (T1) and is said to be the contracting operator system for T1 if: 1)T1 is a contraction, kT1k � 1; 2)E def = ~D1H � ~D2H; ~E def = D1H � D2H; 3) dimT2 ~D1H = dimE; dimD1T2H = dim ~E; 4) operators D1j ~E ; ~D1 �� E ; ~D1T � 2 �� T2 ~D1H ; ~D1 �� E ; T �2D1jD1T2H are boundedly invertible, where Ds = T � s Ts � I; ~Ds = TsT � s � I; s = 1; 2: (1:3) Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 425 V.A. Zolotarev It is easy to see that if fT1; T2g 2 C (T1) ; then the unitary expansion (1) always exists. Let E and ~E be Hilbert spaces de�ned in 2) (1.3). Choose the unitary ope- rators V and ~V , V : T2 ~D1H ! ~D1H; ~V : T �2D1H ! D1H, what is always possible in view of 3) (1.3). De�ne now the invertible operators N1 = ~D1T � 2 V � and ~N1 = ~V T �2D1 in E and ~E (see 4) (1.3)). It is easy to see that the operators �1 = �N��1 1 ~D�11 N1 in E and ~�1 = �D1 in ~E are invertible, selfadjoint and nonnegative in view of 1), 4) (1.3). Consider the following set of operators N = p �1N �1 1 ~D2T � 1 q ��11 ; ~N = p ~�1 ~N �1 1 T �1D2 q ~��11 ; � = p �1N �1 1 � ~D1 � ~D2 �q ��11 ; ~� = p ~�1 ~N �1 1 (D1 �D2) q ~��11 ; � = � q ��11 T1 ~D2T � 1 q ��11 ; ~� = � q ~��11 D2 q ~��11 ; � = �p�1N�1 1 D2N ��1 1 p �1; ~� = �p�1 ~N�1 1 T �1D2T1 ~N ��1 1 p ~�1; ' = PEN1 q ��11 ; = p ~�1P ~E ; K = p ~�1T � 1 T � 2 q ��11 ; where PE and P ~E are orthoprojectors on E and ~E, respectively. It is easy to prove that in this case relations 1.2 are true for � Vs; + Vs �2 1 (1.1). Thus for the commutative operator system fT1; T2g of the class C )(T1) there always exists the unitary isometric expansion (1.1), (1.2). Note that the conditions 1) and 2) (1.2) for the expansions � Vs + Vs �2 1 (1.1) have a standard nature and play an important role in the construction of isometric (unitary) dilations [3, 6, 7]. One should consider relations 3)�5) (1.2) as the conditions of concordance of these expansions which follow from the commutative property of the operator system fT1; T2g. II. Remind the construction of the unitary dilation [3, 6] for a contraction T1. As usually [6, 7], we will denote by l2 M (G) the Hilbert space of G-valued functions uk which assume a value in the Hilbert space G, uk 2 G, where k 2M andM � Z are such that P k2M kukk2 <1. Let H be the Hilbert space of the following type H = D� �H �D+; (1:4) where D� = l2 Z � (E) and D+ = l2 Z+ ( ~E). Specify the dilation U on the vector- functions f = (uk; h; vk) from H (1.4) in the following way: Uf = � PD � uk�1; ~h; ~vk � ; (1:5) 426 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 Scattering Scheme with Many Parameters where ~h = T1h + �u�1, ~v0 = h +Ku�1, ~vk = vk�1 (k = 1; 2; : : :), and PD � is the operator of contraction on D�. The unitary property of U (1.5) in H follows from 1) (1.2). To construct the isometric dilation [8] of a commutative operator system fT1; T2g 2 C (T1), continue the incoming D� and outgoing D+ subspaces D� = l2 Z � (E); D+ = l2 Z+ ( ~E) (1:6) by the second variable �n2�. At �rst, continue functions un1 2 l2 Z � (E) from the semiaxis Z� into the domain ~Z2 � = Z�� (Z� [ f0g) = � n = (n1;n2) 2 Z2 : n1 < 0;n2 � 0 (1:7) using the following Cauchy problem [7, 8]:( ~@2un = � N ~@1 + � � un; n = (n1; n2) 2 ~Z2 �; unjn2=0 = un1 2 l2Z � (E); (1:8) where ~@1un = u(n1�1;n2), ~@2un = u(n1;n2�1). As a result, we obtain the Hilbert space D�(N;�) which is formed by un, the solutions of (1.8), at the same time the norm in D�(N;�) is induced by the norm of initial data kunk = kun1kl2 Z � (E). Similarly, continue functions vn1 2 l2 Z+ ( ~E) from the semiaxis Z+ into the domain Z2 + = Z+� Z+ using the Cauchy problem ( ~@2vn = � ~N ~@1 + ~� � vn; n = (n1; n2) 2 Z2 +; vnjn2=0 = vn1 2 l2Z+(E): (1:9) Thus we obtain the Hilbert space D+( ~N; ~�) that is made of solutions vn (1.9), besides kvnk = kvn1kl2 Z+ ( ~E). Unlike the evident recurrent scheme (1.8) of the layer-to-layer calculation of n2 ! n2�1 for un, in this case, while constructing vn in Z2 +, we are dealing with the implicit linear system of equations for layer-to-layer calculation of n2 ! n2 + 1 for the function vn. Hereinafter, the following lemma plays an important role. The proof of the lemma is given in [8]. Lemma 1.1. Suppose the commutative unitary expansion Vs, + V s (1.1) is such that Ker� = Ker � = f0g: (1:10) Then KerN \ Ker� = f0g given KerK� = f0g, and respectively Ker ~N� \ Ker ~�� = 0 given KerK = f0g. Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 427 V.A. Zolotarev The solvability of the Cauchy problem (1.9) easily follows [8] from the given lemma. Statement 1.1. Let dim ~E < 1 and the assumptions of Lem. 1.1 be true, then the solution vn of the Cauchy problem (1.9) exists and is unique in the domain Z 2 + for all initial data vn1 from l2 Z+ ( ~E). Consider now the operator-function of discrete argument ~�� = � I : � = (1; 0); ~�; � = (0; 1): (1:11) Let Ln0 be the nonincreasing polygon that connects points O = (0; 0) and n = (n1; n2) 2 Z2 + and linear segments of which are parallel to the axes OX (n2 = 0) and OY (n1 = 0). Denote by fPkgN0 all integer-valued points from Z 2 +, Pk 2 Z 2 + (N = n1 + n2) that lie on Ln0 , beginning with (0; 0) and �nishing with the point (n1; n2), that are numbered in nondescending order (of one of the coordinates of Pk). Assuming that P�1 = (�1; 0), de�ne the quadratic form h~�vki2Ln 0 = NX k=0 ~�Pk�Pk�1vPk ; vPk � (1:12) on the vector-functions vk 2 D+( ~N; ~�). Similarly, consider the nondecreasing polygon L�1 m in ~Z2 � (1.7) that connects pointsm = (m1;m2) 2 ~Z2 � and (�1; 0), the straight segments of which are parallel to OX and OY . Let fQsg�1M (M = m1+m2) be all integer-valued points on L�1m , beginning with m = (m1;m2) and �nishing with (�1; 0), that are numbered in nondescending order (of one of the coordinates of Qs). De�ne the metric in D�(N;�), h�uki2L�1m = �1X s=M �Qs�Qs�1 uQs ; uQs � ; (1:13) besides QM � QM�1 = (1; 0), and the operator-function �� is de�ned similarly to ~�� (1.11). Denote by ~L�1�n the polygon in ~Z2 � that is obtained from the curve Ln0 in Z2 + (n 2 Z2 +) using the shift by �n� ~L�1�n = n Qs = (l1; l2) 2 ~Z2 � : (l1 + n1 + 1; l2 + n2) = Pk 2 Ln0 o : (1:14) III. Having now the Hilbert space D�(N;�), that is formed by the solutions of the Cauchy problem (1.8) and the space D+( ~N; ~�), that is formed by the solutions of (1.9), we can de�ne the Hilbert space HN;� = D�(N;�)�H �D+( ~N; ~�); (1:15) 428 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 Scattering Scheme with Many Parameters the norm in which is de�ned by the norm of the initial space H = D� �H �D+ (1.4). Denote by Ẑ2 + the subset in Z2 +, Ẑ 2 + = Z 2 +n(f0g � N) = f(0; 0)g [ (N � Z+); (1:16) that is obviously an additional semigroup. For every n 2 Ẑ 2 + (1.16), de�ne an operator-function U(n) that acts on the vectors f = (uk; h; vk) 2 Hn;� (1.15) in the following way: U(n)f = f(n) = (uk(n); h(n); vk(n)) ; (1:17) where uk(n) = PD � (N;�)uk�n (PD � (N;�) is an orthoprojector that corresponds to the restriction on D�(N;�)); h(n) = y0, besides yk 2 H (k 2 Z2 +) is a solution of the Cauchy problem8< : ~@1yk = T1yk +�u~k; ~@2yk = T2yk +�Nu~k; yn = h; k = (k1; k2) 2 Z2 + (0 � k1 � n1 � 1; 0 � k2 � n2); (1:18) at the same time ~k = k � n when 0 � k1 � n1 � 1, 0 � k2 � n2, and �nally vk(n) = v̂k + vk�n (1:19) and v̂k = Ku~k + yk; where yk is a solution of the Cauchy problem (1.18). It is easy to see that the operator-function U(n) (1.17) maps the space HN;� (1.15) into itself for all n 2 Ẑ2 + (1.16), moreover, the following theorem takes place [8]. Theorem 1.1. Suppose dim ~E < 1 and the suppositions of Lem. 1.1 take place, then the following conservation law is true for the vector-function f(n) = U(n)f (1.17): kh(n)k2 + h~�vk(n)i2Ln̂ 0 = khk2 + h�uki2~L�1 �n (1:20) for all n 2 Ẑ 2 + (1.16) and for all nondecreasing polygons L̂n̂0 that connect points O = (0; 0) and n̂ = (n1 � 1; n2) 2 Z2 +, where ~L�1 �n̂ is a polygon obtained from Ln0 by the shift (1.14) by �n�, at the same time the corresponding �-forms in (1.20) have the form of (1.12) and (1.13). The operator-function U(n) (1.17) is a semigroup, U(n) � U(m) = U(n+m), for all n, m 2 Ẑ2 + (1.16). It follows from [8] and from this theorem that the operator-function U(n) (1.17) is an isometric dilation of the semigroup T (n) = T n11 T n22 ; n = (n1; n2) 2 Z2 +: (1:21) IV. Make the similar continuation of the subspaces D+ and D� (1.6) from the semiaxes Z+ and Z� by the second variable �n2�, corresponding to the dual Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 429 V.A. Zolotarev situation. Denote by D+ � ~N�; ~�� the Hilbert space generated by solutions ~vn of the Cauchy problem( @2~vn = � ~N�@1 + ~�� ~vn; n = (n1; n2) 2 Z2 +; ~vnjn2=0 = vn1 2 l2Z+( ~E); (1:22) in which the norm is induced by the norm of the initial data k~vnk = kvn1kl2 Z+ (E), besides @1~vn = ~v(n1+1;n2), @2~vn = ~v(n1;n2+1). Continue now every function un1 2 l2Z � (E) into the domain ~Z2 � (1.7) using the Cauchy problem� @2~un = (N�@1 +��) ~un; n = (n1; n2) 2 ~Z2 �; ~unjn2=0 = un1 2 l2Z � (E): (1:23) As a result, we obtain the Hilbert space D� (N�;��) generated by ~un, solutions of (1.23), besides k~unk = kun1kl2 Z � (E). Using now Lem. 1.1, we can formulate an analogue of St. 1 [8]. Statement 1.2. Let dimE < 1 and the suppositions of Lem. 1.1 be true, then the solution ~un of the Cauchy problem (1.23) exists and is unique in the domain ~Z2 � (1.7) for all initial data un1 2 l2Z � (E). O b s e r v a t i o n 1.1. The su�cient condition for the simultaneous existence of solutions of the Cauchy problems (1.9) and (1.23), in view of the reversibility of operators K and K�, according to Lem. 1.1, is the following: all hypotheses of Lem. 1.1 are met and dimE = dim ~E <1. Hence we come to the Hilbert space HN�;�� = D� (N�;��)�H �D+ � ~N�; ~�� ; (1:24) where the metric is induced by the norm of the initial space H = D� �H �D+ (1.4). Note that the dual feature of the spaces HN;� (1.15) and HN�;�� (1.24) is that di�erential operators of the Cauchy problems (18) and (1.23) and operators (1.9) and (1.22) also are adjoint with each other respectively in the metric l2. De�ne now the operator-function + U (n) for n 2 Ẑ2 + (1.16) in the space HN�;�� (1.24), which acts on ~f = � ~uk; ~h; ~vk � 2 HN�;�� in the following way: + U (n) ~f = ~f(n) = � ~uk(n); ~h(n); ~v(n) � ; (1:25) where ~vk(n) = P D+( ~N�;~��)~vk+n (PD+( ~N�;~��) is an orthoprojector ontoD+( ~N �; ~��)); 430 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 Scattering Scheme with Many Parameters ~h(n) = ~y(�1;0), besides ~yk (k 2 ~Z2 �) satis�es the Cauchy problem 8< : @1~yk = T �1 ~yk + �~v~k; @2~yk = T �2 ~yk + � ~N�~v~k; ~y(�n1;�n2) = h; k = (k1; k2) 2 ~Z2 � (�n1 � k1 � �1; �n2 � k2 � 0); (1:26) besides ~k = k + n ø (�n1 � k1 � �1; �n2 � k2 � 0); and �nally ~uk(n) = ûk + ~uk+n; (1:27) and ûk = K�~v~k +��~yk; where ~yk is a solution of the system (1.26). Similarly to (1.11), de�ne the operator-function �� = � I; � = (�1; 0); � ; � = (0;�1): (1:28) Denote by L�1m a nondecreasing polygon in ~Z2 � (1.7) with the linear segments that are parallel to the axesOX and OY which connects the pointsm = (m1;m2) 2 ~Z2 � and (�1; 0). Choose now all the points fQsg�1M (M = m1 +m2) on L �1 m that are numerated in nonascending order (of one of the coordinates Qs) beginning with the point (�1; 0) and �nishing with m = (m1;m2) 2 ~Z2 �. De�ne the quadratic form h� ~uki2L�1m = �1X s=M �Qs�Qs+1 ~uQs ; ~uQs � (1:29) in the space D� (N�;��), where Q0 = (0; 0). For the polygon Ln0 in Z 2 +, n = (n1; n2) 2 Z2 +, of the similar type with points fPkgN0 (N = n1 + n2) on L n 0 which are also chosen in nonascending order, de�ne the quadratic form for the functions ~vk 2 D+ � ~N�; ~�� h~�~vki2Ln 0 = NX k=0 ~�Pk�Pk+1~vPk ; ~vPk � ; (1:30) where PN � PN+1 = (�1; 0) and ~�� is de�ned similarly to �� (1.28). Denote by ~Lm0 the polygon in Z2 + obtained from the curve L�1m from ~Z2 � using the shift by �m� ~Lm0 = � Pk = (l1; l2) 2 Z2 + : (l1 +m1; l2 +m2) = Qs 2 L�1m ; (1:31) where m = (m1;m2) 2 ~Z2 �. Similarly to Th. 1.1, the following statement [8] takes place. Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 431 V.A. Zolotarev Theorem 1.2. Suppose that dimE <1 and that the hypotheses of Lem. 1.1 take place, then for the vector-function ~f(n) = + U (n) ~f (1.25) the equality k~h(n)k2 + h� ~uk(n)i2L�1 �n = khk2 + h~�~vki~L�n 0 (1:32) takes place for all n 2 Ẑ 2 + (1.16) and for all polygons L�1�n connecting points �n = (�n1;�n2) 2 ~Z2 � and (�1; 0); where ~L�n0 is a curve in Z 2 + obtained from L�1�n using the shift (1.31) by ��n�, and corresponding � -forms in (1.32) have the form of (1.29) and (1.30). The operator-function + U (n) (1.25) has the semigroup property, + U (n) + U (m) = + U (n+m) for all n, m 2 Ẑ2 + (1.16). The fact that the semigroup + U (n) (1.25) is the isometric dilation of the semigroup T �(n), where T (n) has the form of (1.21), is proved in [8]. In the conclusion of this paragraph, note that the dilations U(n) (1.17) and + U (n) (1.25) are unitary linked, i.e., U� (n1; 0) f = + U (n1; 0) f for all f 2 H (1.4) and for all n1 2 Z+, and the narrowing U (n1; 0) onto H is a unitary semigroup. 2. Scattering Scheme with Many Parameters and Translational Models I. As it is known [3, 6], a translational (as well as a functional) model of the contraction T and its dilation U (1.5) is based on the study of the main properties of the wave operators W� and scattering operator S. In order to construct the wave operatorsW� in the case of many parameters, it is necessary also to continue the vector-functions from l2 Z � ~E � and l2 Z (E) from the axis Z into the domain Z2. Continue every function un1 2 l2Z(E) to the function un, where n = (n1; n2) 2 Z2, using the Cauchy problem( ~@2un = � N ~@1 + � � un; n 2 Z2; unjn2=0 = un1 2 l2Z(E); (2:1) besides kunk = kun1kl2 Z (E). Note that this continuation into the lower semiplane (n2 2 Z�), u (n1; n2) ! u (n1; n2 � 1), has a recurrent nature and continuation into the upper semiplane u (n1; n2)! u (n1; n2 + 1) may be carried out in a non- explicit way, certainly, in the context of suppositions of Lem. 1.1 and dimE <1. As a result, we obtain the Hilbert space l2 N;�(E) the norm of which is induced by the norm of the initial data. De�ne now the shift operator V (p) V (p)un = un�p; (2:2) 432 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 Scattering Scheme with Many Parameters where un 2 l2 N;�(E) for all p 2 Z 2: Obviously, this operator V (p) (2.2) is an iso- metric one. Knowing the perturbed U(n) (1.17) and the free V (n) (2.2) operator semi- groups, it is natural to de�ne the wave operator W�(n), W�(k) = s� lim n!1 U(n; k)PD � (N;�)V (�n;�k) (2:3) for every �xed k 2 Z+; where PD � (N;�) is the orthoprojector of the narrowing onto the component u�n from l2 N;�(E) obtained as a result of continuation into ~Z2 � (1.7) from the semiaxis Z� using the Cauchy problem (2.1). It is obvious that W�(0) =W�, where the wave operator W� corresponds with the dilation U (1.5) and the shift operator V in l2 Z (E) [6]. Thus, W�(k) (2.3) is a natural continuation of the wave operator W� onto the �k-th� horizontal line in Z2 for k 2 Z+. Denote by L10;k the polygon in Z2 + consisting of two linear segments: the �rst one is a vertical segment connecting points O = (0; 0) and (0; k); where k 2 Z+, and the second segment is a horizontal semiline from the point (0; k) to (1; k). Similarly, choose the polygon ~L�1�1;p in ~Z2 � (1.7) that consists also of two linear segments, the �rst of which is a semiline from (�1;�p) to the point (�1;�p); where p 2 Z+; and the second one is a vertical segment from the point (�1;�p) to (�1; 0). In the space HN;� (1.15), specify the following quadratic forms: hfi2 �(p;k) = h�uni2~L�1 �1;p + khk2 + h~�vni2L1 0;k ; hfi2 ~�(k) = kunk2l2 + khk2 + h~�vni2L1 0;k ; hfi2 �(p) = h�uni2~L�1 �1;p + khk2 + kvnk2l2 ; (2:4) where corresponding � and ~� forms in (2.4) are understood in the sense of (1.12) and (1.13). It is easy to see that hfi2 �(0;0) = hfi2 ~�(0) = hfi2 �(0) = kfk2 HN;� and hfi2 �(0;k) = hfi2 ~�(k) , hfi2 �(p;0) = hfi2 �(p) . Similarly to (2.4), specify in l2 N;�(E) the following �-forms: huni2�(p;k) = h�u� n i2~L�1 �1;p + h�u+ n i L 1 0;k ; huni2�+(k) = ku�n k2l2 + h�u+n iL1 0;k ; huni2� � (p) = h�u�n i2~L�1 �1;�p + ku+n k2l2 ; (2:5) where u� n are the continuations of l2 Z � (E) from the semiaxes using the Cauchy problem (2.1). Note that huni2�(0;k) = huni2�+(k); huni 2 �(p;0) = huni2� � (p) and �nally huni2�(0;0) = huni2�+(0) = huni2� � (0) = kunk2l2 . Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 433 V.A. Zolotarev Theorem 2.1. The wave operator W�(k) (2.3) mapping l2 N;�(E) into the space HN;� (1.15) exists for all k 2 Z+, and it is an isometry hW�(k)uni2�(p;k) = huni2�(p;k) (2:6) in metrics (2.4), (2.5) for all p 2 Z+. Moreover, the wave operator W�(k) (2.3) meets the conditions 1) U(1; s)W�(k) =W�(k + s)V (1; s); 2) W�(k)PD � (N;�) = PD � (N;�) (2:7) for all k, s 2 Z+, where PD � (N;�) is an orthoprojector onto D�(N;�). P r o o f. Relation 2) (2.7) is proved exactly in the same way as for W� [6]. The isometric property (2.6) for W�(k) (2.3) follows from Th. 1.1. In order to prove 1) (2.7), consider the identity U(1; s)U(n; k)PD � (N;�)V (�n;�k) = U(n+ 1; k + s)PD � (N;�)V (�n� 1;�k � s)V (1; s); where the limit process leads us to equality 1) when n!1. And since W�(s)V (1; s) = U(1; s)W�(0); thenW�(s) existence follows from the existence ofW�(0) =W� [6] for all s 2 Z+. Note that the equalities 1) U(1; 0)W�(k) =W�(k)V (1; 0); 2) U(1; k)W�(0) =W�(k)V (1; k) (2:8) for all k 2 Z+ follow from 1) (2.7). II. Consider now the continuation of the vector-functions vn1 from l2 Z � ~E � into the domain Z2 using the Cauchy problem8< : ~@2vn = � ~N ~@1 + ~� � vn; n = (n1; n2) 2 Z2; vnjn2=0 = vn1 2 l2Z � ~E � : (2:9) As in the case of problem (2.1), in the semiplane n2 2 Z� we have a recurrent way of the continuation from the axis n2 = 0, n2 ! n2�1 and, when n2 2 Z+; this con- tinuation may be carried out in the context of Supposition 1.1. The Hilbert space obtained in this way may be denoted by l ~N;~� � ~E � , besides kvnk = kvn1kl2 Z( ~E) . 434 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 Scattering Scheme with Many Parameters Similarly to V (p) (2.2), introduce the shift operator ~V (p)vn = vn�p (2:10) for all p 2 Z2 and all vn 2 l2~N;~� � ~E � . De�ne the wave operator W+(p) from HN;� into the space l2~N;~� � ~E � , W+(p) = s� lim n!1 ~V (�n;�p)P D+( ~N;~�)U(n; p) (2:11) for all p 2 Z+, where U(n) has the form of (1.17). It is obvious thatW+(0) =W � +, where W+ is a traditional wave operator [6] corresponding to U (1.5) and to the shift ~V in l2 Z � ~E � . Similarly to Th. 2.1, the following statement is true. Theorem 2.2. For all p 2 Z+, the wave operator W+(p) (2.11) acting from the space HN;� into l2~N;~� � ~E � exists and satis�es the relations 1) W+(p)U(1; s) = ~V (1; s)W+(p+ s); 2) W+(p)PD+( ~N;~�) = P D+( ~N;~�) (2:11) for all p, s 2 Z+, where PD+( ~N;~�) is an orthoprojector onto D+ � ~N; ~� � . The proof of this statement is similar to the proof of Th. 2.1. The equalities 1) W+(p)U(1; 0) = ~V (1; 0)W+(p); 2) W+(0)U(1; p) = ~V (1; p)W+(p) (2:12) for all p 2 Z+ easily follow from 1) (2.11). Knowing the wave operators W�(k) (2.3) and W+(p) (2.11), de�ne the scat- tering operator in a traditional way [6]: S(p; k) =W+(p)W�(k) (2:13) for all p, k 2 Z+. It is obvious that, when p = k = 0; we have S(0; 0) = S; where S is a standard scattering operator S =W � +W� for the dilation U (1.5) [6]. The following statement results from Ths. 2.1 and 2.2. Theorem 2.3. The scattering operator S(p; k) (2.13) represents the bounded operator from l2 N;�(E) into l2~N;~� � ~E � satisfying the following relations: 1) S(p; k)V (1; q) = ~V (1; q)S(p + q; k � q); 2) S(p; k)P�l 2 N;�(E) � P�l 2 ~N;~� � ~E � (2:14) Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 435 V.A. Zolotarev for all p, k, q 2 Z+, 0 � q � k; where P� is the narrowing orthoprojector onto the solutions of the Cauchy problems (2.1) and (2.9) with the initial data on the semiaxis Z� when n2 = 0. Note that the translational invariability of S(p; k) (2.13) with respect to the shift by the �rst variable �n1� S(p; k)V (1; 0) = ~V (1; 0)S(p; q) (2:15) for all p, k 2 Z+ follows from the equality 1) (2.14). Moreover, from 1) it follows that 1) S(p; k)V (1; k) = ~V (1; k)S(p + k; 0) (k = q); 2) S(0; k)V (1; k) = ~V (1:k)S(k; 0) (k = q; p = 0); (2:16) and thus the scattering operator S(p; k) (2.13) is the function of sum (up to the multiplication of V (1; k) and ~V (1; k)) for all p and k from Z+, and it may be obtained from the operator S(k; 0) (or from S(0; k)) using the �bordering� by the shift operators V (1; k) and ~V (1; k). III. Specify now the mapping Pp;k from l2 N;�(E) + l2~N;~� � ~E � into the Hilbert space HN;� (1.15) fp;k = Pp;k (gn) = Pp;k � vn un � =W � +(p)vn +W�(k)un; (2:17) where vn 2 l2~N;~� � ~E � , un 2 l2N;�(E), besides p, k 2 Z+ and W � +(p) adjoined to the operator W+(p) is understood in the sense of Hilbert metric l2. For the commutative operator systems fT1; T2g 2 C (T1) (1.3), the simplicity of the expansion Vs, + V s (1.1) is guaranteed by the operator T1 [4, 8]. Therefore in the case of simplicity of the expansion Vs, + V s (1.1), the functions fp;k = Pp;k (gk) (2.17) form the everywhere dense set in the space HN;� when gn 2 l2~N;~� � ~E � + l2 n;�(E) for all �xed p and k from Z+. And thus �every� function f from the functions of the spaceHN;� (e.g., every �nite one) may have various forms f = fp;k or f = fp0;k0 (p 6= p0, k 6= k0) when one takes di�erent mappings Pp;k (2.17). It is easy to see that hfp;k; fp;kiHN;� = hWp;kgn; gnil2 ; when the weight operator-function Wp;k has the form Wp;k = � W+(p)W � +(p) S(p; k) S�(p; k) W � �(k)W�(k) � ; (2:18) and the scattering operator S(p; k) is de�ned by formula (2.13). 436 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 Scattering Scheme with Many Parameters O b s e r v a t i o n 2.1. All the elements of the weight operator-function Wp;k (2.18) have the translational invariance with respect to the shift by the variable �n1� in view of 1) (2.8), 1) (2.12) and (2.15), and also the unitarity of the operator U(1; 0). So, the mapping Pk;s (2.17) de�nes the isomorphism between the spaces HN;� (1.15) and l2 (Wp;k) = � gn = � vn un � : hWp;kgn; gnil2 <1 � ; (2:19) where un 2 l2N;�(E), vn 2 l2N;� � ~E � and the operator Wp;k has the form of (2.18). It is obvious that the space l2 (Wp;k) (2.19) coincides with the well-known space l2 � I S S I � [6] when p = k = 0. From the relations 1) (2.8), 1) (2.12) and from the unitarity of U(1; 0), it follows that the dilation U(1; 0) in every space l2 (Wp;k) (2.19) is carried out by the shift operator Û(1; 0)gn = � ~V (1; 0) 0 0 V (1; 0) � gn (2:20) for all gn 2 l2 (Wp;k). Study now how the dilation U(1; s) (1.17) acts on the vector-functions fp;k = Pp;k (gn) (2.17) when s 6= 0. First of all, note that it follows from 1) (2.7) that an application of U(1; s) to the wave operator W�(k) (2.3) from the left increases the index k 2 Z+ by s, i.e. k ! k + s, and it follows from the equality 1) (2.11) that an application of the dilation U(1; s) to the wave operatorW+(p) (2.11) from the right also changes the parameter p 2 Z+, namely, p ! p + s. Therefore the dilation U(1; s) maps the element fp;k from HN;� to the representative fp�s;k+s in the space HN;� (1.15), where 0 � s � p. Consider only the case when the dilation U(1; p) (1.17) acts on the vectors of the form fp;0 = Pp;0 (gn) (2.17). So, in view of above, consider the scalar productD U(1; p)fp;0; f̂0;p E ~�(p) = U(1; p)W � +(p)vn;W � +(0)v̂n � ~�(p) + U(1; p)W � +(p)vn;W�(p)ûn � ~�(p) + U(1; p)W�(0)un;W � +(0)v̂n � ~�(p) (2:21) + hU(1; p)W�(0)un;W�(p)ûni~�(p) ; where fp;0 = Pp;0 (gn), f̂0;p = P0;p (ĝn) (2.17). Simplify every element from the right part in (2.21). It is easy to see that the third and the fourth elements have the form U(1; p)W�(0)un;W � +(0)v̂n � ~�(p) = hS(0; p)V (1; p)un; v̂ni~�+(p) ; Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 437 V.A. Zolotarev hU(1; p)W�(0)un;W�(p)ûni~�(p) = hV (1; p)un; ûni�+(p) taking into account property 2) (2.8), the form of the operator S(0; p) (2.13), and the �-isometric condition of the wave operator W�(p) (2.3) by Th. 2.1 and 2) (2.11) used in the �rst relation. In order to simplify the �rst elements in (2.21), use relations 2) (2.11) and 2) (2.12) for the wave operator W+(p) to obtain U(1; p)W � +(p)vn;W � +(0)v̂n � ~�(p) = D ~V (1; p)W+(p)W � +(p)vn; v̂n E ~�+(p) : Finally, taking into account �-isometric property of the dilation U(1; p) (Th. 1.1), for the second element we have U(1; p)W � +(p)vn;W�(p)ûn � ~�(p) = U(1; p)W � +(p)vn; U(1; p)W�(0)V (�1;�p)ûn � ~�(p) = W � +(p)vn;W�(0)V (�1; p)ûn � �(p) = hS�(p; 0)vn; V (�1;�p)ûni� � (p) in view of 2) (2.7). Using now relation 2) (2.16), we obtain that U(1; p)W � +(p)vn;W�(p)ûn � ~�(p) = hV �(�1;�p)S�(p; 0)vn; ûni�+(p) = D S�(0; p) ~V �(�1;�p)vn; ûn E �+(p) : Thus, we can write formula (2.21) in the following way:D U(1; p)fp;0f̂0;p E ~�(p) = �� ~V (1; p)W+(p)W � +(p) ~V �(1; p) S(0; p) S�(0; p) I � � � ~V �(�1;�p) 0 0 V (1; p) � gn; ĝn � ~�+(p);�+(p) ; (2:22) where the bi-linear form in the right part is understood component-wisely in the sense of ~�+(p) and �+(p) (2.5). Let W 0 p;0 = � ~V (1; p)W+(p)W � +(p) ~V �(1; p) S(0; p) S�(0; p) I � ; V̂ (1; p) = � ~V �(�1;�p) 0 0 V (1; p) � : (2:23) 438 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 Scattering Scheme with Many Parameters O b s e r v a t i o n 2.2. Consider the mapping Pp;0 (2.17) and let f 0 p;0 = Pp;0 (g0n) =W � +(p) ~V �(1; p)vn+W�(0)V (�1;�p)un; where un 2 l2N;�(E) and vn 2 l2~N;~� � ~E � . Then it is easy to �nd that f 0 p;0; f 0 p;0 � HN;� = W 0 p;0gn; gn � l2 in view of 2) (2.16). Thus the di�erence between the weight Wp;0 (2.18) and W 0 p;0 (2.23) is that the components vn and un are shifted by ~V �(1; p) and V (�1;�p) respectively after the mapping Pp;0 (2.17). Hence, the dilation U(1; p) (1.7) acts by the shift Û(1; p)gn = V̂ (1; p)gn; (2:24) (V̂ (1; p) has the form of (2.23)) from the Hilbert space l2 � W 0 p;0 � = � gn = � vn un � : W 0 p;0gn; gn � l2 <1 � (2:190) into the space l2 (Wp;0) (2.19). It is obvious that the following subspaces D̂�(N;�) = � 0 P�l 2 N;�(E) � ; D̂+ � ~N; ~� � = P+l 2 ~N;~� � ~E � 0 ! are the prototypes of D�(N;�) and D+ � ~N; ~� � from HN;� (1.15) for the map- ping Pp;k (2.17) (for all p, k 2 Z+). P� and P+ are the orthoprojectors onto the subspaces in l2 N;�(E) and in l2~N;~� � ~E � formed by the solutions of the Cauchy prob- lems (2.1) and (2.9) with the initial data on the semiaxes Z� and Z+; respectively. Therefore the initial space H is isomorphic to the space Ĥp = l2 (Wp;0) P+l 2 ~N;~� � ~E � P�l 2 N;�(E) ! : (2:25) Similar constructions for l2 � W 0 p;0 � (2.190) lead to another space realization of the Hilbert space H Ĥ 0 p = l2 � W 0 p;0 � ~V �(�1;�p)P+l2~N;~� � ~E � V (1; p)P�l 2 N;�(E) ! (2:250) in view of Observation 2.2. It is natural that the spaces Ĥp (2.25) and Ĥ 0 p (2.25 0) are isomorphic one to another. As it is easy to see, the operator Rp : Ĥp ! Ĥ 0 p Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 439 V.A. Zolotarev de�ning this isomorphism has the form Rp = P Ĥ0 p � ~V �(1; p) 0 0 V (�1;�p) � P Ĥp ; (2:26) where P Ĥp and P Ĥ0 p are orthoprojectors onto Ĥp (2.25) and Ĥ 0 p (2.250) in cor- responding spaces. It follows from (2.20) and (2.24) that the operators T1 and T (1; p) = T1T p 2 , p 2 Z+ have the form� T̂1f � n = P Ĥp fn�(1;0); � T̂ (1; p)f � n = P Ĥp V̂ (1; p) (Rpf)n (2:27) for all fn 2 Ĥp (2.25), where P Ĥp is an orthoprojector onto Ĥp (2.25) and the operator Rp has the form (2.26). It is typical that the operator T̂1 has the same form (2.27) in all the spaces Ĥp (2.25) in view of Observation 2.1, and the operator T̂ (1; p) has this form (2.27) only in one speci�c space Ĥp (2.25). Theorem 2.4. Consider the simple [8] commutative unitary expansion Vs, + V s (2.1) corresponding to the commutative operator system fT1; T2g from the class C (T1) (1.3), and let the suppositions of Lem. 1.1 take place, besides dimE = dim ~E < 1. Then the isometric dilation U(1; p) (1.17), p 2 Z+, acting in the Hilbert space HN;� (1.15), is unitary equivalent to the operator Û(1; 0) (2.20) for p = 0 in l2 (Wp;0) (2.19), and to the operator Û(1; p) (2.24), for p 2 N, mapping the space l2 � W 0 p;0 � (2.19 0) into l2 (Wp;0) (2.19). Moreover, the operators T1 and T (1; p) = T1T p 2 (1.21) speci�ed in H are unitary equivalent to the shift operator T̂1 (2.27) in Ĥp (2.25) for all p 2 Z+ and to the operator T̂ (1; p) (2.27) acting in the speci�c space Ĥp (2.25) for p 2 N. IV. Let us now study a dual situation corresponding to the dilation + U (n) (1.25). Similarly to (2.1), continue every vector-function vk 2 l2 Z � ~E � into the domain Z2 using the Cauchy problem8< : @2vn = � ~N�@1 + ~�� vn; n = (n1; n2) 2 Z2; vnjn2=0 = vn1 2 l2Z � ~E � : (2:28) Besides, we have the recurrent way of the continuation v (n1; n2)! v (n1; n2 + 1) into the upper semiplane (n2 2 Z+), and when n2 2 Z�; the continuation v (n1; n2) ! v (n1; n2 � 1) has the nonexplicit nature and may be carried out in the context of suppositions of Lem. 1.1 when dim ~E < 1. Thus, we obtain the Hilbert space l2~N�;~�� ~E � assuming that kvnk = kvn1kl2 Z( ~E) . De�ne the shift operator ~V+(p) in the space l2~N�;~�� ~E � , ~V+(p)vn = vn+p (2:29) 440 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 Scattering Scheme with Many Parameters for all p 2 Z 2. It is obvious that the operator ~V+(q) (2.29) is isometric. Specify now the wave operator ~W+(p) mapping the space l2~N�;~�� ~E � into HN�;�� (1.24) by the following formula: ~W+(p) = s� lim n!1 + U (n; p)P D+( ~N�;~��) ~V+(�n;�p); (2:30) where the number p 2 Z+ is �xed and the operators + U (n) and ~V+(n) are speci�ed by the formulas (1.25) and (2.29), respectively. It is obvious that ~W+(0) = W+; where the operator W+ corresponds to the dilation U (1.5), and so the opera- tor ~W+(p) (2.30) is a continuation of the wave operator W+ onto the ��pth� horizontal line in ~Z2 � (1.7). Consider now the polygon L�1�1;p in ~Z2 � (1.7) formed by the vertical segment connecting points (�1,0) and (�1;�p) and by the horizontal semiline from the point (�1;�p) to (�1;�p); where p 2 Z+. And let ~L10;k be the similar polygon consisting of the rectilinear segments connecting the points (0,0), (0; k) and (1; k) one-by-one in Z2 +. Similarly to (2.4), de�ne the quadratic formsD ~f E2 �(p;k) = h� ~uni2L�1 �1;p + ~h 2 + h~�~vni2~L1 0;k ; D ~f E2 ~�(k) = k~unk2l2 + ~h 2 + h~�vni2~L1 0;k ; (2:31) hfi2 �(p) = h� ~uni2L�1 �1;p + ~h 2 + kvnk2l2 in HN�;�� (1.24), where ~f = � ~un; ~h; ~vn � 2 HN�;�� and respective ~� and � forms are understood in the sense of (1.29) and (1.30). It is easy to see that D ~f E2 �(0;0) =D ~f E2 ~�(0) = D ~f E2 �(0) = ~f 2 HN�;�� and D ~f E2 �(0;k) = D ~f E2 ~�(k) , hfi2 �(p;0) = hfi2 �(p) . As in (2.5), specify the quadratic � -forms, hvni2~�(p;k) = ~�v� n �2 L �1 �1;p + ~�v+ n �2 ~L1 0;k ; hvni2~�+(k) = v�n 2l2 + ~�v+n �2~L1 0;k ; (2:33) hvni2~� � (p) = ~�v� n �2 L �1 �1;p + v+ n 2 l2 ; in the space l2~N�;~�� ~E � , where v� n are the corresponding continuations in the second variable n2 from the semiaxes Z� of the functions of l2 Z � ~E � obtained by using the Cauchy problem (2.28). Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 441 V.A. Zolotarev The following statement, similar to Th. 2.1, is true. Theorem 2.5. The wave operator ~W+(p) (2.30) acting from the space l2~N�;~�� ~E � into the Hilbert space HN�;�� (1.24) exists for all p 2 Z+ and is an isometry D ~W+(p)vn E2 �(p;k) = hvni2~�(p;k) (2:34) in respective metrics (2.32) and (2.33) for all p 2 Z+. Moreover, for all ~W+(p) (2.30) the relations 1) + U (1; s) ~W+(p) = ~W+(p+ s) ~V+(1; s); 2) ~W+(p)PD+( ~N�;~��) = P D+( ~N�;~��) (2:35) are true for all p, s 2 Z+; where PD �( ~N�;~��) is an orthoprojector onto the subspace D+ � ~N�; ~�� . Select two relations that are an immediate corollary of 1) (2.35) and are similar to (2.8), 1) + U (1; 0) ~W+(p) = ~W+(p) ~V+(1; 0); 2) + U (1; p) ~W+(0) = ~W+(p) ~V+(1; p) (2:36) for all p 2 Z+. Continue now each vector-function un1 from the space l2 Z (E) by the second variable n2 into the domain Z2 using the Cauchy problem� @2un = (N�@1 + ��) un; n = (n1; n2) 2 Z2; unjn2=0 = un1 2 l2Z(E): (2:37) As in the case of the Cauchy problem (2.28), the continuation u (n1; n2) ! u (n1; n2 + 1) has the explicit recurrent nature, and the continuation into the lower semiplane n2 2 Z�, u (n1; n2) ! u (n1; n2 � 1) may be done under suppo- sitions of Lem. 1.1 and dimE < 1. The Hilbert space obtained in this way is denoted by l2 N�;��(E), besides kunk def = kun1kl2 Z (E). Similarly to the operator ~V+(p) (2.29), de�ne the shift operator V+(p)un = un+p (2:38) in the space l2 N�;��(E) for all p 2 Z2 and for all un 2 l2N�;��(E). Specify now the wave operator ~W�(k) from the space HN�;�� (1.24) into l2 N�;��(E) ~W�(k) = s� lim n!1 V+(�n;�k)PD � (N�;��) + U (n; k) (2:39) 442 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 Scattering Scheme with Many Parameters for all �xed k 2 Z+, where + U (n) and V+(n) are speci�ed by the formulas (1.25) and (2.38), respectively. It is easy to see that ~W�(0) = W � �, besides W� has the standard form [6]. Theorem 2.6. The wave operator ~W�(k) (2.39) mapping the space HN�;�� (1.24) into l2 N�;��(E) exists for all k 2 Z+ and has the following properties: 1) V+(1; s) ~W�(k + s) = ~W�(k) + U (1; s); 2) ~W�(k)PD � (N�;��) = PD � (N�;��) (2:40) for all k, s 2 Z+; where PD � (N�;��) is an orthoprojector onto D� (N�;��). Select two relations following from equality 1) (2.40): 1) V+(1; 0) ~W�(k) = ~W�(k) + U (1; 0); 2) V+(1; k) ~W�(k) = ~W�(0) + U (1; k) (2:41) for all k 2 Z+. Similarly to (2.13), de�ne now the scattering operator ~S(k; p) from l ~N�;~�� ~E � into the space l2 N�;��(E) ~S(k; p) = ~W�(k) ~W+(p) (2:42) for all k, p 2 Z+; that obviously coincides with S� when k = p = 0. Theorem 2.7. The scattering operator ~S(k; p) (2.42) is a bounded operator from l2~N�;~�� ~E � into the space l2 N�;��(E), besides the following relations 1) ~S(k; p) ~V (1; s) = V+(1; s) ~S(k + s; p� s); 2) ~S(k; p)P+l 2 ~N�;~�� ~E � � P+l 2 N�;��(E) (2:43) take place for all k, p, s 2 Z+, whereas 0 � s � p and P+ is an orthoprojector onto the respective subspaces corresponding to the solutions of the Cauchy problems (2.28) and (2.37) with the initial data on the semiaxis Z+ (n2 = 0). It is obvious that the invariant property of the operator ~S(k; p) with respect to the shift by the coordinate �n1� ~S(k; p) ~V+(1; 0) = V+(1; 0) ~S(k; p) (2:44) follows from 1) (2.43) for all p, k 2 Z+, and 1) ~S(k; p) ~V+(1; p) = V+(1; p) ~S(k + p; 0); p = s; 2) ~S(0; p) ~V+(1; p) = V+(1; p) ~S(p; 0); p = s; k = 0: (2:45) This fact is similar to equalities (2.16). Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 443 V.A. Zolotarev V. De�ne now the mapping ~Pp;k from the direct sum of the Hilbert spaces l2~N�;~�� ~E � + l2 N�;��(E) into the Hilbert space HN�;�� (1.24) in the following way: ~fp;k = ~Pp;k (gn) = ~Pp;k � vn un � = ~W+(p)vn + ~W � �(k)un; (2:46) where vn 2 l ~N�;~�� ~E � , un 2 l2 N�;��(E) for all p, k 2 Z+. As it was noted above (see Paragraph III), the vector-functions ~fp;k form the dense set in the space HN�;�� (1.24) in the case of simplicity of expansion Vs, + V s (1.1), for the �xed p, k 2 Z+. Therefore every vector from the space HN�;�� has di�erent realizations ~fp;k (2.46) for di�erent values of the parameters p and k. It is obvious thatD ~fp;k; ~fp;k E HN�;�� = D ~Wp;kgn; gn E l2 ; where the weight operator ~Wp;k is equal to ~Wp;k = � ~W � +(p) ~W+(p) ~S�(k; p) ~S(k; p) ~W�(k) ~W � �(k) � ; (2:47) besides ~S(k; p) has the form of (2.42). Similarly to Observation 2.1, it is obvious that all blocks of the operator ~Wp;k are translational invariant with respect to the shift by the variable �n1�. Thus, the mapping ~Pp;k (2.46) de�nes the one-to-one unitary correspondence between the space HN�;�� (1.24) and the space l2 � ~Wp;k � = � gn = � vn un � : D ~Wp;kgn; gn E l2 <1 � ; (2:48) where vn 2 l2~N�;~�� ~E � , un 2 l2 N�;��(E). It is easy to see that the given space l2 � ~Wp;k � coincides with l2 � I S S� I � ; when p = k = 0, as in the case of the space l2 (Wp;k) (2.19). It follows from the relations 1) (2.36) and 1) (2.41) and from the unitarity of the + U (1; 0) that the dilation + U (1; 0) acts in every space l2 � ~Wp;k � (2.48) by the shift by the variable �n1� Û+(1; 0)gn = � ~V+(1; 0) 0 0 V+(1; 0) � gn (2:49) for all gn 2 l2 � ~Wp;k � . 444 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 Scattering Scheme with Many Parameters Further, study how the dilation + U (1; s) (1.25) acts on the vectors ~fp;k = ~Pp;k (gn) (2.46). As in the considerations above, study only the case when the dilation + U (1; p) (1.25) acts on the vectors of the type ~f0;p = ~P0;p (gn) (2.46). Similarly to (2.22), it is easy to prove that� + U (1; p) ~f0;p; ~f 0 p;0 � �(p) = �� I ~S�(0; p) ~S(0; p) V+(1; p) ~W�(p) ~W � �(p)V � +(1; p) � � � ~V+(1; p) 0 0 V �+(�1;�p) � gn; g 0 n � ~� � (p);� � (p) ; (2:50) besides, the bi-linear form in the right part is understood component-wisely in the sense of the metrics ~��(p) and ��(p) (2.31). Let ~W 0 0;p = � I ~S�(0; p) ~S(0; p) V+(1; p) ~W�(p) ~W � �(p)V � +(1; p) � ; V̂+(1; p) = � ~V+(1; p) 0 0 V �+(�1;�p) � : (2:51) O b s e r v a t i o n 2.3. Consider the mapping ~P0;p (2.46), denote by ~f 00;p = ~P0;p (g 0 n) = ~W+(0) ~V+(�1;�p)vn + ~W � �V � +(1; p)un; where un 2 l2 N�;��(E), vn 2 l2~N;~� � ~E � . Then D ~f 00;p; ~f 00;p E HN�;�� = D ~W 0 0;pgn; gn E l2 follows from 2) (2.45), that is similar to Observation 2.2. Therefore the dilation + U (1; p) (1.25) acts as a shift operator Û+(1; p) = V̂ (1; p)gn (2:52) from the Hilbert space l2 � ~W 0 0;p � = � gn = � vn un � : D ~W 0 0;pgn; gn E l2 <1 � (2:480) (vn 2 l2~N�;~�� ~E � , un 2 l2N�;��(E)) into the space l2 � ~W0;p � (2.48). It is clear that the subspaces D̂� (N�;��) = � 0 P�l 2 N�;��(E) � ; D̂+ � ~N�; ~�� = P+l ~N�;~�� ~E � 0 ! ; Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 445 V.A. Zolotarev where, as usual, P� and P+ are orthoprojectors in l2 N�;��(E) and in l2~N�;~�� ~E � onto the subspaces of the solutions of the Cauchy problems (2.37) and (2.28) with the initial data on Z� and Z+; respectively, and are the prototypes of the subspaces D� (N�;��) and D+ � ~N�; ~�� from HN�;�� (1.24) for the mapping ~Pp;k (for all p, k 2 Z+). Therefore the space H is isomorphic to Ĥp;+ = l2 � ~W0;p � P+l ~N�;~�� ~E � P�lN�;��(E) ! : (2:53) Using similar considerations for l2 � ~W 0 0;p � (2.53), we obtain a di�erent realization Ĥ 0 p;+ = l2 � ~W 0 p;0 � ~V+(1; p)P+l 2 ~N�;~�� ~E � V �+(�1;�p)P�l2N�;��(E) ! (2:530) in view of Observation 2.3. The spaces Ĥp;+ (2.53) and Ĥ 0 p;+ (2.530) are isomor- phic, besides, the operator Rp;+: Ĥp;+ ! Ĥ 0 p;+ de�ning this isomorphism has the form Rp;+ = P Ĥ 0 p;0 � ~V+(�1;�p) 0 0 V+(1; p) � P Ĥp;+ ; (2:54) where P Ĥp;+ and P Ĥp;+ are orthoprojectors onto Ĥ 0 p;+ (2.530) and onto Ĥp;+ (2.53), respectively. It follows from (2.49) and (2.52) that the operators T �1 and T �(1; p) = T �1 T �p 2 , p 2 Z+, are� T̂ �1 f � n = P Ĥp;+ fn+(1;0); � T̂ �(1; p)f � n = P Ĥp;+ V̂+(1; p) (Rp;+f)n (2:54) for all fn 2 Ĥp;+ (2.53), P Ĥp;+ is an orthoprojector onto Ĥp;+ and the operator Rp;+ is speci�ed by formula (2.54). As in the previous case, the operator T̂ �1 has the same form (2.54) in all spaces Ĥp;+, and the operator T̂ �(1; p) has a given form (1.54) only in one space Ĥp;+ (2.53). Theorem 2.8. Let Vs, + V s (1.1) be the simple [8] commutative unitary expan- sion of the operator system fT1; T2g from the class C (T1) (1.3) and, moreover, the hypotheses of Lem. 1.1 be met, and dimE = dim ~E <1. Then the isometric dilation + U (1; p) (1.25), p 2 Z+, acting in the Hilbert space HN�;�� (1.24), is uni- tary equivalent to the operator Û+(1; 0) (2.49), for p = 0, in l2 � ~W0;p � (1.24) and to the operator Û+(1; p) (2.52), for p 2 N, mapping the space l2 � ~W 0 0;p � (2.48 0) into l2 � ~W0;p � (2.48). Moreover, the operators T �1 and T �(1; p) (1.21) acting in H are unitary equivalent to the shift operator T̂ �1 (2.54) in Ĥp;+ (2.53) for all p 2 Z+ and to the operator T̂ (1; p) (2.54) acting in the �xed Ĥp;+ (2.53) (p 2 N). 446 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 Scattering Scheme with Many Parameters References [1] M.S. Liv�sic, N. Kravitsky, A. Markus, and V. Vinnikov, Theory of Commuting Non- selfadjoint Operators. � Math. Appl. 332 Kluver Acad. Publ. Groups, Dordrecht, 1995. [2] V.A. Zolotarev, Time Cones and Functional Model on the Riemann Surface. � Mat. Sb. 181 (1990), No. 7, 965�995. (Russian) [3] B. Szekefalvi-Nagy and Ch. Foya�s, Harmony Analisys of Operators in the Hilbert Space. Mir, Moscow, 1970. (Russian) [4] V.A. Zolotarev, Model Representations of Commutative Systems of Linear Opera- tors. � Funkts. Anal. i yego Prilozen. (1988), No. 22, 66�68. (Russian) [5] M.S. Brodskiy, Unitary Operator Colligations and their Characteristic Functions. � Usp. Mat. Nauk 33 (1978), No. 4, 141�168. (Russian) [6] V.A. Zolotarev, Analitic Methods of Spectral Representations of Nonselfadjoint and Nonunitary Operators. MagPress, Kharkiv, 2003. (Russian) [7] V.A. Zolotarev, Isometric Expansions of Commutative Systems of Linear Operators. � Mat. �z., analiz, geom. 11 (2004), 282�301. (Russian) [8] V.A. Zolotarev, On Isometric Dilations of Commutative Systems of Linear Opera- tors. � J. Math. Phys., Anal., Geom. 1 (2005), 1922�208. Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 447

Scattering Scheme with Many Parameters and Translational Models of Commutative Operator Systems

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