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Solution of the Boussinesq equation using evolutionary vessels 3 1 Andrey Melnikov 0 Drexel University 2 n January 14, 2013 a J 1 Abstract 1 Inthisworkwepresentasolution oftheBoussinesq equation. Thederivedformulasincludesolitons, ] Schwartz class solutions and solutions, possessing singularities on a closed set Z of R2 ((x,t) domain), P obtained from the zeros of the tau function. The idea for solving the Boussinesq equation is identi- A cal to the (unified) idea of solving the KdV and the evolutionary NLS equations: we use a theory of . h evolutionary vessels. But a more powerful theory of non-symmetric evolutionary vessels is presented, t insertingflexibilityintotheconstructionandallowingtodealwithcomplex-valuedsolutions. Apowerful a scattering theory of Deift-Tomei-Trubowitz for a three dimensional operator, which is used to solve the m Boussinesqequation,fitsintooursettingonlyinaparticularcase. Ontheotherhand,wecreateamuch [ wider class of solutions of theBoussinesq equation with singularities on a closed set Z. 1 v Contents 3 7 5 1 Introduction 2 2 . 2 Scattering theory of the operator L 4 1 2.1 Non-symmetric regular vessel, realizing a scattering theory of L . . . . . . . . . . . . . . . 4 0 2.2 Structureof the moment H (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 0 1 3 Vessels with unbounded operators. Standard construction of a vessel 10 : v i 4 “Uniqueness” of the scattering data 12 X r 5 Choice of the parameters realizing the Boussinesq equation (1) 13 a 5.1 Realizing theBoussinesq equation bya Boussinesq vessel . . . . . . . . . . . . . . . . . . 13 5.2 Standard construction of a Boussinesq vessel . . . . . . . . . . . . . . . . . . . . . . . . . 15 6 Examples of solutions of the Boussinesq equation (1) 15 6.1 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.2 Solutions, belonging tothe Schwartzclass. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.3 General solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1 1 Introduction TheBoussinesq equation [Bou72]1 ∂2 q = [3q 12q2] (1) tt ∂x2 xx− is a foundation of a shallow water theory. A fundamental solution of this equation was presented by P. Deift,C.TomeiandE.Trubowitzin[DTT82]. Thebasicideainthiswork,presentedbyV.E.Zakharov [Zac74], is a developmentof a scattering theory of a three dimensional operator d3 1 d d L=i + (q + q)+p dx3 i dx dx e d2 and using thefact that for Q=i(3 4q), theoperators L,Q constitutea Lax pair: dx2 − e d L=[Q,L]=QL LQ. dt − e e e e It is assumed in [DTT82] that p(x),q(x) are in a Schwartz space, but the theory goes through for q(x),p(x)withonlyafinitenumberofderivativesandafiniteorderofdecay. Thescatteringdatainthis caseconsistsofalistof6functions,whoseevolvingwithtunderananalogueofQisstudiedproducinga solutionoftheBoussinesqequation(1)withagiveninitialvalueq(x,0)=q(x). Sincewewilluseinstead of L its multiplication on i, we define − e d3 d L= iL= 2q (q +ip) (2) − dx3 − dx − x e and a very particular (inverse) scattering of this operator will be researched. More precisely, we will discuss solutions of 2 d3 d Lu= 2q (q′+ip)=k3u, p′(x)=P( q,q,q′,...,q(n)). (3) dx3 − dx − Z Inthissettingq(x)isan“arbitrary”function,but p(x) isderived from q(x) (see formula (38)) usingthe formulap′(x)=P( q,q,q′,...,q(n))forapolynomialP,whichisnotderivedexplicitly,sincewedonot useit. R In order to solve (1), we present a similar to [DTT82] scheme, where we use a special case of the inversescatteringtheory: wecreateinversescatteringof (2),wherep(x)isnotarbitraryandthenevolve q(x) with t. Let us explain first the (inverse) scattering theory. The scattering data is encoded in our setting in a matrix-valued function. In fact, there is an almost “one-to-one” correspondence (Theorem 14)betweenthecoefficientsq(x),definingL(2)and3 3matrix-functionsS(λ)possessing arealization × [BGR90] 0 0 1 S(λ)=I−C0X−01(λI−A)−1B0σ1, σ1 = 0 1 0 , AX0+X0Aζ+B0σ1C0 =0. 1 0 0   ∂2 1ThefullBoussinesqequationqtt= ∂x2[q+qxx−4q2]wasshownbyMcKean[McK78]tobeequivalentto(1),significantly simplifyingalgebracomputations. ∂ ∂ 2we will usually denote by q′ the partial derivative ∂xq, and by q˙ the partial derivative ∂tq. Similarly, for the higher ∂2 derivatives: q′′ standsfor ∂x2q,etc. 2 HereforanauxiliaryHilbertspace thelinearoperatorsactasfollows: C : C3,A ,X ,A: , 0 ζ 0 B :C3 . LetusassumeforthHesimplicityofthepresentation,thatallthHeo→peratorsarebounHde→d.HIn 0 →H 1 0 0 ordertoconstructq(x),uniquelydefinedfromS(λ),usethefollowing 4steps,fixingσ2= 0 0 0 , 0 0 0   0 0 0 γ = 0 0 1 : 0 1 0  −  1. solve for B(x) from B′σ1 = ABσ2 Bγ (5) with initial B(x0)=B0, − − 2. solve for C(x) from σ1C′ = σ2CAζ+Cγ (6) with initial C(x0)=C0, − 3. solve for X(x) from X′(x)=B(x)σ2C(x) (7), X(x0)=X0, 4. define γ (x) = γ+σ2C(x)X−1(x)B(x)σ1 σ1C(x)X−1(x)B(x)σ2 (8), for all points where X(x) is ∗ − invertible. Then we provein Theorem 4 that thefunction S(λ,x)=I C(x)X−1(x)(λI A)−1B(x)σ1 − − d3 is a Bäcklund transformation for the operator L from the trivial L = to a more complicated one 0 dx3 3 d2 L (2), in which q(x)= −2dx2 lndet X−01X(x) and p(x) is defined from q(x) up to a constant. In the regular case, explained here thecoeffi(cid:0)cients q(x(cid:1)),p(x) are analytic functions at all points, where X(x) is invertible. LettingtheoperatorsC,B,Xfurtherevolvewithrespecttot,wewillobtainasolutionoftheBoussi- nesq equation (1). In order to show it, we take threeadditional matrices (48) 0 i 0 0 0 0 − σ1 =σ1, σ2 = i 0 0 , γ = 0 0 0  0 0 0 0 0 i e e   e   and follow thesesteps (last step is the same as the previousfourth step): 4˙. solve for B(x,t) from B˙σ = ABσ Bγ (50) (see footnote 2) with initial B(x,t )=B(x), 1 2 0 − − 5˙. solve for C(x,t) from σ C˙ = σ CA +Cγ (51) with initial C(x,t )=C(x), 1e − 2 e ζ e 0 6˙. solve for X(x,t) from X˙(x)=B(x,t)σ C(x,t) (7), X(x,t )=X(x), e e 2 e 0 7.=4. wdehfienreeXγ∗((xx,,t)t)is=inγve+rtiσb2leC.(x,t)X−1(xe,t)B(x,t)σ1 −σ1C(x,t)X−1(x,t)B(x,t)σ2 (8), for all points 3 ∂2 Itturnsout that q(x,t)=−2∂x2 lndet X−01X(x,t) (37)and satisfies theBoussinesq equation (1) (see Theorem 17 for details). Construction o(cid:0)f the coeffic(cid:1)ient q(x) from a realized function S(λ) is called the standard construction of a vessel and is presented in Section 3. The simplicity and richness of this construction is best revealed in soliton formulas (Section 6.1). By choosing the inner space = C - the one dimensional Hilbert space, we create a classical soliton H 9µ2 18e2√3µ(x+2tµ)µ2 q(x,t) = (58) and another one q(x,t) = (59). Here µ is − √3 −(e2√3xµ+e4√3tµ2)2 2cosh2( µ(x+tµ)) 2 an arbitrary complex parameter. This construction seems to be more concrete and suitable for the solution of the Boussinesq equation (1), because it uses just “enough” of the very powerful and complicated inverse scattering theory, 3 developed in [DTT82]. Notice that in this later work the coefficients q(x),p(x) are quite arbitrary. In our work, on the other hand, the coefficient p(x) is uniquely determined (up to a constant) from q(x). Still,theformulasenabletoproducesolutions, applyingthestandardconstruction ofavessel, explained above, to different S(λ). Moreover, at the same section 3 we show that much more general classes of solutionsarise,asweimposeasfewaspossiblerestrictionsonS(λ). Thesolutions,presentedin[DTT82] are either in the Schwartz class or exponentially decaying and correspond in our setting to an analytic S(λ),possessing jumpsalong thereal negative axis. Finally, this work presents a very general setting for a construction of solutions of (1). We find necessaryregularityassumptions(61)ontheoperatorB(x,t)suchthatwecancreateavessel,andhence asolution of (1). This is proved in Theorem 19. Following theremark after Theorem 19, one can easily construction a solution, which fails to be five times x-differentiable (four times are necessary for the existenceof (1)). 2 Scattering theory of the operator L Westart from thedefinitionofthevesselparameters, whichcreateaninversescatteringtheoryofL(2). Definition 1. The vessel parameters are defined as follows 0 0 1 1 0 0 0 0 0 σ1 = 0 1 0 ,σ2= 0 0 0 ,γ = 0 0 1 . 1 0 0 0 0 0 0 1 0      −  2.1 Non-symmetric regular vessel, realizing a scattering theory of L Definition 2. A (regular, non-symmetric) vessel, associated to vessel parameters (see Definition 1) isa collection of operators, spaces and an interval I V =(C(x),A ,X(x),A,B(x);σ ,σ ,γ,γ (x); ,C3;I), (4) reg ζ 1 2 ∗ H where the bounded operators C(x): C3, A ,X(x),A: , B(x):C3 and a 3 3 matrix ζ H→ H →H →H × function γ (x) satisfy the following vessel conditions: ∗ ∂ ∂xB =−(ABσ2+Bγ)σ1−1, (5) ∂ ∂xC =σ1−1(γC−σ2CAζ), (6) ∂ X =Bσ C, (7) ∂x 2 γ =γ+σ2CX−1Bσ1 σ1CX−1Bσ2, (8) ∗ − AX+XA = Bσ C. (9) ζ 1 − The operator X(x) is assumed to be invertible on the interval I. If Aζ =A∗ and C =B∗ we call such a vessel symmetric. Remarks: 1. Notice that the operators C(x),X(x),B(x) are globally defined for all x R: C(x),B(x) aresolutionsofoperator-valueddifferentialequationswithconstantcoefficientsandX(x)∈isobtainedfrom thembyasimpleintegration. 2. Forthedefinitionofthematrix-functionγ (x)weneedtheinvertability of the operator X(x), so in general we may suppose that γ (x) is defined fo∗r all x R, except for those points where X(x) is not invertible. For simplicity, we tak∗e an interval I and in a∈more general setting we consider γ (x)on R, except fro thepoints where X(x) is not invertible. ∗ 4 Theorem3(permanencyoftheLyapunovequation). SupposethatB(x),C(x),X(x)satisfy (5),(6)and (7) respectively, then if the Lyapunov equation (9) AX(x)+X(x)A +B(x)σ C(x)=0 ζ 1 holds for a fixed x I, then it holds for all x I. 0 ∈ ∈ Proof: By differentiating theleft hand side of (9), we will obtain that it is zero. By the definition,thetransfer function of thevessel V is definedas follows: reg S(λ,x)=I C(x)X−1(x)(λI A)−1B(x)σ1. (10) − − NoticethatpolesandsingularitiesofSwithrespecttoλaredeterminedbyAonly. Wewouldliketoshow that the function S(λ,x) realizes a Bäcklund transformation of the corresponding LDEs: multiplication bythefunctionS(λ,x)maps[Ls01,AMV12,Mel11]asolution oftheinputLinearDifferentialEquation (LDE) with thespectral parameter λ ∂ λσ u(λ,x) σ u(λ,x)+γu(λ,x)=0. (11) 2 − 1∂x toa solution of the outputLDE with thesame spectral parameter ∂ λσ y(λ,x) σ y(λ,x)+γ (x)y(λ,x)=0. (12) 2 − 1∂x ∗ The function γ is defined by the Linkage condition (8). The fundamental solutions of (11) and (12), ∗ which are equal to I (theidentity matrix matrix at x=0) are denoted usually by Φ(λ,x) and Φ (λ,x). ∗ Inother words, these are matrix functions satisfying: ∂ λσ Φ(λ,x) σ Φ(λ,x)+γΦ(λ,x)=0, Φ(λ,0)=I, (13) 2 − 1∂x ∂ λσ Φ (λ,x) σ Φ (λ,x)+γ (x)Φ (λ,x)=0, Φ (λ,0)=I. (14) 2 ∗ − 1∂x ∗ ∗ ∗ ∗ Theorem 4 (Vessel=Bäcklund transformation [Ls01, AMV12, Mel11]). Suppose that u(λ,x) satisfies (11), then y(λ,x)=S(λ,x)u(λ,x) satisfies (12). Remark: Notice that it is enough to provethat the transfer function satisfies thefollowing ODE: ∂ ∂xS(λ,x)=σ1−1(σ2λ+γ∗(x))S(λ,x)−S(λ,x)σ1−1(σ2λ+γ). (15) Proof: First we calculate d dx C(x)X−1(x) =σ1−1(γC−σ2CAζ)X−1−CX−1Bσ2CX−1 = by (9) (cid:0) (cid:1) ===σσσ111−−−111σγσ2C2CCXXX−−−111+AAσ++1−σσ1σ11−−211Cγ(γ∗XC+−X1σA−21C+.Xσ−1−11Bσσ2C1−X−σ11BCσX1−C1XB−σ12)−CCXX−−11=Bσb2yC(X8−)1 Sowe obtain that d dx CX−1 =σ1−1σ2CX−1A+σ1−1γ∗CX−1. (16) (cid:0) (cid:1) 5 Let us differentiate nextthe transfer function using (10): d d d dxS(λ,x) =−dx CX−1 (λI−A)−1Bσ1−CX−1(λI−A)−1dxBσ1=by (5), (16) ==σ(cid:0)σ1−1−1γ1(cid:0)σ∗(2SC−X−I(cid:1)1)A−+(Sσ1−−1Iγ)∗σC1−X1−γ1+(cid:1)(σλ1−I1−σ2AC)X−1−B1Aσ1(λ−IC−XA−)1−(1λBIσ−1A−)C−1X(−A1B(λσI2−+AB)γ−)1ABσ2 =insert A=A λI and expand =σ1−+1γC∗X(S−1−BIσ)2±−(λSC−XI−)1σ(1−λI1γ+Aσ)1−−11Bσ2σC2X−1Bσ1−λσ1−1σ2CX−1(λI−A)−1Bσ1− − − =using (8), (10) ==σσ11−−11(γλ∗σ(S2+−γI∗))−S(−SS−σI1−)1σ(1−λσ1γ2++γσ)1−.1(γ∗−γ)+λσ1−1σ2S−Sσ1−1λσ2 Expanding the transfer function S(λ,x) into a Taylour series around λ = , we obtain a notion of ∞ themoment: S(λ,x)=I ∞ Hn(x)σ1, − λn+1 nX=0 where bythe definition the n-th moment H (x) of thefunction S(λ,x) is n Hn(x)=C(x)X−1(x)AnB(x). (17) Usingthezeromoment,forexample,weobtainthatthesocalled“linkagecondition”(8)isequivalentto γ (x)=γ+σ H (x)σ σ H (x)σ . 2 0 1 1 0 2 ∗ − There is also a recurrent relation between the moments H (x),arising from (15): n Theorem 5. The following recurrent relation between the moments of the vessel V holds reg σ1−1σ2Hn+1−Hn+1σ2σ1−1 =(Hn)′x−σ1−1γ∗Hn+Hnγσ1−1. (18) H (x)σ Proof: Follows from thedifferential equation (15) by plugging S(λ,x)=I− ∞n=0 λnn+1 1. uP 1 Let us investigate more carefully theLDEs (11) and (12). Denoteu= u2 , then (11) becomes u  3  λu1 u′3 0 0  0 − u′2 + u3 = 0 .  0   u′1   −u2   0  Solvingthis we obtain that u2 =−u′1,  uu3′1′′==u−′2λ=u1−.u′1′, (19) Wecanseethatactuallythisequationisequivalenttoathird-orderdifferentialequationwiththespectral parameter λ: u′1′′ =−λu1. (20) 6 π π π 11 12 13 In order to analyze (12), we denote first moment H0(x)=[πij]= π21 π22 π23 , and as a result, π π π  31 32 33  thelinkage condition (8) becomes π π π π 13 31 12 11 − γ =γ+ π21 0 0 . ∗ − π 0 0  − 11  y 1 Denotenext y= y2  and plugging theexpression for γ just derived into (12), we will obtain that ∗ y  3  λy1 y3′ (π13−π31)y1+π12y2+π11y3 0  0 − y2′ + −π21y1+y3 = 0   0   y1′   −π11y1−y2   0  or solving this: y2 =−π11y1−y1′,  yy13′′′=−π22q1yy1′1−+(yq2′′+=pπ)2y11y1=−−yλ1yπ11′,1−π11y1′ −y1′′, (21) where  q(x)= π121+π12+2π21−2π1′1, p(x)=−i(−π13+π31+π11(π12−π21)− π1′2−2 π2′1). (22) Inother words theequation (12) is equivalent to y1′′′−2qy1′ −(q′+ip)y1=−λy1. (23) Nowwe are ready to justify theterm “scattering matrix” attached to S(λ,0). Anindependent set y (λ,x) 1 y(λ,x)= y2(λ,x)  y (λ,x)  3  of solutions of (2), can be derived from (12) in thefollowing form 1 1 y(λ,x)=S(λ,x)Φ(λ,x) 0 =Φ (λ,x)S(λ,0) 0 . ∗ 0 0     ThefundamentalmatrixΦ(λ,x),solving(11)withtheinitialconditionΦ(λ,0)=I (-theidentitymatrix) is obtained from (19) and can be explicitly written as R R Φ(λ,x)= 3α12  kRR13 Rk21 −−kR2k32  (24)  k2R kR R   − 2 − 3 1  R =α2(E +E +E ), 1 1 2 3 R =α2E +αE +E , 2 1 2 3 R =α2E +E +αE 3 1 2 3 whereE1 =e−kx,E2 =e−αkx,E3=e−α2kx forα=e2πi/3 (α3 =1). Thismatrixisanalyticinλ,because examiningTaylorseries of E we will cometotheconclusion thatall theentries ofΦdependon k3 =λ. i ThestructureofS(λ,x)isalsoknownfrom(10),sowecanstudysolutionsof (2)orequivalentlyof (12), creating in thismanner the(inverse) scattering of L (2). 7 2.2 Structure of the moment H (x) 0 Let us examine the recurrence relation (18). We will research for the simplicity of the presentation the structureof thefirst moment H (x),butalmost thesame structurewill actually apply for all moments. 0 Let us denote g g g 11 12 13 H1(x)= g21 g22 g23 . g g g  31 32 33  Then theleft hand side of (18) becomes 0 0 g 11 σ1−1σ2H1(x)−H1(x)σ2σ1−1= 0 0 −−g21  g g g g  11 12 13− 31  which must beequal to (H0)′x−σπ1−1211γ+∗Hπ102++Hπ02γ1σ+1−π11′=1 π11π12−π13+π22+π1′2 π11π13+π23+π1′3 = π11π21−π31+π22+π2′1 π12π21−π23−π32+π2′2 Res(23) ,  −π11π13−π12π21+π32+π3′1 Res(32) Res(33)  where Res(23)=π13π21−π33+π2′3, Res(32)=−π12(π13−π31+π22)−π11π32−π33+π3′2, Res(33)=−π12π23+π13(−π13+π31)−π11π33+π3′3. Equatingwe obtain that g11 =−π11π13−π23−π1′3, g21 =−Res(23), g12 =Res(32), g13−g31 =Res(33) (25) and thefollowing system of equations: π121+π12+π21+π1′1 =0,  πππ111112πππ122211−−−ππ(π133123+++πππ222232++)+ππ12′′π212′==2 =00,,0,  −π11π13−π23−π1′3 =−π11π13−π12π21+π32+π3′1(=g11). or rearranging π21 =−(π12+π121+π1′1),  πππ322222 ===π−−1ππ211π112ππ121−12++π2ππ331+13−−π2′ππ221′′,12,, (26) Pluggingthefourthequationof (26)iπn3t2o=thπe1l2aπs2t1o−neπ,2w3e−wπi3l′l1o−btπa1′i3n.thatthelaterbecomesπ3′1+π1′3 = −π2′2 or requiring a normalization π (x )+π (x )= π (x ) (27) 31 0 13 0 22 0 − we obtain π +π = π . (28) 31 13 22 − 8 In a similar manner, one can solve for some other entires and we summarize these intermediate results in the next lemma. Additional relations between the entries of H (x) are obtained if we consider (18) 0 for n=1. Forexample, formulas similar to (26) are as follows: π11g11+g12+g21+g1′1 =0,  πgg111121ππg122211−−−gg(g133123+++ggg222232++)+gg12′′g212′==2 =00,,0, From these formul−asπf1o1lglo13w−thge23fo−lllgo1′w3i=ng−rπel1a1tgi3o1n+s g11(π31−π13)−π12g21+g32+g3′1. π11g11+g12+g21+g1′1 =0, (29) (cid:26) π11g12−π11g21−(g13−g31)+g1′2−g2′1 =0, whichcan berewritten as relations on theentriesof H (x)in viewof (25). Otherthreerelations, which 0 are obtained are π11g12−g13+g22+g1′2+g11π21−g31+g22+g2′1 =0,  g−1π2π112g11−3−(gg2233+−gg321′3)+=g−2′2π1=1g03,1+g11(π31−π13)−π12g21+g32+g3′1,  whichservetofindg2′2,g23+g32,g1′3+g3′1. Itturnsoutthatactuallytherelations(25)arenotindependent andaformulaforπ12π1′1 isderivedfromthem. AlltheseresultsaresummarizedinthenextLemmaand we notice that theexact formulas for π2′3,π3′3 are omitted, because we are not interested in their form: Lemma 6. The following relations between the entries of the first moment H (x) hold 0 π21 =−(π12+π121+π1′1), (30) π22 =−π11π12+π13−π1′2, (31) π31 =π13−π131−π11(2π12+3π1′1)−2π1′2−π1′′1, (32) 3 1 π32 =π12π21−π23−π12π1′1− 2((π1′1)2−π11π1′′1)+ 2π1′′1′, (33) 3 3 1 π1′3 =−2(π1′1)2+π11π1′2+ 2π11π1′′1+π1′′2+ 2π1′′1′, (34) 6π12π1′1 =−(6π11+15π1′1)π1′1+3π11π1′′1+π1′′1′. (35) The relations (29) result in formulas for π2′3,π3′3. Proof: Notice that we haveobtained 5 relations on theentries of H (x)in (26), 2 relations in (29) and 0 onemore relation from thefact that (25) together with (29) are overdetermined. Itis possible to derive theseformulas using any symbolic computation software. It would be interesting to derive formulas for the entries of H (x), relying on the structure of the 0 matrices σ ,σ ,γ only, without referring to tedious direct computations. It can be probably done using 1 2 thefollowing notion, appearing first in [Mel11]: Definition 7. Tau-function of the vessel V is defined as follows: τ(x)=det(X−1(x0)X(x)), (36) where x I is an arbitrary point. 0 ∈ The fact that this object is well-defined follows from an equivalentto (7) equation x X(x)=X + B(y)σ C(y)dy, 0 2 Z 0 so that X−01X(x)=I+T(x)for a trace-class operator T. Using this notion we obtain thefollowing 9 Theorem 8. The coefficient q(x) possesses the following formula: 3 d2 3 d q(x) = lnτ(x)= π . (37) −2dx2 −2dx 11 The derivative of the coefficient p(x) is a differential polynomial in π : 11 p′(x)=P(π11,π1′1,...), P - a polynomial. (38) Proof: Inserting (30) into theformula (22) for q(x) we find that q(x)= π121+π12+2π21−2π1′1 =−23π1′1. Usinga formula for thedeterminant of an operator, we obtain that (see [GK69, AMV12] for details) ττ′((xx)) =tr(X′(x)X−1(x))=tr(B(x)σ2B∗(x)X−1(x))=tr(σ2H0(x))=π11 and theresult follows for q(x). As for p(x),we differentiate theformula for p(x),appearing in (22): p′(x)=−i(−π1′3+π3′1)−iddx(π11(π12−π21)− π1′2−2 π2′1). Then using (32), (34), (30) and (35) we will obtain a differential polynomial in π . 11 One can also derive the following formulas, corresponding to the symmetric case C =B∗, Aζ =A∗, plugging thedefinition (36) and using thefact that H (x) is self-adjoint: 0 Lemma 9. For the symmetric case, C =B∗, Aζ =A∗, the followingrelations between the entries ofthe first moment H0(x)=H0∗(x) hold π = 1τ′′, π = 1τ′′′, π = 1τ′′′, ℜ 12 −2 τ 22 3 τ ℜ 13 −6 τ ℑπ13 =π11ℑπ12+ℑπ1′2, π2 2 1 1τ(4) ℜπ23 = ℑ212 + 9(q(x)2− 4q′′(x))+ 8 τ 3 Vessels with unbounded operators. Standard construction of a vessel TheideaspresentedinthisSectioncanbefoundin[Mela]forthesymmetriccase. Theclassoffunctions serving as “initial conditions” for the transfer functions of vessels is defined as follows Definition 10. Class R(σ ) consist of p p matrix-valued functions S(λ) of the complex variable λ, 1 × possessing the following representation: S(λ)=I−C0X−01(λI−A)−1B0σ1 (39) where for an auxiliary Hilbert space there are defined operators C : C3, A ,X ,A : , 0 ζ 0 B :C3 . Ageneral matrix-functiHonS(λ),representable insuchaformHi→scalledrealized. MHore→oveHr, 0 →H the operators are subject to the following assumptions: 1. the operators A,A have dense domains D(A),D(A ). A,A are generators of C semi-groups on ζ ζ ζ 0 . Denote the resolvents as follows R(λ)=(λI A)−1, Rζ(λ)=(λI+Aζ)−1, H − 2. the operator B satisfies R(λ)B e for all λ spec(A),e C3, 0 0 ∈H 6∈ ∈ 10