GBDT of discrete skew-selfadjoint Dirac systems and explicit solutions of the corresponding non-stationary problems
aa r X i v : . [ m a t h . C A ] D ec GBDT of discrete skew-selfadjoint Diracsystems and explicit solutions of thecorresponding non-stationary problems
Alexander Sakhnovich
Dedicated to Rien Kaashoek on the occasion of his 80thanniversary
Abstract
Generalized B¨acklund-Darboux transformations (GBDTs) of dis-crete skew-selfadjoint Dirac systems have been successfully used forexplicit solving of direct and inverse problems of Weyl-Titchmarshtheory. During explicit solving of direct and inverse problems, weconsidered GBDT of the trivial initial systems. However, GBDT ofarbitrary discrete skew-selfadjoint Dirac systems is important as welland we introduce these transformations in the present paper. Theobtained results are applied to the construction of explicit solutionsof the interesting related non-stationary systems.
MSC(2010): 34A05, 39A06, 39A12.Keywords:
Discrete skew-selfadjoint Dirac system, generalized B¨acklund-Darboux transformation, fundamental solution, non-stationary system, ex-plicit solution.
We consider a discrete skew-selfadjoint Dirac system y k +1 ( z ) = (cid:18) I m + i z C k (cid:19) y k ( z ) , C k = U ∗ k jU k ( k ∈ I ) , (1.1)1here I m is the m × m identity matrix, U k are m × m unitary matrices, j = (cid:20) I m − I m (cid:21) ( m , m ∈ N , m + m = m ) , (1.2)and I is either N or the set { k ∈ N : 0 ≤ k < N < ∞} ( N ∈ N ). Here, asusual, N denotes the set of natural numbers and N = 0 ∪ N .The relations C k = C ∗ k , C k = I m (1.3)are immediate from the second equality in (1.1).This paper is a certain prolongation of the papers [3, 10], where directand inverse problems for Dirac systems (1.1) have been solved explicitlyand explicit solutions of the isotropic Heisenberg magnet model have beenconstructed. We would like to mention also the earlier papers on the casesof the continuous Dirac systems (see, e.g., [8, 9]).The GBDT version of the B¨acklund-Darboux transformations have beenused in [3, 8–10]. B¨acklund-Darboux transformations and related commuta-tion methods (see, e.g., [1, 2, 6, 7, 11, 14, 17] and numerous references therein)are well-known tools in the spectral theory and in the construction of explicitsolutions. In particular, the generalized B¨acklund-Darboux transformations(i.e., the GBDT version of the B¨acklund-Darboux transformations) were in-troduced in [12] and developed further in a series of papers (see [14] fordetails).Whereas GBDTs of the trivial initial systems have been used in [3, 8–10](in particular, initial systems (1.1) where C k ≡ j have been consideredin [3, 10]), the case of an arbitrary initial discrete Dirac system (1.1) is con-sidered here. In Section 2, we introduce GBDT, construct the so calledDarboux matrix and give representation of the fundamental solution of thetransformed system (see Theorem 2.2). Note that an explicit representationof the fundamental solutions of the transformed systems in terms of the so-lutions of the initial systems is one of the main features and advantages ofthe Darboux transformations.One of the recent developments of the GBDT theory is connected with itsapplication to the construction of explicit solutions of important dynamical2ystems (see, e.g., [5, 13]). In Section 3 of this article, we use the same ap-proach in order to construct explicit solutions of the non-stationary systemscorresponding to the systems (1.1).In the paper, C stands for the complex plane and C + stands for the openupper halfplane. The notation σ ( α ) stands for the spectrum of the matrix α and the notation diag { d , d , . . . } stands for the block diagonal matrix withthe blocks d , d , . . . on the main diagonal. Each GBDT of the initial system (1.1) is determined by some triple { α, S , Λ } of the n × n matrices α and S = S ∗ and the n × m matrix Λ ( n ∈ N ) suchthat αS − S α ∗ = iΛ Λ ∗ . (2.1)The initial skew-selfadjoint Dirac system has the form (1.1) and the trans-formed (i.e., GBDT-transformed) system has the form e y k +1 ( z ) = (cid:18) I m + i z e C k (cid:19) e y k ( z ) ( k ∈ I ) , (2.2)where the potential { e C k } ( k ∈ I ) is given by the relationsΛ k +1 = Λ k + i α − Λ k C k , (2.3) S k +1 = S k + α − S k ( α ∗ ) − + α − Λ k C k Λ ∗ k ( α ∗ ) − , (2.4) e C k = C k + Λ ∗ k S − k Λ k − Λ ∗ k +1 S − k +1 Λ k +1 , k ∈ I . (2.5)Here and further in the text we assume thatdet α = 0 , (2.6)and suppose additionally in (2.5) thatdet S k = 0 (2.7)3or k ∈ N or for 0 ≤ k ≤ N depending on the choice of the interval I , onwhich the Dirac system is considered.Similar to the proof of [3, (3.7)], using the equality C k = I m from (1.3)and relations (2.1)–(2.4) one easily proves by induction that αS k − S k α ∗ = iΛ k Λ ∗ k . (2.8) Remark 2.1
Clearly, S k = S ∗ k and e C k = e C ∗ k . Further in the text, in Propo-sition 2.3 we show that e C k = I m . In Theorem 2.5, we show that underconditions S > and , i σ ( α ) ( σ ( α ) is the spectrum of α ) we have S k > and e C k = e U ∗ k j e U k , (2.9) where j is given in (1.2) and the matrices e U k are unitary. The equality (2.9) means that the transformed system (2.2) is again a skew-selfadjoint Diracsystem in the sense of the definition (1.1) . Before Theorem 2.5 we considerthe GBDT-transformed system (2.2) without the requirement (2.9) . The fundamental solutions of (1.1) and (2.2) are denoted by w ( k, z ) and e w ( k, z ), respectively, and are normalized by the conditions e w (0 , z ) = w (0 , z ) = I m . (2.10)In other words, y k = w ( k, z ) and e y k = e w ( k, z ) are m × m matrix solutionsof the initial and transformed systems, respectively, which satisfy the ini-tial conditions (2.10). The so called Darboux matrix corresponding to thetransformation of the system (1.1) into (2.2) is given by the transfer matrixfunction w α in Lev Sakhnovich form: w α ( k, z ) = I m − iΛ ∗ k S − k ( α − zI n ) − Λ k . (2.11)See [15, 16] as well as [14] and further references therein for the notion andproperties of this transfer matrix function. The statement that the Darbouxmatrix has the form (2.11) may be formulated as the following theorem.4 heorem 2.2 Let the initial Dirac system (1.1) and a triple { α, S , Λ } ,which satisfies the relations (2.1) , (2.6) , (2.7) and S = S ∗ , be given. Then,the fundamental solution w of the initial system and fundamental solution e w of the transformed system (2.2) ( determined by the triple { α, S , Λ } viarelations (2.3) – (2.5)) satisfy the equality e w ( k, z ) = w α ( k, − z ) w ( k, z ) w α (0 , − z ) − ( k ≥ , (2.12) where w α has the form (2.11) . P r o o f. The following equality is crucial for our proof w α ( k + 1 , z ) (cid:18) I m − i z C k (cid:19) = (cid:18) I m − i z e C k (cid:19) w α ( k, z ) . (2.13)(It is easy to see that the important formula [3, (3.16] is a particular caseof (2.13).) In order to prove (2.13), note that according to (2.11) formula(2.13) is equivalent to the formula1 z (cid:16) e C k − C k (cid:17) = (cid:18) I m − i z e C k (cid:19) Λ ∗ k S − k ( zI n − α ) − Λ k − Λ ∗ k +1 S − k +1 ( zI n − α ) − Λ k +1 (cid:18) I m − i z C k (cid:19) . (2.14)Using the Taylor expansion of ( zI n − α ) − at infinity we see that (2.14) is inturn equivalent to the set of equalities: e C k − C k = Λ ∗ k S − k Λ k − Λ ∗ k +1 S − k +1 Λ k +1 , (2.15)Λ ∗ k +1 S − k +1 α p Λ k +1 − iΛ ∗ k +1 S − k +1 α p − Λ k +1 C k = Λ ∗ k S − k α p Λ k − i e C k Λ ∗ k S − k α p − Λ k ( p > . (2.16)Equality (2.15) is equivalent to (2.5) and it remains to prove (2.16). From(2.3), taking into account C k = I m we have α Λ k +1 − iΛ k +1 C k = α Λ k +1 − iΛ k C k + α − Λ k = α Λ k + α − Λ k . (2.17)Substituting (2.17) into the left hand side of (2.16) and using simple trans-formations, we rewrite (2.16) in the form Z k α p − Λ k = 0 , Z k := Λ ∗ k +1 S − k +1 ( α + I n ) − Λ ∗ k S − k α + i e C k Λ ∗ k S − k α. ∗ k +1 S − k +1 ( α + I n ) = Λ ∗ k S − k α − i e C k Λ ∗ k S − k α, (2.18)that is, Z k = 0. Relation (2.18) is of interest in itself, since it is an analogueof (2.3) (more precisely of the relation adjoint to (2.3)) when Λ ∗ r S − r is takeninstead of Λ ∗ r . Such analogues are useful in continuous and discrete GBDTas well as in the construction of explicit solutions of dynamical systems (see,e.g. [5, 13, 14] and references therein).Taking into account (2.5) and (2.3), we rewrite (2.18) in the formΛ ∗ k +1 S − k +1 ( α + I n ) − Λ ∗ k S − k α + i (cid:0) C k + Λ ∗ k S − k Λ k − Λ ∗ k +1 S − k +1 Λ k +1 (cid:1) Λ ∗ k S − k α = Λ ∗ k +1 S − k +1 ( α + I n ) − iΛ ∗ k +1 S − k +1 (Λ k + i α − Λ k C k )Λ ∗ k S − k α − Λ ∗ k S − k α + i (cid:0) C k + Λ ∗ k S − k Λ k (cid:1) Λ ∗ k S − k α = 0 . (2.19)Since (2.8) yields iΛ k Λ ∗ k S − k = α − S k α ∗ S − k , (2.20)we rewrite the third line in (2.19) and see that (2.19) (i.e., also (2.18)) isequivalent to Λ ∗ k +1 S − k +1 ( I n + α − Λ k C k Λ ∗ k S − k α + S k α ∗ S − k α ) − Λ ∗ k S − k α + i (cid:0) C k + Λ ∗ k S − k Λ k (cid:1) Λ ∗ k S − k α = 0 . (2.21)Formula (2.4) implies that( I n + α − Λ k C k Λ ∗ k S − k α + S k α ∗ S − k α ) = S k +1 α ∗ S − k α. Hence, (2.21) is equivalent toΛ ∗ k +1 α ∗ S − k α − Λ ∗ k S − k α + i (cid:0) C k + Λ ∗ k S − k Λ k (cid:1) Λ ∗ k S − k α = 0 . (2.22)Using again (2.20), we rewrite (2.22) as (cid:0) Λ ∗ k +1 − Λ ∗ k + i C k Λ ∗ k ( α ∗ ) − (cid:1) α ∗ S − k α = 0 . (2.23)6he equality (2.23) is immediate from (2.3), and so (2.18) is also proved.Thus, (2.13) is proved as well.Next, (2.12) is proved by induction. Clearly (2.10) yields (2.12) for k = 0.If (2.12) holds for k = r , using (2.12) for k = r and relations (1.1), (2.2) and(2.13) we write e w ( r + 1 , z ) = (cid:18) I m + i z e C r (cid:19) e w ( r, z )= (cid:18) I m + i z e C r (cid:19) w α ( r, − z ) w ( r, z ) w α (0 , − z ) − = w α ( r + 1 , − z ) (cid:18) I m + i z C r (cid:19) w ( r, z ) w α (0 , − z ) − = w α ( r + 1 , − z ) w ( r + 1 , z ) w α (0 , − z ) − . (2.24)Thus, (2.12) holds for k = r + 1, and so (2.12) is proved. (cid:4) Using (2.13) we prove the next proposition.
Proposition 2.3
Assume that the matrices C k satisfy the second equality in (1.1) and the triple { α, S , Λ } satisfies the relations (2.1) , (2.6) , (2.7) , and S = S ∗ . Then, the transformed matrices e C k given by (2.3) – (2.5) have thefollowing property : e C k = I m . (2.25)P r o o f. It easily follows from (2.8) and (2.11) (see, e.g. [15] or [14, Corollary1.13]) that w α ( r, z ) w α ( r, z ) ∗ ≡ I m . (2.26)Since C k = I m , we have (cid:18) I m − i z C k (cid:19) (cid:18) I m + i z C k (cid:19) = (cid:18) z (cid:19) I m . (2.27)In view of (2.26) and (2.27) we derive w α ( k + 1 , z ) (cid:18) I m − i z C k (cid:19) (cid:18) I m + i z C k (cid:19) w α ( k + 1 , z ) ∗ = (cid:18) z (cid:19) I m . (2.28)7n the other hand (2.26) yields (cid:18) I m − i z e C k (cid:19) w α ( k, z ) w α ( k, z ) ∗ (cid:18) I m + i z e C k (cid:19) = (cid:18) I m − i z e C k (cid:19) (cid:18) I m + i z e C k (cid:19) = I m + 1 z e C k . (2.29)According to (2.13), the left hand sides of (2.28) and (2.29) are equal, andso we derive I m + z e C k = (cid:0) z (cid:1) I m , that is, (2.25) holds. (cid:4) Now, we introduce the notion of an admissible triple { α, S , Λ } and showafterwards that the admissible triples determine S k > k ∈ N ). Thedefinition of an admissible triple differs somewhat from the correspondingdefinition in [3], and the proof that S k > Definition 2.4
The triple { α, S , Λ } is called admissible if , i σ ( α ) , S > and the matrix identity (2.1) is valid. Theorem 2.5
Let an initial Dirac system (1.1) and an admissible triple { α, S , Λ } be given. Then, the conditions of Theorem 2.2 are satisfied. More-over, we have S k > k ∈ N ) , (2.30) and the transformed system (2.2) is skew-selfadjoint Dirac, that is, (2.9) isvalid. P r o o f. In order to prove (2.30) consider the difference S k +1 − (cid:0) I n − i α − (cid:1) S k (cid:0) I n + i( α − ) ∗ (cid:1) = S k +1 − S k − α − S k ( α − ) ∗ + i (cid:0) α − S k − S k ( α − ) ∗ (cid:1) . (2.31)Using (2.4), (2.8) and the second equality in (1.1), we rewrite (2.31) andderive a useful inequality: S k +1 − (cid:0) I n − i α − (cid:1) S k (cid:0) I n + i( α − ) ∗ (cid:1) = α − Λ k C k Λ ∗ k ( α − ) ∗ + α − Λ k Λ ∗ k ( α − ) ∗ ≥ . (2.32)8ince 0 , i σ ( α ), the sequence (cid:0) I n − i α − (cid:1) − k S k (cid:0) I n + i( α − ) ∗ (cid:1) − k ( k ∈ N ) iswell-defined. In view of (2.32), this sequence is nondecreasing. Hence, takinginto account S > (cid:0) I n − i α − (cid:1) − k S k (cid:0) I n + i( α − ) ∗ (cid:1) − k >
0, and so(2.30) holds.Similar to [3, Lemma A.1] one can show that σ ( α ) ⊂ C + . (2.33)That is, one rewrites (2.1) in the form (cid:16) S − / αS / (cid:17) − (cid:16) S − / αS − / (cid:17) ∗ = i S − / Λ Λ ∗ S − / , and from S − / Λ Λ ∗ S − / ≥
0, the relation σ ( α ) = σ ( S − / αS / ) ⊂ C + follows. Clearly, (2.33) yields − i σ ( α ). Therefore, we may set z = − iin (2.13) and (taking into account the second equality in (1.1) and formula(2.26)) we obtain I m + e C k = w α ( k + 1 , − i)( I m + C k ) w α ( k, i) ∗ = 2 w α ( k + 1 , − i) U ∗ k (cid:20) I m (cid:21) (cid:2) I m (cid:3) U k w α ( k, i) ∗ . (2.34)In the same way, setting in (2.13) z = i we obtain I m − e C k = 2 w α ( k + 1 , i) U ∗ k (cid:20) I m (cid:21) (cid:2) I m (cid:3) U k w α ( k, − i) ∗ . (2.35)According to (2.34) and (2.35) the dimension of the subspace of e C k corre-sponding to the eigenvalue λ = − m and the dimensionof the subspace of e C k corresponding to the eigenvalue λ = 1 is more or equalto m . Thus, the representation (2.9) is immediate. (cid:4) Remark 2.6
It follows from (2.9) that I m + e C k ≥ and that I m + e C k hasrank m . Hence, (2.34) yields (cid:2) I m (cid:3) U k w α ( k + 1 , − i) ∗ = ˘ q k (cid:2) I m (cid:3) U k w α ( k, i) ∗ , ˘ q k > for some matrix ˘ q k . In the same way, formulas (2.9) and (2.35) imply that (cid:2) I m (cid:3) U k w α ( k + 1 , i) ∗ = ˆ q k (cid:2) I m (cid:3) U k w α ( k, − i) ∗ , ˆ q k > . (2.37)9 ow, setting e U k = W k := diag { ˘ q / k , ˆ q / k } (cid:2) I m (cid:3) U k w α ( k, i) ∗ (cid:2) I m (cid:3) U k w α ( k, − i) ∗ (2.38) we provide expressions for some suitable unitary matrices e U k in the repre-sentations (2.9) of the matrices e C k .Indeed, according to the definition of W k in (2.38) and to the relations (2.34) – (2.37) we have e C k = 12 (cid:16) ( I m + e C k ) − ( I m − e C k ) (cid:17) = W ∗ k jW k , (2.39) and it remains to show that W k is unitary. In view of (2.26) and (2.38) , itis easy to see that W k W ∗ k is a block diagonal matrix : W k W ∗ k = diag { ˘ ρ k , ˆ ρ k } > . (2.40) Hence, for R k > from the polar decomposition W k = R k V k ( V k V ∗ k = I m ) wehave R k = diag { ˘ ρ k , ˆ ρ k } ( and, in particular, R k is block diagonal ) . Therefore, (2.39) may be rewritten in the form e C k = V ∗ k diag { ˘ ρ k , − ˆ ρ k } V k ( ˘ ρ k > , ˆ ρ k > . Comparing (2.9) with the formula above, we see that all the eigenvalues of ˘ ρ k and ˆ ρ k equal , that is, R k = R k = I m , and so W k = V k . In other words, W k is unitary. Recall the equalities (2.18):Λ ∗ k +1 S − k +1 ( α + I n ) = Λ ∗ k S − k α − i e C k Λ ∗ k S − k α, k ∈ N , (3.1)10hich are basic for the construction of explicit solutions of non-stationarysystems. Introduce the semi-infinite shift block matrix S and diagonal blockmatrix e C : S := I m . . . I m . . . I m . . .. . . . . . . . . . . . . . . , e C := diag { e C , e C , e C , . . . } . (3.2)The semi-infinite block column Ψ( t ) is given by the formulaΨ( t ) = Y e i tα , Y = { Y k } ∞ k =0 , Y k = Λ ∗ k S − k . (3.3)It is easy to see that equalities (3.1) and Theorem 2.5 yield the followingresult. Theorem 3.1
Let the initial Dirac system (1.1) and an admissible triple { α, S , Λ } be given. Then, the matrices Λ k , S k and e C k are well-defined via (2.3) – (2.5) for k ∈ N . Moreover, the block vector function Ψ( t ) constructedin (3.3) satisfies the non-stationary system ( I − S )Ψ ′′ + e C Ψ ′ + S Ψ = 0 , (3.4) where I is the identity operator and Ψ ′ = ddt Ψ . Acknowledgments.
This research was supported by the Austrian ScienceFund (FWF) under Grant No. P29177.
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