On asymptotic behaviour of solutions of the Dirac system and applications to the Sturm-Liouville problem with a singular potential
aa r X i v : . [ m a t h . SP ] M a r On asymptotic behaviour of solutions of theDirac system and applications to theSturm–Liouville problem with a singularpotential
Łukasz Rzepnicki
Faculty of Mathematics and Computer Science Nicolaus Copernicus UniversityChopina 12/18, 87-100 Toruń, Poland [email protected]
Alexander Gomilko
Faculty of Mathematics and Computer Science Nicolaus Copernicus UniversityChopina 12/18, 87-100 Toruń, Poland [email protected]
Abstract
The main focus of this paper is the following matrix Cauchy problemfor the Dirac system on the interval [0 ,
1] : D ′ ( x )+ (cid:20) σ ( x ) σ ( x ) 0 (cid:21) D ( x ) = iµ (cid:20) − (cid:21) D ( x ) , D (0) = (cid:20) (cid:21) , where µ ∈ C is a spectral parameter, and σ j ∈ L [0 , , j = 1 , . Wepropose a new approach for the study of asymptotic behaviour of its solu-tions as µ → ∞ and | Im µ | ≤ d . As an application, we obtain new, sharpasymptotic formulas for eigenfunctions of Sturm–Liouville operators withsingular potentials. keyword: Sturm–Liouville problem, singular potential, Dirac system, Sturm–Liouville op-erator
MSC [2010] 34L20, 34E10
The motivation to study the Dirac systems came from our interest in the spectralproblem for Sturm–Liouville equation with a singular potential, in particularthe asymptotic behaviour with respect to a spectral parameter of its solutions.We found a way to transform the spectral problem into a special form of theperturbed Dirac system. The known results related to asymptotic behaviour ofthe solutions of the Dirac system were not precise enough for our aims. Thereforewe attracted our attention to the Dirac system itself. In this way we obtained1ew results about Dirac systems and gained a new tool to spectral analysis ofSturm–Liouville problems.The paper thus is naturally divided into two parts: one is devoted to Diracsystems and the other one is focused on the applications of the results to spectralproblems for Sturm–Liouville equations with singular potentials. We start withthe presentation of the background concerning Dirac systems.Consider the following matrix Cauchy problem for the Dirac system: D ′ ( x ) + J ( x ) D ( x ) = iµJ D ( x ) , x ∈ [0 , , D (0) = I, (1.1)where J = (cid:20) − (cid:21) , J ( x ) = (cid:20) σ ( x ) σ ( x ) 0 (cid:21) , I := (cid:20) (cid:21) , (1.2) µ ∈ C is a spectral parameter, and σ j ∈ L [0 , , j = 1 , , are complex-valuedfunctions. By a solution of (1.1), as usual, we understand a matrix D withentries from the space of absolutely continuous on [0 , functions (i.e. from theSobolev space W [0 , ) satisfying (1.1) for a.e. x ∈ [0 , . In fact here bothconditions yield that D has entries from W [0 , In recent years the study of asymptotic behavior of fundamental solutions toDirac systems has attracted a lot of attention, and we may mention e.g. [1], [3],[5], [6], [8], [7], [13], [14], [16], [18], and [21], as samples. In particular, there is avariety of asymptotic formulas for the solutions to (1.1) in different settings andunder a wide range of assumptions on the potentials. However, most of thoseformulas contain just a leading term. In this paper, by a new and comparativelysimple argument, we find sharp asymptotic for solutions to Dirac systems, thusgeneralizing most of similar relations from the literature.More specifically, we study the asymptotic behaviour of solutions D ( x ) = D ( x, µ ) to the Cauchy problem (1.1) as µ → ∞ , µ ∈ P d , where P d := { µ ∈ C : | Im µ | ≤ d } . The main results related to this issue are placed in Section 2 in Lemma 2.4 andCorollary 2.5.Our results can be analyzed wider since there exists the connection betweenthe Dirac system presented in this paper and more general formulations of theproblem, often studied in literature. Recall that any solution V = { v , v } T , v j ∈ W [0 , , of the next Cauchy problem for the Dirac system: V ′ ( x ) + J ( x ) V ( x ) = iµJ V ( x ) , x ∈ [0 , , (1.3) V (0) = { c , c } T ∈ C , is of the form V ( x ) = D ( x ) V (0) , x ∈ [0 , . Note that a system formally more general than (1.3): Y ′ ( x ) + P ( x ) Y ( x ) = iµJ Y ( x ) , P ( x ) = (cid:20) p ( x ) p ( x ) p ( x ) p ( x ) (cid:21) , (1.4)2here x ∈ [0 , and Y = { y , y } T , p jj ∈ L [0 , , p ij ∈ L [0 , , i, j = 1 , , i = j , can be reduced to (1.3) by a simple transformation given in [16]. Indeed,setting r j ( x ) = Z x p jj ( t ) dx, j = 1 , , it is easy to see that Y is a solution of (1.4) if and only if a vector-function V ( x ) = (cid:20) e r ( x ) e r ( x ) (cid:21) Y ( x ) satisfies (1.3) with σ ( x ) = p ( x ) e r ( x ) − r ( x ) , σ ( x ) = p ( x ) e r ( x ) − r ( x ) . On the other hand, the classical Dirac system (see. e. g. [13, Ch. VII, § ]) BZ ′ ( x ) + Q ( x ) Z ( x ) = µZ ( x ) , x ∈ [0 , , (1.5)where B = (cid:20) − (cid:21) , Q ( x ) = (cid:20) q ( x ) q ( x ) q ( x ) q ( x ) (cid:21) , q ij ∈ L [0 , , can be transformed to (1.4) by writing Z ( x ) = U Y ( x ) , U = (cid:20) − i − i (cid:21) . The idea of our method, originating from [15, Ch. 1, § , Problem 5], relieson a special integral representation for D ( x ) = D ( x, µ ) . The representation hasthe form D ( x ) = e xA µ + Z x e ( x − t ) A µ [ J ( t ) + Q ( x, t )] dt, (1.6)where A µ := iµJ = (cid:20) iµ − iµ (cid:21) , (1.7)and Q is a matrix function continuous on ∆ := { ( x, t ) ∈ R : 0 ≤ t ≤ x ≤ } , (1.8)and, moreover, Q is a unique solution in the space of continuous functions on ∆ of the integral equation Q ( x, t ) = Z x − t J ( t + ξ ) J ( ξ ) dξ − Z x − t J ( t + ξ ) Q ( t + ξ, ξ ) dξ. (1.9)It is crucial to observe that the kernel Q does not depend on µ . More detailscan be found in Lemma 2.2. This approach allows us to obtain much sharperasymptotic formulas for solutions to Dirac systems than those in the literature.3s a consequence, we get fine asymptotics for solutions to perturbed Diracsystems of the form ˜ D ′ ( x ) + J ( x ) ˜ D ( x ) = iµJ ˜ D ( x ) + P ( x ) µ ˜ D ( x ) , x ∈ [0 , , ˜ D (0) = I, (1.10)where µ = 0 and the entries of P are in L [0 , . Note that such an approachcan be extended to a more general case when σ j ∈ L p [0 , , j = 1 , , p ≥ .Perturbed Dirac systems are analyzed in Section 3, where the main result isTheorem 3.2.It should be mentioned that the representation (1.6) is not quite new. In [1]the authors considered a matrix Dirac equation BU ′ ( x ) + (cid:20) q ( x ) q ( x ) q ( x ) − q ( x ) (cid:21) U ( x ) = λU ( x ) , x ∈ [0 , , U (0) = I, (1.11)where q j , j = 1 , are real-valued functions in L p [0 , , p ∈ [1 , ∞ ) . By [1,Theorem 2.1] one has U ( x, λ ) = e − λxB + Z x e − λ ( x − s ) P ( x, s ) ds, (1.12)where for every x ∈ [0 , the entries p ij ( x, · ) , i, j = 1 , of P ( x, · ) , belong to L p [0 , and the mapping x p ij ( x, · ) is continuous on [0 , . The representation(1.12) was derived in [1] via the method of successive approximations by solvingthe integral equation U ( x, λ ) = e − λxB + Z x e − λ ( x − t ) B BQ ( t ) U ( t, λ ) dt, x ∈ [0 , , Q = { q ij } i,j =1 , and using the property e − λxB Q ( t ) = Q ( t ) e λxB , x, t ∈ [0 , . In the Hilbert spacesetting, when p = 2 , we follow a different strategy for obtaining (1.6). Namely,we deduce an integral equation for Q and investigate this equation directly. As aresult, we obtain more detailed information on Q and this allows to derive moreprecise asymptotic formulas for the solutions of the Dirac system (see Lemma2.4).Note that in [18] and [21] the Dirac system (1.5) was studied in a situationwhen q ij , i, j = 1 , are complex-valued functions in L p [0 , , p ∈ [1 , ∞ ) . Bymeans of Prüfer substitution [9], in a rather complicated manner, “short” asymp-totic formulas (as µ → ∞ , µ ∈ P d ) were obtained in [18] and [21] for solutions Z of (1.5) with the initial conditions Z (0) = { , } T and Z (0) = { , } T . Theseformulas (with p = 2 ) correspond to our Corollary 2.5.We show in Section 4 and 5 that our results apply to the study of asymptoticbehaviour of the system of fundamental solutions to the differential equation l ( y ) + λy = 0 , l ( y ) := y ′′ + q ( x ) y, x ∈ [0 , , (1.13)where a potential q belongs to W − [0 , . In other words, q ( x ) = σ ′ ( x ) , σ ∈ L [0 , , where the derivative is understood in the sense of distributions. Such a classof singular potentials includes Dirac δ -type and Coulomb /x -type interactions,4hich are frequently used in quantum mechanics and mathematical physics. Fora comprehensive treatment of physical models with potentials from negativeSobolev spaces we refer to [12]. A short history and different approaches to thestudy of singular Sturm–Liouville operators can be found in [19].Following the regularization method described in [4] and [19], given y weformally introduce a quasi-derivative y [1] of y : y [1] ( x ) := y ′ ( x ) + σ ( x ) y ( x ) , so that (1.13) can be rewritten in the form ( y [1] ) ′ ( x ) − σ ( x ) y [1] ( x ) + σ ( x ) y ( x ) + λy ( x ) = 0 , x ∈ [0 , . (1.14)Our main idea is to transform the equation (1.14) into an appropriate per-turbed Dirac system (1.10). We show that if µ = 0 then the transformation (cid:18) yy [1] (cid:19) = (cid:20) iµ − iµ (cid:21) (cid:18) v v (cid:19) (1.15)“recasts” (1.14) as the following perturbed Dirac system for V = { v , v } T , x ∈ [0 ,
1] : V ′ ( x ) + (cid:20) σ ( x ) σ ( x ) 0 (cid:21) V ( x ) = iµJ V ( x ) + iσ ( x )2 (cid:20) − − (cid:21) V ( x ) . In this way, using our results on asymptotic of Cauchy’s matrix for (1.10), weobtain very precise information on asymptotic behavior of fundamental solutionsto (1.13) and on asymptotic distribution of the eigenvalues and eigenfunctionsto the spectral problem l ( y ) + λy = 0 , y (0) = 0 , y (1) = 0 . (1.16)The main results Theorem 4.1 and Corollary 4.2 concerning the asymptoticbehavior of fundamental solutions to (1.13) are placed in Section 4.In Section 5 we concentrate on spectral problem (1.16). The problem (1.16)was widely studied in the literature. Basic facts concerning the asymptotics of itseigenvalues and eigenfunctions can be found in [10] and [11]. The authors of [10]and [11] employed the method of transformation operators and used a factorizedform of l to obtain asymptotic formulas with remainder terms expressed in termsof Fourier sine coefficients. Their results were precise enough to study the inversespectral problem for (1.16), see, for instance, [1] and [2]. More detailed formulas,stated in Theorem 5.1, were obtained in [17], [19] and [20]. The approach ofthe latter papers relied on the Prüfer substitution, and it allowed one to deriveasymptotic formulas for eigenfunctions y n of (1.16) in the form πny n ( x ) = y ,n ( x ) + ψ n ( x ) , n ≥ , (1.17)where y ,n is a known function (the explicit formulation of it is given by (5.6)),and remainders ψ ,n satisfying sup x ∈ [0 , ∞ X n =1 | ψ n ( x ) | < ∞ . (1.18)5his method was further developed in a number of subsequent articles, see e.g.[22], [23], [24], and [25]. Our main result generalizes and sharpens Theorem 5.1essentially. In particular, we provide similar asymptotic formulas to (1.17) i.e. πny n ( x ) = y ,n ( x ) + y ,n ( x ) + ˜ ψ n ( x ) , n ≥ , (1.19)where y ,n are known functions (see (5.9)) and remainder terms ˜ ψ n satisfy astrengthened form of (1.18): ∞ X n =1 sup x ∈ [0 , | ˜ ψ n ( x ) | < ∞ . Note that our method of reduction to the perturbed Dirac system applies ina setting more general than that of this paper. For example (see Remark 2), atransformation similar to (1.15) allows one to treat the equations of the form ( a ( x ) y ′ ( x )) ′ + q ( x ) y ( x ) + λc ( x ) y ( x ) = 0 , x ∈ [0 , , where q = u ′ , u ∈ L [0 , , and a ∈ W [0 , , c ∈ W [0 , , a ( x ) > , c ( x ) > , x ∈ [0 , . The paper is organized as follows. Section 2 is devoted to the Dirac system(1.1). We derive there a special representation and establish asymptotic formu-las of the solutions. In Section 3 we use the facts from the previous section toanalyze the asymptotic behaviour of the solutions of the perturbed Dirac sys-tem. Section 4 is focused on the application of the results related to the Diracsystem to investigation of Sturm–Liouville equations with singular potentials,in particular we present there how to transform a Sturm–Liouville equation intothe Dirac system of a special form. The Dirichlet spectral problem for Sturm–Liouville equation with singular potential is studied in Section 5. For the clarityof exposition some part of auxiliary and technical results necessary in the proofsof Sections 2 and 3 are placed in the Appendix.In the paper we adopt the following convention: when the domain of afunction is not given, it is assumed to be [0 , . For example, we use the notation L p := L p [0 , . This section is devoted to the matrix Cauchy problem D ′ ( x ) + J ( x ) D ( x ) = A µ D ( x ) , D (0) = I, x ∈ [0 , , (2.1)where A µ = iµJ (see (1.7)) and σ j ∈ L [0 , , j = 1 , .We obtain an important integral representation for the solutions of (2.1),given by (1.6) and (1.9).First, we introduce a necessary notation. Denote the space of continuousfunctions on ∆ with the supremum norm k · k C (∆) by C (∆) . Define l := max {k σ k L , k σ k L } , l := k σ k L · k σ k L , l := k σ k L + k σ k L , (2.2)6nd ˜ l := max {k σ k L , k σ k L } , ˜ l := k σ k L · k σ k L , l = k σ k L + k σ k L . (2.3)Moreover, let σ ( x ) := | σ ( x ) | + | σ ( x ) | ∈ L [0 , . (2.4)Let X be a Banach space, then M ( X ) will stand for the Banach space of × matrices with entries from X and the norm k Q k M ( X ) := X k,j =1 k Q jk k X , Q = [ Q jk ] j,k =1 . In our analysis of D we will need several auxiliary results addressing the twointegrals in (1.9). If J and J are defined by (1.2), then J = I, J J ( x ) + J ( x ) J = 0 , a.e. x ∈ [0 , . (2.5)Moreover, let the matrix function ˜ J be given by ˜ J ( x, t ) := Z x − t J ( t + ξ ) J ( ξ ) dξ = Z xt J ( s ) J ( s − t ) ds, ( x, t ) ∈ ∆ . (2.6)Note that ˜ J ( x, t ) = (cid:18) ˜ σ ( x, t ) 00 ˜ σ ( x, t ) (cid:19) , where ˜ σ ( x, t ) := Z x − t σ ( t + ξ ) σ ( ξ ) dξ, ˜ σ ( x, t ) := Z x − t σ ( t + ξ ) σ ( ξ ) dξ. (2.7)Using Cauchy–Schwarz inequality, for ( x, t ) ∈ ∆ and ( x + ǫ, t + δ ) ∈ ∆ , wehave | ˜ σ ( x + ǫ, t + δ ) − ˜ σ ( x, t ) | = (cid:12)(cid:12)(cid:12) Z x − t + ǫ − δ σ ( t + δ + s ) σ ( s ) ds − Z x − t σ ( t + s ) σ ( s ) ds (cid:12)(cid:12)(cid:12) ≤ Z x − t + ǫ − δx − t | σ ( s ) | ds ! / · (cid:18)Z x + ǫx + δ | σ ( s ) | ds (cid:19) / + k σ k L (cid:18)Z xt | σ ( δ + s ) − σ ( s ) | ds (cid:19) / . Clearly, a similar estimate holds for ˜ σ as well. Thus ˜ σ j ∈ C (∆) , j = 1 , , and k ˜ σ j k C (∆) ≤ ˜ l, j = 1 , . (2.8)To treat the other integral from (1.9), define a linear operator T σ on C (∆) by ( T σ f )( x, t ) = Z x − t σ ( t + ξ ) f ( t + ξ, ξ ) dξ = Z xt σ ( s ) f ( s, s − t ) ds, (2.9)7here σ ∈ L [0 , . Let ( x, t ) ∈ ∆ and ǫ ∈ R , δ ∈ R so that ( x + ǫ, t + δ ) ∈ ∆ .Then | ( T σ f )( x + ǫ, t + δ ) − ( T σ f )( x, t ) |≤ Z x + ǫt + δ σ ( s ) | f ( s, s − t − δ ) − f ( s, s − t ) | ds + (cid:12)(cid:12)(cid:12)(cid:12)Z x + ǫt + δ σ ( s ) f ( s, s − t ) ds − Z xt σ ( s ) f ( s, s − t ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ k σ k L sup s ∈ [0 , sup τ ,τ ∈ [0 ,s ] , | τ − τ |≤ δ | f ( s, τ ) − f ( s, τ ) | + 2 k f k C (∆) sup
One has k T nkj f k C (∆) ≤ l n n ! k f k C (∆) , f ∈ C (∆) , n ∈ N , k, j = 1 , , k = j. (2.12) Proof.
Consider the operator T . Define η ∈ C [0 , by η ( x ) := Z x | σ ( s ) | (cid:18)Z s | σ ( τ ) | dτ (cid:19) ds, x ∈ [0 , . It suffices to prove that for all ( x, t ) ∈ ∆ and n = 1 , , . . . , | ( T n f )( x, t ) | ≤ k f k C (∆) n ! η n ( x ) , f ∈ C (∆) . (2.13)If n = 1 , then the estimate (2.13) follows directly from (2.11). Arguing byinduction, suppose that (2.13) holds for some n ∈ N . Then for ( x, t ) ∈ ∆ and8 ∈ C (∆) we have | ( T n +112 f )( x, t ) | ≤ Z xt | σ ( s ) | Z ss − t | σ ( τ ) || ( T n f )( τ, τ − s + t ) | dτ ds ≤ k f k C (∆) n ! Z x | σ ( s ) | Z ss − t | σ ( τ ) | η n ( τ ) dτ ds ≤ k f k C (∆) n ! Z x | σ ( s ) | Z s | σ ( τ ) | dτ η n ( s ) ds = k f k C (∆) n ! Z x η n ( s ) dη ( s ) = k f k C (∆) ( n + 1)! η n +1 ( x ) . Therefore (2.13) and then (2.12) hold true.Now we are ready to establish a first crucial property of the solutions to(2.1). The proof of the next statement follows an idea from [15, Ch. 1, § ]. Lemma 2.2. If σ j ∈ L [0 , , j = 1 , and µ ∈ P d , then the unique solution D = D ( x, µ ) of (2.1) can be represented as D ( x, µ ) = e xA µ + Z x e ( x − t ) A µ [ J ( t ) + Q ( x, t )] dt, (2.14) where Q ∈ M ( C (∆)) is the unique solution of the integral equation Q ( x, t ) = ˜ J ( x, t ) + Z x − t J ( t + ξ ) Q ( t + ξ, ξ ) dξ, (2.15) where ˜ J ∈ M ( C (∆)) is given by (2.6) . Moreover, k Q k M ( C (∆)) ≤ l (1 + l ) e l , (2.16) and k D k M ( C [0 , ≤ e d a, a := 1 + l + 2˜ l (1 + l ) e l . (2.17) Proof.
The uniqueness of solutions comes from [26, Thm. 1.2.1]. We look forsolutions of (2.1) in the form D ( x, µ ) = e xA µ U ( x, µ ) , U (0 , µ ) = I. (2.18)The relation J ( x ) e xA µ = e − xA µ J ( x ) , a. e. x ∈ [0 , (2.19)implies that U satisfies the Cauchy problem U ′ ( x, µ ) + e − xA µ J ( x ) U ( x, µ ) = 0 , x ∈ [0 , , U (0 , µ ) = I, which is equivalent to the integral equation U ( x, µ ) = I − Z x e − tA µ J ( t ) U ( t, µ ) dt, x ∈ [0 , . (2.20)Now, we look for solutions of (2.20) in the form U ( x, µ ) = I + Z x e − tA µ Q ( x, t ) dt, (2.21)9here Q ∈ M ( L (∆)) does not depend on µ . Substituting (2.21) into (2.20),we obtain Z x e − tA µ Q ( x, t ) dt = − Z x e − tA µ J ( t ) dt − Z x e − tA µ J ( t ) Z t e − sA µ Q ( t, s ) ds dt. Then, by (2.5), we have Z x e − tA µ J ( t ) Z t e − sA µ Q ( t, s ) ds dt = Z x e − tA µ Z t e sA µ J ( t ) Q ( t, s ) ds dt = Z x e − tA µ Z x − t J ( t + ξ ) Q ( t + ξ, ξ ) dξdt, hence Z x e − tA µ Q ( x, t ) dt = − Z x e − tA µ (cid:18) J ( t ) + Z x − t J ( t + ξ ) Q ( t + ξ, ξ ) dξ (cid:19) dt for all x ∈ [0 , . Therefore, U is a solution of (2.20) if and only if Q ∈ M ( L (∆)) is a solution of Q ( x, t ) = − J ( t ) − Z x − t J ( t + ξ ) Q ( t + ξ, ξ ) dξ. (2.22)Next, setting Q ( x, t ) = − J ( t ) + Q ( x, t ) , ( x, t ) ∈ ∆ , and using (2.15), we infer that Q satisfies (2.22). The latter equation can bewritten in an operator form Q = ˜ J + ˜ T Q, ˜ T = − (cid:20) T σ T σ (cid:21) , (2.23)where the operators T σ and T σ , defined by (2.9), are linear and bounded on C (∆) . In particular, by (2.10), we have k ˜ T F k M ( C (∆)) ≤ max {k σ k L , k σ k L }k F k M ( C (∆)) , F ∈ M ( C (∆)) . A more detailed analysis of ˜ T can be found in the Appendix.Next observe that ˜ T n = (cid:20) T n T n (cid:21) , n ∈ N , therefore, by (2.12), k ˜ T n F k M ( C (∆)) ≤ l n n ! k F k M ( C (∆)) , F ∈ M ( C (∆)) . So (2.15) has a unique solution Q ∈ C (∆) of the form Q = ∞ X n =0 ˜ T n ˜ J = ∞ X n =0 ˜ T n ( I + ˜ T ) ˜ J, (2.24)10nd moreover k Q k M ( C (∆)) ≤ (1 + l ) e l k ˜ J k M ( C (∆)) . (2.25)Thus, (2.24) and (2.8) imply (2.16).Finally, in view of (2.14) and (2.16), we obtain k D k M ( C [0 , ≤ e d (cid:16) l + k Q ( x, t ) k M ( C (∆)) (cid:1) , µ ∈ P d , and (2.17) follows.By means of simple calculations, one obtains the following representation ofthe matrix Q defined by (2.24). Corollary 2.3. If Q is given by (2.24) , then Q = (cid:20) Z − T σ Z − T σ Z Z (cid:21) , Z = ˜ σ + ∞ X k =1 T k ˜ σ , Z = ˜ σ + ∞ X k =1 T k ˜ σ . (2.26) Moreover, k Z j k C (∆) ≤ ˜ l exp l, (2.27) k T σ j Z l k C (∆) ≤ k σ j k L k Z l k C (∆) , j = 1 , . Going back to the formulas (2.14) and (2.24) for D = D ( x, µ ) , x ∈ [0 , , µ ∈ P d , note that D ( x, µ ) = e xA µ + Z x e ( x − t ) A µ [ J ( t ) + Q ( x, t )] dt = e xA µ + Z x e ( x − t ) A µ J ( t ) dt + Z x e ( x − t ) A µ ˜ J ( x, t ) dt + Z x e ( x − t ) A µ ( ˜ T ˜ J )( x, t ) dt + Z x e ( x − t ) A µ ∞ X n =2 ( ˜ T n ˜ J )( x, t ) dt. (2.28)Now, we are ready to investigate the asymptotic behaviour of D ( x, µ ) definedby (2.28) for µ ∈ P d and fixed d > .In what follows we will use different types of estimates for reminders. Forfixed σ j ∈ L , j = 1 , , and µ ∈ C define γ ( µ ) := X j =1 (cid:18)(cid:13)(cid:13)(cid:13) Z x e − iµt σ j ( t ) dt (cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13) Z x e iµt σ j ( t ) dt (cid:13)(cid:13)(cid:13) L (cid:19) , (2.29) γ ( x, µ ) := X j =1 (cid:18)(cid:12)(cid:12)(cid:12) Z x e − iµt σ j ( t ) dt (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z x e iµt σ j ( t ) dt (cid:12)(cid:12)(cid:12)(cid:19) , x ∈ [0 , . (2.30)and γ ( µ ) := Z σ ( s ) γ ( s, µ ) ds, γ ( µ ) := l γ ( µ ) + l γ ( µ ) . (2.31)11t is easy to see that if µ ∈ P d , then k γ ( x, µ ) k L ≤ γ ( µ ) , γ ( µ ) ≤ e d l , γ ( x, µ ) ≤ e d l , x ∈ [0 , , (2.32)and γ ( µ ) ≤ e d l ( l + l ) , γ ( µ ) ≤ l e d ( l + 2 l e d k σ k L ) γ ( µ ) , (2.33)For the sequel, define N ( x, t ) := ( ˜ J + ˜ T ˜ J )( x, t ) ∈ C (∆) , (2.34)and observe that N ( x, t ) = (cid:20) ˜ σ ( x, t ) − ( T σ ˜ σ )( x, t ) − ( T σ ˜ σ )( x, t ) ˜ σ ( x, t ) (cid:21) ( x, t ) ∈ ∆ . Lemma 2.4.
Let σ j ∈ L , j = 1 , . If D ( x, µ ) is a solution of (2.1) then D ( x, µ ) = e xA µ + D (0) ( x, µ ) + D (1) ( x, µ ) , (2.35) where D (0) ( x, µ ) = Z x e ( x − t ) A µ J ( t ) dt + Z x e ( x − t ) A µ N ( x, t ) dt, and for all µ ∈ P d and x ∈ [0 , , k D (1) ( x, µ ) k M ( C [0 , ≤ e d exp ( e d l ) γ ( µ ) . Proof.
By (2.28) we have D (1) ( x, µ ) = Z x e ( x − t ) A µ ∞ X n =2 ( ˜ T n ˜ J )( x, t ) dt. and using the inequality (6.7) proved in the Appendix, we infer that k D (1) ( x, µ ) k M ( C [0 , ≤ ∞ X n =2 (cid:13)(cid:13)(cid:13)(cid:13)Z x e ( x − t ) A µ ( ˜ T n ˜ J )( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C [0 , ≤ e d γ ( µ ) ∞ X n =2 e nd l n − ( n − e d exp ( e d l ) γ ( µ ) , for all x ∈ [0 , and µ ∈ P d .The above lemma leads to asymptotic formulas for D ( x, µ ) , µ ∈ P d . Let usstart with several simple observations. Note that from (cid:13)(cid:13)(cid:13)(cid:13)Z x e ( x − t ) A µ J ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ≤ e d γ ( x, µ ) , x ∈ [0 , , (6.4), and (6.5) it follows that (cid:13)(cid:13)(cid:13) D (0) ( x, µ ) (cid:13)(cid:13)(cid:13) M (C) ≤ e d γ ( x, µ ) + 2 e d (1 + ˜ l e d )˜ l γ ( µ ) , x ∈ [0 , . (cid:13)(cid:13)(cid:13)(cid:13)Z x e ( x − t ) A µ N ( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C [0 , ≤ e d ˜ l
9! + e d ˜ l ) γ ( µ ) . Combining these inequalities with Lemma 2.4 and the estimates from (2.33), weobtain the following representations for D . Corollary 2.5.
For every d > there exist a j = a j ( d, k σ k L , k σ k L ) , j =0 , , such that for all x ∈ [0 , and µ ∈ P d ,D ( x, µ ) = e xA µ + R ( x, µ ) , (2.36) where k R ( x, µ ) k M ( C [0 , ≤ a , k R ( x, µ ) k M (C) ≤ a ( γ ( µ ) + γ ( x, µ )) , x ∈ [0 , . Moreover, D ( x, µ ) = e tA µ + D ( x, µ ) + R ( x, µ ) , (2.37) where D ( x, µ ) := Z x e ( x − t ) A µ J ( t ) dt and k R ( x, µ ) k M ( C [0 , ≤ a ( γ ( µ ) + γ ( µ )) , x ∈ [0 , . Remark 1.
Note that the explicit formula for D is the following D ( x, µ ) = (cid:20) q ( x, µ ) q ( x, µ ) 0 (cid:21) ,q ( x, µ ) := Z x e iµ ( x − t ) σ ( t ) dt, q ( x, µ ) := Z x e − iµ ( x − t ) σ ( t ) dt. Let us consider the matrix Cauchy problem ˜ D ′ ( x ) + J ( x ) ˜ D ( x ) = A µ ˜ D ( x ) + P ( x ) µ ˜ D ( x ) , x ∈ [0 , , ˜ D (0) = I, (3.1)where the entries of the matrix P ( x ) = [ p k,j ( x )] k,j =1 are complex-valued func-tions from L [0 , . Let µ = 0 and let D be a solution of (2.1). Then the solutionsof (3.1) also satisfy the integral equation ˜ D = D + 1 µ A ˜ D, (3.2)where the linear operator A = A ( µ ) acting on M ( C [0 , is given by ( A ˜ D )( x ) = D ( x )( A ˜ D )( x ) , ( A ˜ D )( x ) := Z x D − ( t ) P ( t ) ˜ D ( t ) dt, x ∈ [0 , . (3.3)13ote that by Liouville’s theorem, det D ( x ) = det D (0) = 1 , x ∈ [0 , , hence D − ( x ) = − ˜ J [ D ( x )] T ˜ J , ˜ J := (cid:20) − (cid:21) , x ∈ [0 , . (3.4)As a consequence, for every x ∈ [0 , , k D ( x ) k M ( C ) = k D − ( x ) k M ( C ) . (3.5)The identity (3.5) and the estimate (2.17) imply that kA ( µ ) k M ( C [0 , ≤ k D k M ( C [0 , k P ( t ) k M ( L [0 , ≤ be d , µ ∈ P d , (3.6)where b := a k P k M ( L [0 , , and a constant a is defined in (2.17).The following statement relates the solutions of the perturbed matrix Cauchyproblem (3.1) to the solutions of the Cauchy problem (2.1). Corollary 3.1.
For every µ ∈ P d , | µ | > be d , the solution ˜ D ( x, µ ) of (3.1) admits the representation ˜ D ( x, µ ) = D ( x, µ ) + ∞ X k =1 µ − k A k ( µ ) D ( x, µ ) , (3.7) where the series converges in M ( C [0 , . Moreover, for µ ∈ P d , and | µ | > be d , k ˜ D − D k M ( C [0 , ≤ a | µ | , a = 2 abe d (3.8) and ˜ D ( x, µ ) = D ( x, µ ) + 1 µ A ( µ ) D ( x, µ ) + ˜ D ( x, µ ) , (3.9) where k ˜ D ( x, µ ) k M ( C [0 , ≤ ∞ X k =2 | µ | − k kA k ( µ ) D ( x, µ ) k M ( C [0 , ≤ a | µ | , (3.10) and a = 2 ab e d . Now we focus on the representation (3.9). In particular we investigate thebehaviour of the second term /µ A ( µ ) D ( x, µ ) in (3.9). Due to (3.4) and (2.36),we have D − ( t ) P ( t ) D ( t ) = − ˜ J D T ( t ) ˜ J P ( t ) D ( t )= − ˜ J [ e tA µ + R T ( t, µ )] ˜ J P ( t )[ e tA µ + R ( t, µ )]= e − tA µ P ( t ) e tAµ + R ( t, µ ) , (3.11)14here for almost all t ∈ [0 , , k R ( t, µ ) k M ( C ) ≤ ( k R k M ( C ) + 2 e d k R k M ( C ) ) k P ( t ) k M ( C ) ≤ a ( γ ( µ ) + γ ( t, µ )) k P ( t ) k M ( C ) , with a constant a = a ( d, k σ k L , k σ k L ) . Note that e − tA µ P ( t ) e tAµ = (cid:20) p ( t ) e − iµt p ( t ) e iµt p ( t ) p ( t ) (cid:21) . Furthermore, we have (cid:13)(cid:13)(cid:13)(cid:13)Z x R ( t, µ ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C [0 , ≤ a ( k P k M ( L [0 , γ ( µ ) + k P ( µ )) , where k P ( µ ) := Z k P ( t ) k M ( C ) γ ( t, µ ) dt. (3.12)Using (2.37), we obtain that A ( µ ) D ( x, µ ) = (cid:2) e A µ x + D ( x, µ ) + R ( x, µ ) (cid:3) Z x e − tA µ P ( t ) e tAµ dt + D ( x, µ ) Z x R ( t, µ ) dt = e xA µ Z x e − tA µ P ( t ) e tAµ dt + D ( x, µ ) Z x e − tA µ P ( t ) e tAµ dt + R ( x, µ ) , where R ( x, µ ) = R ( x, µ ) Z x e − tA µ P ( t ) e tAµ dt + D ( x, µ ) Z x R ( t, µ ) dt and k R ( x, µ ) k M ( C [0 , ≤ a (cid:16) k P k M ( L [0 , (cid:0) γ ( µ ) + γ ( µ ) (cid:1) + k P ( µ ) (cid:17) , µ ∈ P d , with a constant a = a ( d, k σ k L , k σ k L ) . In this way, combining the observations following Corollary 3.1 with (3.9),we derive the following assertion yielding sharp asymptotics for ˜ D . Theorem 3.2. If µ ∈ P d , | µ | > be d , then ˜ D ( x, µ ) = R ( x, µ ) + R ( x, µ ) , x ∈ [0 , , where R is given by R ( x, µ ) := e xA µ + D (0) ( x, µ ) + 1 µ e xA µ Z x e − tA µ P ( t ) e tAµ dt + 1 µ D ( x, µ ) Z x e − tA µ P ( t ) e tAµ dt, here R ( x, µ ) = D (1) ( x, µ ) + R ( x, µ ) µ + ˜ D ( x, µ ) , with D (1) given by (2.35) and the reminder satisfies kR ( x, µ ) k M ( C ) ≤ a ( γ ( µ ) + γ ( µ ) + | µ | − k P ( µ ) + | µ | − ) , with a constant a = a ( d, k σ k L , k σ k L , k P k M ( L ) ) . In this section we consider a Sturm–Liouville equation y ′′ ( x ) + q ( x ) y ( x ) + µ y ( x ) = 0 , x ∈ [0 , , (4.1)where the potential q is a complex-valued distribution of the first order, i.e. q = σ ′ , σ ∈ L [0 , , and µ ∈ C is a spectral parameter. Following the regularization method, weintroduce the quasi-derivative of y ∈ W [0 , as y [1] ( x ) := y ′ ( x ) + σ ( x ) y ( x ) , y ′ ( x ) = y [1] ( x ) − σ ( x ) y ( x ) , and rewrite (4.1) as ( y [1] ( x )) ′ − σ ( x ) y [1] ( x ) + σ ( x ) y ( x ) + µ y ( x ) = 0 , x ∈ [0 , . (4.2)We say that y is a solution of (4.2) if y ∈ D := { y ∈ W [0 , , y [1] ( x ) ∈ W [0 , } , and (4.2) is satisfied for a.e. x ∈ [0 , .Note that (4.2) can be written in a matrix form L (cid:18) yy [1] (cid:19) := ddx (cid:18) yy [1] (cid:19) + (cid:20) σ − σ + µ − σ (cid:21) (cid:18) yy [1] (cid:19) = 0 . (4.3)Let us now relate the equation (4.2) to the perturbed system (3.1). For µ = 0 define S ( µ ) := (cid:20) iµ (cid:21) S , S := (cid:20) − (cid:21) , S − ( µ ) = 12 iµ (cid:20) iµ − iµ − (cid:21) . If (cid:18) yy [1] (cid:19) = S ( µ ) (cid:18) v v (cid:19) , (cid:18) v v (cid:19) = S − ( µ ) (cid:18) yy [1] (cid:19) , (4.4)then by (4.3), L (cid:18) yy [1] (cid:19) = S ( µ ) L (cid:18) v v (cid:19) , L (cid:18) v v (cid:19) := (cid:18) v v (cid:19) ′ + 12 iµ (cid:20) µ + σ iµσ + σ iµσ − σ − µ − σ (cid:21) (cid:18) v v (cid:19) . Summarizing, we showed that y is a solution of (4.2) if and only if V = { v , v } T ,defined by (4.4), is a solution of the system V ′ ( x ) + σ ( x ) (cid:20) (cid:21) V ( x ) = iµJ V ( x ) + iσ ( x )2 µ (cid:20) − − (cid:21) V ( x ) , x ∈ [0 , , (4.5)and this is precisely (3.1) with J ( x ) = (cid:20) σ ( x ) σ ( x ) 0 (cid:21) , σ ∈ L [0 , ,P ( x ) = i (cid:20) τ ( x ) τ ( x ) − τ ( x ) − τ ( x ) (cid:21) , τ = σ ∈ L [0 , . (4.6)Note that in this case we have k τ k L = k σ k L .As a result, if y is a solution of (4.2) with the initial conditions y (0) = c , y [1] (0) = c , (4.7)and µ = 0 then we can apply Theorem 3.2 to (4.5). In this case e − tA µ P ( t ) e tAµ dt = i (cid:20) e − iµt − e − iµt (cid:21) , and D ( x, µ ) = Z x (cid:20) e iµ ( x − t ) e − iµ ( x − t ) (cid:21) σ ( t ) dt,D (0) ( x, µ ) = D ( x, µ ) + Z x e ( x − t ) A µ (cid:20) ˜ σ ( x, t ) − ( T σ ˜ σ )( x, t ) − ( T σ ˜ σ )( x, t ) ˜ σ ( x, t ) (cid:21) dt, where ˜ σ ( x, t ) = Z x − t σ ( t + ξ ) σ ( ξ ) dξ,N σ ( x, t ) := ( T σ ˜ σ )( x, t ) = Z x − t σ ( t + ξ )˜ σ ( t + ξ, ξ ) dξ. (4.8)Theorem 3.2 and the transformation (4.4) yield the identity (cid:18) yy [1] (cid:19) = S ( µ ) [ R ( x, µ ) + R ( x, µ )] S − ( µ ) (cid:18) c c (cid:19) . Thus we proved the following statement in which and in what follows we willwrite ρ ( µ ) := γ ( µ ) + γ ( µ ) + | µ | − k σ ( µ )] + | µ | − , (4.9)where all necessary definitions are given by (2.29), (2.30) (2.31) and (3.12) with σ = σ = σ . 17 heorem 4.1. Let d > and Ω d ( σ ) := { µ ∈ P d : | µ | > be d } , b = k σ k L (cid:16) k σ k L + e k σ k L k σ k L (cid:17) . (4.10) If y = y ( x, µ ) , µ ∈ Ω d ( σ ) , is a solution of (4.2) with the initial conditions (4.7) ,then (cid:18) y ( iµ ) − y [1] (cid:19) = W ( x, µ ) ~c ( µ ) + W ( x, µ ) ~c ( µ ) , where ~c ( µ ) = S − ( µ ) (cid:18) c c (cid:19) = 12 (cid:18) c + c iµ c − c iµ (cid:19) , and W ( x, µ ) = (cid:20) e iµx e − iµx e iµx − e − iµx (cid:21) + (cid:20) d ( x, µ ) + d ( x, − µ ) d ( x, µ ) + d ( x, − µ ) d ( x, µ ) − d ( x, − µ ) d ( x, µ ) − d ( x, − µ ) (cid:21) + i µ Z x σ ( t ) (cid:20) e iµx − e − iµx e iµt e iµx e − iµt − e − iµx e iµx + e − iµx e iµt e iµx e − iµt + e − iµx (cid:21) dt + i µ (cid:20) q ( x, − µ ) q ( x, µ ) − q ( x, − µ ) q ( x, µ ) (cid:19) Z x σ ( t ) (cid:18) e − iµt − e iµt − (cid:21) dt,q ( x, µ ) = − Z x e iµ ( x − t ) σ ( t ) dt, d ( x, µ ) = Z x e iµ ( x − t ) ˜ σ ( x, t ) dt,d ( x, µ ) = q ( x, µ ) − Z x e i ( x − t ) N σ ( x, t ) dt, where N σ is defined by (4.8) . Moreover, for µ ∈ Ω d ( σ ) , k W ( x, µ ) k M ( C [0 , ≤ Cρ ( µ ) , (4.11) where a constant C > depends only on d > and k σ k L . If in Theorem 4.1 we set ~c = (0 , T , then using the relations Z x cos(2 µt )˜ σ ( x, t ) dt = Z x Z x cos(2 µ ( t − s )) σ ( s ) σ ( t ) dsdt = (cid:18)Z x cos(2 µt ) σ ( t ) dt (cid:19) + (cid:18)Z x sin(2 µt ) σ ( t ) dt (cid:19) , (4.12) Z x sin(2 µt )˜ σ ( x, t ) dt = Z x Z s σ ( t ) σ ( s ) sin(2 µ ( s − t )) dtds = Z x σ ( t ) cos(2 µt ) dt Z x σ ( s ) sin(2 µs ) ds − Z x Z s σ ( t ) σ ( s ) cos(2 µs ) sin(2 µt ) dtds. (4.13)and simple calculations, we get the next corollary.18 orollary 4.2. If y = y ( x, µ ) is a solution of (4.2) with the initial conditions y (0 , µ ) = 0 , y [1] (0 , µ ) = 1 , (4.14) then for µ ∈ Ω d ( σ ) , given by (4.10) , we have µy ( x, µ ) = sin( µx ) + sin( µx ) Z x cos(2 µt ) σ ( t ) dt − cos( µx ) Z x sin(2 µt ) σ ( t ) dt + 2 cos( µx ) Z x Z t σ ( s ) σ ( t ) cos(2 µt ) sin(2 µs ) dsdt − µ cos( µx ) Z x cos(2 µt ) σ ( t ) dt − µ sin( µx ) Z x sin(2 µt ) σ ( t ) dt + 12 µ cos( µx ) Z x σ ( t ) dt + Z x sin( µ ( x − t )) N σ ( x, t ) dt − cos( µx ) Z x σ ( t ) sin(2 µt ) dt Z x σ ( s ) cos(2 µs ) ds + 12 sin( µx ) (cid:18)Z x cos(2 µt ) σ ( t ) dt (cid:19) + (cid:18)Z x sin(2 µt ) σ ( t ) dt (cid:19) ! − µ Z x σ ( t ) dt Z x cos(2 µ ( x − s )) σ ( s ) ds + 12 µ cos( µx ) Z x Z x cos(2 µ ( t − s )) σ ( t ) σ ( s ) dtds − µ sin( µx ) Z x Z x sin(2 µ ( t − s )) σ ( t ) σ ( s ) dtds + φ ( x, µ ) , (4.15) y [1] ( x, µ ) = cos( µx ) + cos( µx ) Z x cos(2 µt ) σ ( t ) dt + sin( µx ) Z x sin(2 µt ) σ ( t ) dt − µx ) Z x Z t σ ( s ) σ ( t ) cos(2 µt ) sin(2 µs ) dsdt + 12 µ sin( µx ) Z x cos(2 µt ) σ ( t ) dt − µ cos( µx ) Z x sin(2 µt ) σ ( t ) dt − µ sin( µx ) Z x σ ( t ) dt + Z x cos( µ ( x − t )) N σ ( x, t ) dt + sin( µx ) Z x σ ( t ) sin(2 µt ) dt Z x σ ( s ) cos(2 µs ) ds + 12 cos( µx ) (cid:18)Z x cos(2 µt ) σ ( t ) dt (cid:19) + (cid:18)Z x sin(2 µt ) σ ( t ) dt (cid:19) ! + 12 µ Z x σ ( t ) dt Z x sin( µ ( x − s )) σ ( s ) ds − µ sin( µx ) Z x Z x cos(2 µ ( t − s )) σ ( t ) σ ( s ) dtds − µ cos( µx ) Z x Z x sin(2 µ ( t − s )) σ ( t ) σ ( s ) dtds + φ ( x, µ ) . (4.16) where N σ is given by (4.8) and k φ j ( x, µ ) k M ( C [0 , ≤ Cρ ( µ ) , j = 1 , , (4.17)19 nd ρ is defined by (4.9) . In this section we apply Corollary 4.2 to the spectral problem y ′′ ( x ) + q ( x ) y ( x ) + λy ( x ) = 0 , x ∈ [0 , , (5.1) y (0) = 0 , y (1) = 0 . (5.2)where q = σ ′ and σ ∈ L [0 , .Recall that basic facts on asymptotic of eigenvalues and eigenfunctions of theproblem (5.1)-(5.2) can be found in [18] and [19]. For the reader’s convenienceand for the matter of comparison with our results, we formulate some of thembelow. Theorem 5.1. If ( λ n ) n ≥ are the eigenvalues of the spectral problem (5.1) - (5.2) , then λ n = µ n , µ n = πn + µ ,n + r n , n ∈ N , (5.3) where µ ,n := Z sin(2 πnt ) σ ( t ) dt − Z Z t σ ( t ) σ ( s ) sin(2 πns ) cos(2 πnt ) dsdt, − πn Z (1 − cos(2 πnt )) σ ( t ) dt, (5.4) and ( r n ) n ≥ ∈ l . The eigenfunctions ( y n ) n ≥ of the spectral problem (5.1) - (5.2) satisfy πny n ( x ) = y ,n ( x ) + ψ ,n ( x ) , (5.5) y ,n ( x ) = sin( πnx ) Z x cos(2 πns ) σ ( s ) ds − πn Z x sin(2 πnt ) σ ( t ) dt ! + cos( πnx ) µ ,n x − Z x sin(2 πns ) σ ( s ) ds + 12 πn Z x (1 − cos(2 πnt )) σ ( t ) dt + 2 Z x Z t σ ( t ) σ ( s ) sin(2 πns ) cos(2 πnt ) dsdt ! , (5.6) and y [1] n ( x ) = y [1]0 ,n ( x ) + ψ ,n ( x ) , (5.7) y [1]0 ,n ( x ) = cos( πnx ) Z x cos(2 πns ) σ ( s ) ds − πn Z x sin(2 πnt ) σ ( t ) dt ! − sin( πnx ) µ ,n x − Z x sin(2 πns ) σ ( s ) ds + 12 πn Z x (1 − cos(2 πnt ) σ ( t ) dt + 2 Z x Z t σ ( t ) σ ( s ) sin(2 πns ) cos(2 πnt ) dsdt ! , (5.8)20 here sup x ∈ [0 , ∞ X n =1 | ψ j,n ( x ) | < ∞ , j = 1 , . Our main aim in this section is to provide sharp asymptotic formulas for ( y n ) n ≥ . Theorem 5.2.
Let µ ,n be defined by (5.4) and let y ,n ( x ) , y [1]0 ,n ( x ) , n ≥ , be given by (5.6) , (5.8) . The eigenfunctions of the spectral problem (5.1) - (5.2) admit the representation πny n ( x ) = y ,n ( x ) + y ,n ( x ) + ˜ ψ ,n ( x ) ,y [1] n ( x ) = y [1]0 ,n ( x ) + y [1]1 ,n ( x ) + ˜ ψ ,n ( x ) , where y ,n ( x ) = sin( πnx ) A n ( x ) + cos( πnx ) B n ( x ) (5.9) y [1]1 ,n ( x ) = cos( πnx ) A n ( x ) − sin( πnx ) B n ( x ) (5.10) and A n ( x ) = 12 (cid:18)Z x cos(2 πnt ) σ ( t ) dt (cid:19) + 12 (cid:18)Z x sin(2 πnt ) σ ( t ) dt (cid:19) + Z x cos(2 πnt ) N σ ( x, t ) dt − µ ,n Z x sin(2 πnt ) σ ( t ) tdt + Z x sin(2 πnt ) N σ ( x, t ) tdt ! + µ ,n x Z x sin(2 πnt ) σ ( t ) dt − Z x Z t σ ( t ) σ ( s ) sin(2 πns ) cos(2 πnt ) dsdt + Z x sin(2 πns ) σ ( s ) ds Z x cos(2 πnt ) σ ( t ) dt + Z x sin(2 πnt ) N σ ( x, t ) dt ! ,B n ( x ) = µ ,n x Z x cos(2 πnt ) σ ( t ) dt + 12 (cid:18)Z x cos(2 πnt ) σ ( t ) dt (cid:19) + 12 (cid:18)Z x sin(2 πnt ) σ ( t ) dt (cid:19) + Z x cos(2 πnt ) N σ ( x, t ) dt ! − Z x sin(2 πns ) σ ( s ) ds Z x cos(2 πnt ) σ ( t ) dt − Z x sin(2 πnt ) N σ ( x, t ) dt + µ ,n Z x cos(2 πnt ) σ ( t ) tdt + 2 Z x Z x cos (cid:0) πn ( s − t ) (cid:1) σ ( s ) sσ ( t ) dsdt − Z x cos(2 πnt ) N σ ( x, t ) tdt ! , here N σ is given by (4.8) and ∞ X n =1 sup x ∈ [0 , | ˜ ψ j,n ( x ) | < ∞ , j = 1 , . Proof.
Using the relations sin(2 tµ n ) − sin(2 πnt ) = 2 µ ,n t cos(2 πnt ) + s n ( t ) , and cos(2 tµ n ) − cos(2 πnt ) = − µ ,n t sin(2 πnt ) + c n ( t ) , where ∞ X n =1 sup t ∈ [0 , ( | s n ( t ) | + | c n ( t ) | ) < ∞ , it suffices to insert the formulas (5.3) and (5.4) into the identities (4.15) and(4.16), and to finish the proof by showing that ∞ X n =1 ρ ( µ n ) < ∞ , (5.11)where ρ is given by (4.9). In what follows we will use a basic formula foreigenvalues µ n = πn + ˜ µ n , where (˜ µ n ) ∈ l . To prove (5.11), using Parseval’s identity and a simple inequal-ity | e iz − | ≤ | z | e d , z ∈ P d , we infer that ∞ X n =1 (cid:12)(cid:12)(cid:12) Z x e ± iµ n t σ ( t ) dt (cid:12)(cid:12)(cid:12) ≤ ∞ X n =1 (cid:12)(cid:12)(cid:12)(cid:12)Z x e ± πint σ ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) + 2 ∞ X n =1 (cid:18)Z x | e i ˜ µ n t − || σ ( t ) | dt (cid:19) ≤ k σ k L [0 ,x ] + 4 e d k σ k L [0 ,x ] ∞ X n =1 | ˜ µ n | ≤ m < ∞ , (5.12)for any x ∈ [0 , , where m := 2(1 + 2 e d k ˜ µ n k l ) k σ k L . It follows from (5.12)that ∞ X n =1 γ ( x, µ ) ≤ m, x ∈ [0 , (5.13)22inally, by (5.13), we have ∞ X n =1 ρ ( µ n ) ≤ Z (1 + | σ ( s ) | ) ∞ X n =1 γ ( s, µ n ) ! ds + 2 Z | σ ( s ) | ∞ X n =1 | µ n | − | γ ( s, µ n ) ! ds + ∞ X n =1 | µ n | − ≤ Z (1 + | σ ( s ) | + k σ k L ) ∞ X n =1 γ ( s, µ n ) ! ds + (1 + k σ k L ) ∞ X n =1 | µ n | − < ∞ , thus (5.11) follows.Note that, in the same way, we can study a more general Sturm–Liouvilleequation ( p ( x ) y ′ ( x )) ′ + q ( x ) y ( x ) + µ p ( x ) y ( x ) = 0 , x ∈ [0 , , (5.14)where q = u ′ , u ∈ L [0 , , and the coefficient p is such that p ∈ W [0 , , p ( x ) > , x ∈ [0 , . Indeed, rewrite (5.14) as ( y [1] ( x )) ′ − u ( x ) p ( x ) y [1] ( x ) + u ( x ) p ( x ) y ( x ) + µ p ( x ) y ( x ) = 0 , x ∈ [0 , , (5.15)where y [1] ( x ) := p ( x ) y ′ ( x ) + u ( x ) y ( x ) , y ′ ( x ) = y [1] ( x ) − u ( x ) y ( x ) p ( x ) . We say that y is a solution of (5.15) if y ∈ D := { y ∈ W [0 ,
1] : y [1] ∈ W [0 , } and (5.15) is satisfied for a.e. x ∈ [0 , . We will show that (5.15) can betransformed into the perturbed Dirac system (1.1) with appropriate unknownfunctions v , v and coefficients J ∈ M ( L [0 , and P ∈ M ( L [0 , .The matrix form of (5.15) is ˜ L (cid:18) yy [1] (cid:19) := ddx (cid:18) yy [1] (cid:19) + M ( x, µ ) (cid:18) yy [1] (cid:19) = 0 , (5.16) M ( x, µ ) := 1 p (cid:18) u − u + µ p − u (cid:19) . (5.17)Define w ( x ) := u ( x ) + p ( x ) g ( x ) ∈ L [0 , , g ( x ) := Z x u ( t ) p ′ ( t ) p ( t ) dt ∈ W [0 , , S ( x, µ ) = 1 √ p (cid:18) iµ − g ) p − ( iµ + g ) p (cid:19) , ( ˜ S ( x, µ )) − = 12 iµ √ p (cid:18) ( iµ + g ) p iµ − g ) p − (cid:19) , µ = 0 . Since √ p ) ′ = p ′ / √ p, g ′ = up ′ /p , we have ddx ˜ S ( x, µ ) = p ′ p √ p (cid:18) − − iµ − g ) p/ − u − ( iµ + g ) p/ − u (cid:19) . For µ = 0 and y ∈ D , we introduce a transformation y
7→ { v , v } T by Y = ˜ S ( x, µ ) V, Y := (cid:18) yy [1] (cid:19) , V := (cid:18) v v (cid:19) . Then straightforward calculations reveal that ˜ LY = ˜ S ( µ ) ˜ L V ˜ L V := V ′ + M V, where M = ˜ S − ( ˜ S ′ + M ˜ S ) = − iµJ − σ ( x ) J − iτ ( x )2 µ S, x ∈ [0 , . and σ ( x ) = 2 w − p ′ p ∈ L [0 , , τ ( x ) = w ( p ′ − w ) p ∈ L [0 , . (5.18)Summarizing the above, y ∈ D is a solution of (5.14) if and only if V = { v , v } T is a solution of (1.1) with coefficients σ and τ defined by (5.18). Thenone can derive asymptotic formulas for the fundamental system of solutions to(5.14) (as µ → ∞ , | Im µ | ≤ c ). We do not give any details here since they arequite technical. Remark 2.
We can also consider the equation ( a ( x ) y ′ ( x )) ′ + q ( x ) y ( x ) + µ c ( x ) y ( x ) = 0 , x ∈ [0 , , (5.19) where q = u ′ , u ∈ L [0 , , and the coefficients a , c ar such that a ∈ W [0 , , c ∈ W [0 , , a ( x ) > , c ( x ) > , x ∈ [0 , . In this setting, our results are also applicable, however one needs to employ anadditional change of variables.Let t = 1 d Z x c / ( s ) a / ( s ) ds ∈ [0 , , d = Z c / ( s ) a / ( s ) ds, z ( t ) = y ( x ( t )) , t ∈ [0 , . Then from (5.19) it follows that ( p ( t ) z ′ ( t )) ′ + q ( t ) z ( t ) + ˜ µ p ( t ) z ( t ) = 0 , t ∈ [0 , , where ˜ µ = dµ, p ( t ) = p a ( x ( t )) c ( x ( t )) ∈ W [0 , , q ( t ) = d p a ( x ( t )) p c ( x ( t )) q ( x ( t )) , and q ( t ) = u ′ ( t ) , u ( t ) = du ( x ( t )) ∈ L [0 , . This is exactly the situation described above. Appendix
In this section we include series of results related to the operator ˜ T , which isused in Section 2, and it is defined by (2.23). These inequalities are crucial inthe proofs concerning the asymptotic behaviour of the solutions to (1.1). Proposition 6.1. If σ j ∈ L [0 , , and F ∈ M ( C (∆)) , then Z x e − iµt ( ˜ T F )( x, t ) dt = − Z x e − iµs J ( s ) Z s e iµξ F ( s, ξ ) dξds. (6.1) Moreover, Z x e − iµt ( ˜ T ˜ J )( x, t ) dt = − Z x e iµy Z xy J ( z ) e − iµz dz Z y J T ( τ ) e − iµτ dτ ! J T ( y ) dy. (6.2) Proof.
Indeed, we have Z x e − iµt ( ˜ T F )( x, t ) dt = − Z x e − iµt Z xt J ( s ) F ( s, s − t ) dsdt = − Z x J ( s ) Z s e − iµt F ( s, s − t ) dtds = − Z x e − iµs J ( s ) Z s e iµξ F ( s, ξ ) dξ ds. and (6.1) is true. Moreover, Z x e − iµt ( ˜ T ˜ J )( x, t ) dt = − Z x e − iµt Z x − t J ( t + ξ ) Z t J T ( τ ) J T ( ξ + τ ) dτ dξdt = − Z x Z x − ξ e − iµt J ( t + ξ ) Z t J T ( τ ) J T ( ξ + τ ) dτ dtdξ = − Z x Z x − ξ Z xξ + τ e − iµz e iµξ J ( z ) J T ( τ ) J T ( ξ + τ ) dzdτ dξ = − Z x Z xτ Z xy e − iµz e iµy e − iµτ J ( z ) J T ( τ ) J T ( y ) dzdydτ = − Z x e iµy Z xy J ( z ) e − iµz dz Z y J T ( τ ) e − iµτ dτ ! J T ( y ) dy, where we first changed the order of integration, then passed to new variables z = t + ξ and y = τ + ξ and changed the order of integration again.Note that if F ∈ M ( C (∆)) then (cid:13)(cid:13)(cid:13)(cid:13)Z x e − tA µ F ( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z x e − iµt F ( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) + (cid:13)(cid:13)(cid:13)(cid:13)Z x e iµt F ( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) , (6.3)for every x ∈ [0 , . 25 emma 6.2. If µ ∈ P d , then (cid:13)(cid:13)(cid:13)(cid:13)Z x e − tA µ ˜ J ( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C [0 , ≤ e d ˜ l γ ( µ ) , (6.4) (cid:13)(cid:13)(cid:13)(cid:13)Z x e − tA µ ( ˜ T ˜ J )( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C [0 , ≤ e d ˜ l γ ( µ ) , (6.5) (cid:13)(cid:13)(cid:13)(cid:13)Z x e − tA µ ( ˜ T ˜ J )( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ≤ l + 1) e d (cid:16) γ ( µ ) γ ( x, µ )+ γ ( µ ) (cid:17) , x ∈ [0 , , (6.6) (cid:13)(cid:13)(cid:13)(cid:13)Z x e − tA µ ( ˜ T n ˜ J )( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C [0 , ≤ e nd l n − ( n − γ ( µ ) , n ≥ . (6.7) Proof.
Observe that (cid:13)(cid:13)(cid:13)(cid:13)Z x e − iµt ˜ J ( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) = (cid:13)(cid:13)(cid:13)(cid:13)Z x J ( s ) e − iµs Z s J ( ξ ) e iµξ dξds (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) = (cid:12)(cid:12)(cid:12)(cid:12)Z x e − iµs σ ( s ) Z s e iµξ σ ( ξ ) dξds (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z x e − iµs σ ( s ) Z s e iµξ σ ( ξ ) dξds (cid:12)(cid:12)(cid:12)(cid:12) ≤ e d n k σ k L (cid:13)(cid:13)(cid:13) Z s e iµξ σ ( ξ ) dξ (cid:13)(cid:13)(cid:13) L + k σ k L (cid:13)(cid:13)(cid:13) Z s e iµξ σ ( ξ ) dξ (cid:13)(cid:13)(cid:13) L o ≤ e d max {k σ k L , k σ k L } γ ( µ ) , x ∈ [0 , . Hence, by (6.3), the estimate (6.4) follows.Next, in view of (6.1), if µ ∈ P d , x ∈ [0 , and F ∈ M ( C (∆)) , then (cid:13)(cid:13)(cid:13)(cid:13)Z x e − iµt ( ˜ T F )( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ≤ e d Z x (cid:13)(cid:13)(cid:13)(cid:13) J ( s ) Z s e iµξ F ( s, ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ds, (6.8)and (cid:13)(cid:13)(cid:13)(cid:13)Z x e − iµt ( ˜ T F )( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C [0 , ≤ e d l (cid:13)(cid:13)(cid:13)(cid:13)Z s e iµξ F ( s, ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) M ( C [0 , . (6.9)Using (6.9) and (6.4), we obtain that (cid:13)(cid:13)(cid:13)(cid:13)Z x e − iµt ( ˜ T ˜ J )( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C [0 , ≤ e d l (cid:13)(cid:13)(cid:13)(cid:13)Z s e iµξ ˜ J ( s, ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) M ( C [0 , ≤ e d ˜ l γ ( µ ) , thus, by (6.3), the estimate (6.5) holds.26y (6.3) and the estimate (cid:12)(cid:12)(cid:12)(cid:12)Z x σ ( s ) γ ( y, µ ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ k σ k L k γ ( y, µ ) k L ≤ ( l + 1) γ ( µ ) , the inequality (6.6) holds if (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z x e − iµt ( ˜ T ˜ J )( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ≤ e d (cid:18) γ ( µ ) + γ ( x, µ ) Z x σ ( s ) γ ( y, µ ) dy (cid:19) . (6.10)On the other hand, using (6.2), we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z x e − iµt ( ˜ T ˜ J )( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ≤ e d Z x (cid:13)(cid:13)(cid:13)(cid:13)Z xy e − iµz J ( z ) dz Z y e − iµτ J T ( τ ) dτ J T ( y ) (cid:13)(cid:13)(cid:13)(cid:13) M (C) dy ≤ e d Z x σ ( y ) (cid:13)(cid:13)(cid:13)(cid:13)Z xy e − iµz J ( z ) dz (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) (cid:13)(cid:13)(cid:13)(cid:13)Z y e − iµτ J ( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) dy ≤ e d Z x σ ( y ) (cid:13)(cid:13)(cid:13)(cid:13)Z y e − iµτ J ( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) dy + e d (cid:13)(cid:13)(cid:13)(cid:13)Z x e − iµz J ( z ) dz (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) Z x σ ( s ) (cid:13)(cid:13)(cid:13)(cid:13)Z y e − iµτ J ( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) dy ≤ e d γ ( µ ) + e d γ ( x, µ ) Z x σ ( s ) γ ( y, µ ) dy, and (6.10) follows.To prove (6.7) it suffices to show that for all n ≥ and any x ∈ [0 , , (cid:13)(cid:13)(cid:13)(cid:13)Z x e − iµt ( ˜ T n ˜ J )( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ≤ e nd ( n − (cid:18)Z x σ ( s ) ds (cid:19) n − γ ( µ ) . (6.11)We prove (6.11) by induction. Using (6.8) for F = ˜ T ˜ J and (6.10), we note that (cid:13)(cid:13)(cid:13) Z x e − iµt ( ˜ T ˜ J )( x, t ) dt (cid:13)(cid:13)(cid:13) M ( C ) ≤ e d Z x σ ( s ) (cid:13)(cid:13)(cid:13)(cid:13)Z s e iµξ ( ˜ T ˜ J )( s, ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ds ≤ e d Z x | σ ( s ) (cid:18) γ ( s, µ ) Z s σ ( y ) γ ( y, µ ) dy + γ ( µ ) (cid:19) ds ≤ e d Z x σ ( s ) γ ( s, µ ) Z s σ ( y ) γ ( y, µ ) dy ds + e d l γ ( µ ) ≤ e d (cid:0)R x σ ( s ) γ ( s, µ ) ds (cid:1) e d l γ ( µ ) ≤ e d l γ ( µ ) + e d l γ ( µ ) . Therefore, (6.11) holds for n = 2 . 27uppose now that (6.11) is true for some n ≥ . Then, once again using(6.8), we have (cid:13)(cid:13)(cid:13) Z x e − iµt ( ˜ T n +1 ˜ J )( x, t ) dt (cid:13)(cid:13)(cid:13) M ( C ) ≤ e d Z x σ ( s ) (cid:13)(cid:13)(cid:13)(cid:13)Z s e iµξ ( ˜ T n ˜ J )( s, ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ds ≤ e n +1) d ( n − γ ( µ ) Z x σ ( s ) (cid:18)Z s σ ( τ ) dτ (cid:19) n − ds = e n +1) d γ ( µ )( n − (cid:18)Z x σ ( τ ) dτ (cid:19) n − , x ∈ [0 , , so (6.11) holds also for n + 1 , and the proof of (6.11) is finished. References [1] S. Albeverio, R. Hryniv, and Y. Mykytyuk. Inverse spectral problemsfor Dirac operators with summable potentials.
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