Asymptotic behavior of solutions of the Dirac system with an integrable potential
aa r X i v : . [ m a t h . SP ] N ov Asymptotic behavior of solutions of the Diracsystem with an integrable potential
August 26, 2020
Łukasz Rzepnicki
Faculty of Mathematics and Computer ScienceNicolaus Copernicus Universityul. Chopina 12/18, 87-100 ToruńPoland [email protected]
Abstract
We consider the Dirac system on the interval [0 , with a spectral pa-rameter µ ∈ C and a complex-valued potential with entries from L p [0 , ,where ≤ p < . We study the asymptotic behavior of its solutions ina stripe | Im µ | ≤ d for µ → ∞ . These results allows us to obtain sharpasymptotic formulas for eigenvalues and eigenfunctions of Sturm–Liouvilleoperators associated with the aforementioned Dirac system. keywords: Dirac system, spectral problem, integrable potential, Sturm–Liouville operator
MSC [2010] Primary 34L20, Secondary 34E05
Acknowledgement
The author was supported by NCN grant no. UMO-2017/27/B/ST1/00078.
Let consider for x ∈ [0 , , a Cauchy problem D ′ ( x ) + J ( x ) D ( x ) = A µ D ( x ) , D (0) = I, (1.1)where A µ = iµJ , and 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 for j = 1 , complex-valued functions σ j belong to L p [0 , , where ≤ p < . We study the asymptotic behavior of itssolutions D ( x ) = D ( x, µ ) with respect to µ from a horizontal stripe P d := { µ ∈ C : | Im µ | ≤ d } . µ → ∞ .The solution of (1.1), is a matrix D with entries from the space of absolutelycontinuous on [0 , functions (i.e. from the Sobolev space W [0 , ) satisfying(1.1) for a.e. x ∈ [0 , . In our case, this conditions together with the equationyield that D has entries from W p [0 , .This article is an addendum to the paper [7], where the problem (1.1) wasanalyzed for σ j ∈ L [0 , , j = 1 , . In that text one can find background forDirac systems and their connection with Sturm–Liouville problems.We relay here on the same method as in [7] to use all of its advantages andobtain sharp asymptotic formulas for D and consequently for spectral problemsassociated with (1.1). In the case when σ j ∈ L p [0 , , j = 1 , , p > one canuse the results from [7] due to the obvious embedding between L p [0 , spaces.Thus, in this text we restrict ourselves only to ≤ p < .We are interested in the following spectral problem: Y ′ ( x ) + J ( x ) Y ( x ) = A µ Y ( x ) , x ∈ [0 , , (1.3)where Y = [ y , y ] T and y (0) = y (0) , y (1) = y (1) . (1.4)Conditions (1.4) are an example of strongly regular boundary conditions.The Dirac-type systems or equation (1.3) with a general formulation of regular or strongly regular conditions have been studied recently in many papers anddifferent method.In [15] A. M. Savchuk and A. A. Shkalikov derived for p ≥ basic asymptoticformulas for eigenvalues and for fundamental solutions of the Dirac-type systemonly with the leading term and the reminders expressed by γ and γ given by(2.22) and (2.23). They obtained their results applying Prüfer’s substitution.Their result is equivalent to first thesis (2.31) of corollary 2.3. Note that nextstatement (2.32) is a significant extension of the previous result. Its version for p = 1 may be found in remark 2.5. The most general result is the content oflemma 2.2.Using our method it is also possible to obtain very detailed formulas foreigenvalues and eigenfunctions. In case of the spectral problem associated with(1.4) the eigenvalues admit the representation (3.15)-(3.16) with remainderssatisfying (3.17) and (3.18) for p = 1 and < p < respectively. In literature(for instance in [15]) for < p < one may found results which state thateigenvalues are of the form πn + r n , where ( r n ) ∈ l q , and q is conjugated to p . Here it is worth to underline that beside the leading term in our asymptoticformulas there occur Fourier coefficients of known functions and the reminder,which belongs to l q/ . Additionally, for p = 1 we extend known formulas with | r n | < c Γ( πn ) (where Γ is defined in (2.25)) into more detailed one with theremainder satisfying | r n | < c Γ ( πn ) .In the same spirit theorem 3.3 and corollary 3.4 related to eigenfunctionsgeneralize significantly those from literature.Our method is applicable not only to the spectral problem (1.3)-(1.4) but itworks as well for different cases of strongly regular boundary conditions. Whatis more it may be used to deal with the class of regular boundary conditions (inthe sense of Birkhoff). 2he articles of A. M. Savchuk and I. V. Sadovnichaya: [11], [12], [13] and[14] may be regarded as a continuation of method from [15] and its applicationfor p = 1 to problems from the fields of asymptotics formulas and basis proper-ties. Almost all aforementioned works prove or use the same type of results asmentioned before since they deal with the Riesz basis property and very detailedformulas are not needed.The same aims had M. M. Malamud, A. V. Agibalova and L. L. Oridorogain [1] for p = 2 and latter two in [8] for p = 1 . Here the authors used the methodof transformation operators.Whereas in order to study inverse spectral problems S. Albeverio, R. Hrynivand Y. Mykytyuk in [2] investigated a direct spectral problem for the Diracsystem in the form 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 j ∈ L p [0 , , j = 1 , , with p ≥ . They proved also short formulas for fundamental system of solu-tions, where reminders were expressed in terms of Fourier coefficients for un-known functions from L p . Furthermore, for the operators associated with thesystem (1.5) with two kinds of conditions z j (1) = z (0) = 0 , j = 1 , , (1.6)they presented basic formulas for eigenvalues with the same type of reminders.That class of results can be directly derived from our approach with the help oftransformation Z = U Y , where U = (cid:20) − i − i (cid:21) . It leads to the system (1.1) with σ = q + iq and σ = q − iq with appropriateconditions. The relation between different formulations of Dirac systems isexplained deeper in [7].More results concerning different type of problems for the Dirac system withmay be found in the series of paper of P. Djakov and B. Mityagin: [4], [5] and[6] or D. V. Puyda [10].We start with the section concerning asymptotic behavior for solutions ofDirac system. Next, in section 3 we apply these results to the aforementionedspectral problem. For the clarity of exposition some technical results are placedat the end in appendix. In this section we study the matrix Cauchy problem (1.1) and the behavior of itssolution in a special integral form. The idea of this approach was taken from [9,Ch. 1, § ] and developed in [7]. We follow it here directly for similar operatorsbut in different function spaces. 3irst, we introduce a necessary notation. We use throughout the text astandard symbol L p [0 , , p ≥ to denote the space of measurable complexfunctions integrable with p -th power with the classical norm k f k L p = (cid:16) Z | f ( x ) | p dx (cid:17) /p . We write l p , p ≥ for the space of complex sequences summable with p -thpower and endowed with the norm k ( x n ) k p = (cid:16) ∞ X n =1 | x n | p (cid:17) /p .W p [0 , is a standard Sobolev space with the derivative in L p [0 , .If X is a Banach space, then M ( X ) stands for the Banach space of × matrices with entries from X and the norm k Q k M ( B ) := X k,j =1 k Q jk k B , Q = [ Q jk ] j,k =1 . We assume throughout the text that ≤ p < . Morever if < p < ,then let q and p be conjugate exponents and r be the number from Young’sconvolution inequality ie. p + 1 q = 1 and r = 22 − p . (2.1)If p = 1 , then q = ∞ and r = 1 . Let ∆ := { ( x, t ) ∈ R : 0 ≤ t ≤ x ≤ } (2.2)and B := { f : [0 , × [0 , → C a.e. : ∀ x ∈ [0 , f ( x, · ) ∈ C ([0 , , L r ) , supp f ⊂ ∆) } . We equip B with the norm k f k B := sup x ∈ [0 , k f ( x, · ) k L r [0 ,x ] , so that B is a Banach space. In particular, directly from the definition if f ∈ B ,then f ( x, t ) = 0 for ≤ x < t ≤ . This comment allows us to underline theproperty which will be used in the text i.e. for f ∈ B there holds Z x f ( x, t ) dt = Z f ( x, t ) dt ∈ C [0 , . (2.3)We will use the series of constants connected with functions σ j , j = 1 , inour estimations: a := max {k σ k L , k σ k L } , a := k σ k L · k σ k L , a := k σ k L + k σ k L , (2.4)4nd ˜ a := max {k σ k L p , k σ k L p } , ˜ a := k σ k L p · k σ k L p , a := k σ k L p + k σ k L p . (2.5)Moreover, let σ ( x ) := | σ ( x ) | + | σ ( x ) | ∈ L p [0 , . (2.6)Now we are ready to establish a first crucial property of the solutions to(1.1). The proof of the following lemma relays on technical results related tocertain integral operators, which are placed in appendix. Lemma 2.1.
Let σ ∈ L p [0 , , ≤ p < .a) The unique solution D = D ( x, µ ) of Cauchy problem (1.1) can be repre-sented as D ( x, µ ) = e xA µ + Z x e ( x − t ) A µ [ J ( t ) + Q ( x, t )] dt, (2.7) where Q ∈ M ( B ) 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, (2.8) with ˜ J ∈ M ( B ) is given by ˜ J ( x, t ) := Z x − t J ( t + ξ ) J ( ξ ) dξ = Z xt J ( s ) J ( s − t ) ds, ( x, t ) ∈ ∆ . (2.9) b) The following estimates hold: k Q k M ( B ) ≤ c, k D k M ( C [0 , ≤ c, µ ∈ P d (2.10) with certain constants c = c ( d, σ , σ ) .Proof. Note that the uniqueness of solutions comes from general results onSturm–Liouville equations (for instance [16, Thm. 1.2.1]). We look for solu-tions of (1.1) in a special form D ( x, µ ) = e xA µ U ( x, µ ) , U (0 , µ ) = I. (2.11)The identity J ( x ) e xA µ = e − xA µ J ( x ) , a. e. x ∈ [0 , (2.12)yield that U satisfies the Cauchy problem U ′ ( x, µ ) + e − xA µ J ( x ) U ( x, µ ) = 0 , x ∈ [0 , , U (0 , µ ) = I, and this is equivalent to the integral equation U ( x, µ ) = I − Z x e − tA µ J ( t ) U ( t, µ ) dt, x ∈ [0 , . (2.13)We will seek for solutions of (2.13) in the form U ( x, µ ) = I + Z x e − tA µ Q ( x, t ) dt, (2.14)5here Q ∈ M ( B ) does not depend on µ . Inserting (2.14) into (2.13), 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. Due to the fact J = I, J J ( x ) + J ( x ) J = 0 , a.e. x ∈ [0 , , (2.15)we get 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, thus 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 , . We conclude that U is a solution of (2.13) if and only if Q ∈ M ( B ) is a solution of Q ( x, t ) = − J ( t ) − Z x − t J ( t + ξ ) Q ( t + ξ, ξ ) dξ. (2.16)Next, setting Q ( x, t ) = − J ( t ) + Q ( x, t ) , ( x, t ) ∈ ∆ , and using (2.8), we infer that Q satisfies (2.16). For Q the equation (2.16) canbe rewritten in an operator form Q = ˜ J + ˜ T Q, ˜ T = − (cid:20) T σ T σ (cid:21) , for the operators T σ and T σ , defined on B 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, (2.17)where σ ∈ L p [0 , .Observe 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. (2.18)According to lemma 4.1 ˜ J ∈ M ( B ) . What is more, the operators T σ are linearand bounded on B due to lemma 4.3. In particular, we have k ˜ T F k M ( B ) ≤ a k F k M ( B ) , F ∈ M ( B ) . ˜ T n = (cid:20) T n T n (cid:21) , n ∈ N , for bounded linear operators T and T on B given by T := T σ T σ , T := T σ T σ . Therefore by (4.4), we derive k ˜ T n F k M ( B ) ≤ a n n ! k F k M ( B ) , F ∈ M ( B ) . We thus see that (2.8) has a unique solution Q ∈ M ( B ) of the form Q = ∞ X n =0 ˜ T n ˜ J = ∞ X n =0 ˜ T n ( I + ˜ T ) ˜ J, (2.19)and moreover k Q k M ( B ) ≤ (1 + a ) e a k ˜ J k M ( B ) . (2.20)Then (2.19) and (4.1) imply (2.10).Note that from (2.7) and (2.3) we have D ∈ C [0 , . Adding together of (2.7)and (2.10), we obtain k D k M ( C [0 , ≤ e d (cid:16) a + k Q ( x, t ) k M ( B ) (cid:1) , µ ∈ P d . (2.21)We now proceed to derivation of asymptotic formulas for D with the use ofthe previous lemma. In what follows we will use different types of estimates forreminders. For fixed σ j ∈ L p , p ≥ , 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 q + (cid:13)(cid:13)(cid:13) Z x e iµt σ j ( t ) dt (cid:13)(cid:13)(cid:13) L q (cid:19) , (2.22)where /q + 1 /p = 1 . We will need also 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 , (2.23)and γ ( µ ) := Z σ ( s ) γ ( s, µ ) ds, γ ( µ ) := l γ ( µ ) + l γ ( µ ) . (2.24) Γ( µ ) := X j =1 sup x ∈ [0 , (cid:12)(cid:12)(cid:12) Z x e − iµt σ j ( t ) dt (cid:12)(cid:12)(cid:12) + sup x ∈ [0 , (cid:12)(cid:12)(cid:12) Z x e iµt σ j ( t ) dt (cid:12)(cid:12)(cid:12)! . (2.25)Note that Γ is nothing else than γ for p = 1 and q = ∞ .7t is easy to see that if µ ∈ P d then γ ( x, µ ) ≤ e d a , k γ ( x, µ ) k L q ≤ γ ( µ ) , γ ( µ ) ≤ e d a , x ∈ [0 , , (2.26)and γ ( µ ) ≤ e d a ( a + a ) , γ ( µ ) ≤ a e d ( a + 2 a e d k σ k L p ) γ ( µ ) , (2.27)In the following lemma we will need N ( x, t ) := ( ˜ J + ˜ T ˜ J )( x, t ) ∈ B. (2.28)Observe that the explicit form of N is N ( x, t ) = (cid:18) ˜ σ ( x, t ) − ( T σ ˜ σ )( x, t ) − ( T σ ˜ σ )( x, t ) ˜ σ ( x, t ) (cid:19) . The very basic but crucial result use mainly the description of some integralsconnected with the operator ˜ T and its powers stated in lemma 4.6. Lemma 2.2.
Let σ j ∈ L p , ≤ p < for j = 1 , . If D ( x, µ ) is a solution of (1.1) then D ( x, µ ) = e xA µ + D (0) ( x, µ ) + D (1) ( x, µ ) , (2.29) 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 , ≤ cγ ( µ ) , where c = c ( d, σ , σ ) .Proof. Going back to the formulas (2.7) and (2.19) 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 + D (1) ( x, µ ) (2.30)where D (1) ( x, µ ) = Z x e ( x − t ) A µ ∞ X n =2 ( ˜ T n ˜ J )( x, t ) dt. Using (2.30) and the inequality (4.12) proved in 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 a n − ( n − e d exp ( e d a ) γ ( µ ) , for all x ∈ [0 , and µ ∈ P d . 8he above lemma leads to sharp asymptotic formulas for D , which are themain result of this section. Corollary 2.3.
For every d > there exist c j = c j ( d, σ , σ ) , j = 0 , , suchthat for all x ∈ [0 , and µ ∈ P d ,D ( x, µ ) = e xA µ + R ( x, µ ) , (2.31) where k R ( x, µ ) k M ( C [0 , ≤ c , k R ( x, µ ) k M (C) ≤ c ( γ ( µ ) + γ ( x, µ )) . Moreover, D ( x, µ ) = e xA µ + D ( x, µ ) + R ( x, µ ) , (2.32) where D ( x, µ ) := Z x e ( x − t ) A µ ( J ( t ) + ˜ J ( t )) dt and k R ( x, µ ) k M ( C ) ≤ c ( γ ( µ ) γ ( x, µ ) + γ ( µ )) , x ∈ [0 , . Proof.
Let us start with several simple observations. First of all, remark that k D (0) ( x, µ ) k M ( C ([0 , ≤ e d ( k J k M ( L [0 , + k ˜ J k M ( C (∆)) + k ˜ T ˜ J k M ( C (∆)) ) ≤ e d ( a + (1 + a ) k ˜ J k M ( C (∆)) ) ≤ e d ( a + 2(1 + a )˜ l ) . Note also 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 , , (4.9), and (4.10) 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 + ˜ a )˜ a γ ( µ ) , x ∈ [0 , . Furthermore, by (4.11), (cid:13)(cid:13)(cid:13)(cid:13)Z x e iµ ( x − t ) ˜ T ˜ J ( x, t ) dt (cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ≤ e d (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 ) ≤ e d (cid:16) ( a + 1) (cid:0) γ ( µ ) γ ( x, µ ) + γ ( µ ) (cid:1)(cid:17) , where x ∈ [0 , and µ ∈ P d . Combining all these inequalities with Lemma 2.2and the estimates from (2.27), we obtain the required representations for D . Remark . Note that the explicit formula for D is the following D ( x, µ ) = (cid:18) r ( x, µ ) q ( x, µ ) q ( x, µ ) r ( x, µ ) (cid:19) ,q ( x, µ ) := Z x e iµ ( x − t ) σ ( t ) dt, q ( x, µ ) := Z x e − iµ ( x − t ) σ ( t ) dtr ( x, µ ) := Z x e iµ ( x − t ) ˜ σ ( x, t ) dt, r ( x, µ ) := Z x e − iµ ( x − t ) ˜ σ ( x, t ) dt and ˜ σ j are given by (2.18). 9 emark . If p = 1 , then the remainder R from (2.32) satisfies k R ( x, µ ) k M ( C ) ≤ c Γ ( µ ) , where Γ is given by (2.25). We consider spectral problem Y ′ ( x ) + J ( x ) Y ( x ) = A µ Y ( x ) , x ∈ [0 , , (3.1)associated with the matrix problem (1.1) where Y = [ y , y ] T and y (0) = y (0) , y (1) = y (1) . (3.2)Let c = c ( x, µ ) = [ c , c ] T and s = s ( x, µ ) = [ s , s ] T be the solutions of(3.1) satisfying c (0) = 1 , c (0) = 0 and s (0) = 0 , s (0) = 1 . Then due toconditions (3.2) we find that the eigenvalues are the zeros of Φ( λ ) = c (1 , λ ) + s (1 , λ ) − c (1 , λ ) − s (1 , λ ) . (3.3)The eigenfunctions will be of the form: Y = [ y , y ] T = [ c ( · , µ n ) + s ( · , µ n ) , c ( · , µ n ) + s ( · , µ n )] T . (3.4)The analysis of zeros of (3.3) will now lead us to characterization of eigenvalues.The standard approach is to derive first basic formula for eigenvalues andthen using sharp asymptotic results derive more accurate form. We thus needresults related to functions s and c from (2.7). We derive that Φ( µ ) = 2 i sin µ + Z e (1 − t ) iµ (cid:16) Q (1 , t ) + Q (1 , t ) + σ ( t ) (cid:17) dt − Z e − (1 − t ) iµ (cid:16) Q (1 , t ) + Q (1 , t ) + σ ( t ) (cid:17) dt. (3.5)Changing variables in integrals we may write Φ( µ ) = 2 i sin( µ ) + V ( µ ) , (3.6)where V ( µ ) = Z − e iµs f ( s ) ds − Z − e − iµs g ( s ) ds (3.7)and f , g are certain function from L p [ − , .Note that the identities (3.5) and (3.6) are true not only for µ ∈ P d butfor all µ ∈ C . It is a standard procedure (see for instance [3]) to derive usingRouche Theorem that zeros of Φ are in the form µ n = πn + e µ n , where ( e µ n ) is bounded. This conclusion yield that eigenvalues lie in a certain horizontalstripe of the complex plane. We may continue and investigate more precise thebehavior of ( e µ n ) . 10he formula for Φ gives us sin( e µ n ) = ( − n +1 i R ( πn + e µ n ) . (3.8)This expression converges to zero since the convergence of the integral in (3.7)follows from Lebesgue–Riemann Lemma and the fact that e µ n are bounded. Thus e µ n → when n → ∞ . Here ends the reasoning and first claim for p = 1 .For < p < we may continue in order to obtain more information. Using sin x = x + O ( x ) , x → , and the fact that e µ n → we obtain e µ n = ( − n +1 i Z − e i e µ n s e iπns f ( s ) ds (3.9) − ( − n +1 i Z − e − i e µ n s e − iπns g ( s ) ds + O ( | e µ n | ) . Next, the expansion of the exponential function e µt = 1 + µt + O ( | µ | ) , µ → , | t | ≤ leads to a conclusions for one of the integrals (cid:12)(cid:12)(cid:12) Z − e i e µ n s e iπns f ( s ) ds (cid:12)(cid:12)(cid:12) = Z − e iπns f ( s ) ds + i e µ n Z − e iπns sf ( s ) ds + O ( | e µ n | ) . Note that second integral is a product of e µ n and a Fourier coefficient of thefunction from L p , hence it would give a sequence from l q , which converges tozero. Consequently, we go back to (3.9) and conclude that e µ n is a sum of Fouriercoefficients for functions from L p , hence ( e µ n ) ∈ l q . Summarizing, we showedthat the eigenvalues µ n of our spectral problem satisfy µ n = πn + e µ n , ( e µ n ) ∈ l q . (3.10)This representation for < p < and the fact that for p = 1 the remaindergoes to zero allows us to find in both cases more accurate description of eigen-values. Recall we showed eigenvalues lie in P d for a certain d > , thus we canuse asymptotic formulas true in a stripe. The main tool will be the formulas for c and s and consequently for Φ from corollary 2.3.We infer that Φ( µ ) = 2 i sin µ + Z e (1 − t ) iµ σ ( t ) dt + Z e (1 − t ) iµ ˜ σ (1 , t ) dt − Z e − (1 − t ) iµ σ ( t ) dt − Z e − (1 − t ) iµ ˜ σ (1 , t ) dt + r ( µ ) , (3.11)where | r ( µ ) | ≤ c ( γ ( µ ) γ (1 , µ ) + γ ( µ )) ≤ c ( γ ( µ ) + γ (1 , µ ) + γ ( µ )) . i ( − n +1 e µ n = − Z e − πint σ ( t ) dt + 2( − n +1 Z Z t σ ( t ) σ ( ξ ) e − πint e πinξ dξdt + Z e πint σ ( t ) dt + r ( µ n ) , (3.12)For p = 1 we have here | r ( µ n ) | ≤ c Γ ( πn ) .Our last aim is to prove that for < p < there holds ( r ( µ n )) ∈ l q/ .In what follows we will use a basic formula for eigenvalues (3.10), a simpleinequality | e iz − | ≤ | z | e d , z ∈ P d and the Hausdorff–Young inequality. Weinfer for σ ∈ L p [0 , that ∞ X n =1 (cid:12)(cid:12)(cid:12) Z x e ± µ n t σ ( t ) dt (cid:12)(cid:12)(cid:12) q ≤ c q ∞ 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) q (3.13) + c q ∞ X n =1 (cid:18)Z x | e i ˜ µ n t − || σ ( t ) | dt (cid:19) q ≤ c q k σ k qL p [0 , + c k σ k qL [0 , ∞ X n =1 | ˜ µ n | q ≤ m < ∞ , for any x ∈ [0 , . It follows from (3.13) that sup x ∈ [0 , ∞ X n =1 γ q ( x, µ n ) < ∞ , (3.14)Note that by (3.14) ∞ X n =1 γ q ( µ n ) ≤ c Z ∞ X n =1 γ q ( s, µ n ) ds < ∞ . and k γ ( µ n ) k l q/ ≤ Z | σ ( s ) |k γ ( s, µ n ) k l q/ ds = Z | σ ( s ) | ∞ X n =1 γ q ( s, µ n ) ! /q ds ≤ c k σ k L . Finally, we obtain ∞ X n =1 | r n | q/ < ∞ . Summarizing the discussion above we proved the following fact.12 heorem 3.1.
The eigenvalues of the spectral problem (3.1) - (3.2) lie in a cer-tain stripe P d and admit the representation µ n = πn + µ ,n + ρ n , n = 1 , , . . . (3.15) with µ ,n = ( − n i Z e − πint σ ( t ) dt + ( − n +1 i Z e πint σ ( t ) dt (3.16) − i Z Z t σ ( t ) σ ( ξ ) e − πint e πinξ dξdt and for p = 1 there holds | ρ n | < c Γ ( πn ) , (3.17) where Γ is defined in (2.25) , whereas for < p < it is true that ∞ X n =1 | ρ n | q/ < ∞ . (3.18) Remark . Recall that according to lemma 4.1 for every x ∈ [0 , functions ˜ σ j ( x, · ) are from L r . If < p ≤ , then < r ≤ and Fourier coefficients of ˜ σ j ( x, · ) are from l q/ . Then the representation (3.15) with ∞ X n =1 | ρ n | q/ < ∞ is true but with µ ,n given by i ( − n +1 µ ,n = − Z e − πint σ ( t ) dt + Z e πint σ ( t ) dt. (3.19)Now, we can proceed to eigenfunctions. We are going to combine resultsfrom the previous theorem with lemma 2.2 and corollary 2.3. Theorem 3.3.
Let < p < and F ( x, t ) = σ ( t ) + ˜ σ ( x, t ) − ( T σ ˜ σ )( x, t ) F ( x, t ) = σ ( t ) + ˜ σ ( x, t ) − ( T σ ˜ σ )( x, t ) . The eigenfunctions of the spectral problem (3.1) - (3.2) admit the representation y ( x, µ n ) = e iπnx (1 + iµ ,n x ) Z x e − πint F ( x, t ) dt ! − iµ ,n e iπnx Z x e − πint tF ( x, y ) dt + r ( x, n ) ,y ( x, µ n ) = e − iπnx (1 − iµ ,n x ) Z x e πint F ( x, t ) dt ! + 2 iµ ,n e − iπnx Z x e πint tF ( x, t ) dt + r ( x, n ) , here ∞ X n =1 sup x ∈ [0 , | r j ( x, n ) | q/ < ∞ , Proof.
According to (3.4) eigenfunctions are express by solutions c and s in thefollowing way y ( x, µ n ) = c ( x, µ n ) + s ( x, µ n ) and y ( x, µ n ) = c ( x, µ n ) + s ( x, µ n ) . The results of lemma 2.29 yield that y ( x, µ n ) = e iµ n x + Z x e ( x − t ) iµ n σ ( t ) dt + Z x e ( x − t ) iµ n ˜ σ ( x, t ) dt − Z x e iµ ( x − t ) ( T σ ˜ σ )( x, t ) dt + α ( x, µ n ) ,y ( x, µ n ) = e − iµ n x + Z x e − ( x − t ) iµ n σ ( t ) dt + Z x e − ( x − t ) iµ n ˜ σ ( x, t ) dt − Z x e − iµ ( x − t ) ( T σ ˜ σ )( x, t ) dt + β ( x, µ n ) , where | α ( x, µ n ) | + | β ( x, µ n ) | ≤ cγ ( µ n ) . Repeating once more all argument used in order to derive formulas for eigen-values, we obtain the thesis with claimed estimates for reminders.It is possible to obtain shorter but less precise formulas for eigenfunctions.This time we use the representation (2.32) and comments from lemma 4.2 toprove the following fact.
Corollary 3.4.
Let ≤ p < , then the eigenfunctions of the spectral problem (3.1) - (3.2) admit the representation y ( x, µ n ) = e iπnx (cid:16) iµ ,n x + Z x e − πint σ ( t ) dt + Z x Z s σ ( s ) σ ( ξ ) e − iµs e iµξ dξds (cid:17) + r ( x, n ) ,y ( x, µ n ) = e − iπnx (cid:16) − iµ ,n x + Z x e πint ( σ ( t ) dt + Z x Z s σ ( ξ ) σ ( s ) e iµs e − iµξ dξds (cid:17) + r ( x, n ) , (3.20) where for < p < we have sup x ∈ [0 , ∞ X n =1 | r j ( x, n ) | q/ < ∞ , whereas for p = 1 there holds | r j ( x, n ) | ≤ c Γ ( πn ) . Appendix
Lemma 4.1.
For every x ∈ [0 , and j = 1 , the functions ˜ σ j ( x, · ) belong to L r [0 , , therefore ˜ σ j ∈ B , j = 1 , , and ˜ J ∈ M ( B ) .Proof. We take ( x, t ) ∈ ∆ . Let b σ , b σ denote the extension of σ and σ by zerooutside [0 , . Note that for every x ∈ [0 , we get Z x − t σ ( t + ξ ) σ ( ξ ) dξ = Z ∞−∞ b σ ( t + ξ ) b σ ( ξ ) χ ( x − ( t + ξ )) dξ = Z ∞−∞ b σ ( t − s ) b σ ( − s ) χ ( x − ( t − s )) ds = (cid:16) ( b σ ( · ) χ ( x − · )) ∗ ( b σ ( −· )) (cid:17) ( t ) . We thus have (cid:13)(cid:13)(cid:13) Z x − t σ ( t + ξ ) σ ( ξ ) dξ (cid:13)(cid:13)(cid:13) L r [0 , ≤ (cid:13)(cid:13)(cid:13) Z x − t b σ ( t + ξ ) b σ ( ξ ) dξ (cid:13)(cid:13)(cid:13) L r ( R ) ≤ k ( b σ ( · ) χ ( x − · )) k L p ( R ) k b σ ( −· ) k L p ( R ) ≤ k σ k L p [0 , k σ k L p [0 , , hence k ˜ σ j k B ≤ ˜ a, j = 1 , . (4.1)Clearly, a similar estimate holds for ˜ σ as well.Therefore, if we consider ǫ and x such that ≤ t ≤ x + ǫ ≤ , then repeatingthe reasoning from the latter inequality, we obtain (cid:13)(cid:13)(cid:13) Z x + ǫ − t σ ( t + ξ ) σ ( ξ ) dξ − Z x − t σ ( t + ξ ) σ ( ξ ) dξ (cid:13)(cid:13)(cid:13) L r [0 , ≤ k ( b σ ( · ) (cid:2) χ ( x + ǫ − · ) − χ ( x − · ) (cid:3) k L p ( R ) k b σ ( −· ) k L p ( R ) ≤ Z R | b σ ( s ) | p | χ ( x + ǫ − s ) − χ ( x − s ) | p ds k b σ k L p ( R ) . The integral in the last line converges to zero, if ǫ → , because of Lebesgue The-orem, hence the mapping x ˜ σ j ( x, · ) ∈ L r [0 , is continuous for j = 1 , . Lemma 4.2.
The following identity holds Z x e − tiµ ˜ σ ( x, t ) dt + Z x e tiµ ˜ σ ( x, t ) dt = Z x e − iµξ σ ( ξ ) dξ Z x σ ( s ) e iµs ds. Moreover, we have (cid:12)(cid:12)(cid:12) Z x e − tiµ ˜ σ ( x, t ) dt + Z x e tiµ ˜ σ ( x, t ) dt (cid:12)(cid:12)(cid:12) ≤ cγ ( x, µ ) and Z x e − tiµ ˜ σ ( x, t ) dt − Z x e tiµ ˜ σ ( x, t ) dt = − Z x Z s σ ( ξ ) σ ( s ) e iµs e − iµξ dξds + α ( µ ) , x e tiµ ˜ σ ( x, t ) dt − Z x e − tiµ ˜ σ ( x, t ) dt = − Z x Z s σ ( s ) σ ( ξ ) e − iµs e iµξ dξds + α ( µ ) , where α j ( µ ) = O ( γ ( x, µ )) for j = 1 , .Proof. Note that Z x e − tiµ ˜ σ ( x, t ) dt = Z x Z s σ ( s ) σ ( ξ ) e − iµs e iµξ dξds Z x e tiµ ˜ σ ( x, t ) dt = Z x Z s σ ( ξ ) σ ( s ) e iµs e − iµξ dξds. Observe that the change of variables yield Z x e − tiµ ˜ σ ( x, t ) dt = Z x Z xξ σ ( ξ ) σ ( s ) e − iµs e iµξ dsdξ = Z x Z xs σ ( s ) σ ( ξ ) e − iµξ e iµs dξds, thus Z x e − tiµ ˜ σ ( x, t ) dt + Z x e tiµ ˜ σ ( x, t ) dt = Z x Z x σ ( ξ ) σ ( s ) e iµs e − iµξ dξds = Z x e − iµξ σ ( ξ ) dξ Z x σ ( s ) e iµs ds. This step shows that (cid:12)(cid:12)(cid:12) Z x e − tiµ ˜ σ ( x, t ) dt + Z x e tiµ ˜ σ ( x, t ) dt (cid:12)(cid:12)(cid:12) ≤ cγ ( x, µ ) . What is more, then Z x e − tiµ ˜ σ ( x, t ) dt − Z x e tiµ ˜ σ ( x, t ) dt = Z x e − iµξ σ ( ξ ) dξ Z x σ ( s ) e iµs ds − Z x Z s σ ( ξ ) σ ( s ) e iµs e − iµξ dξds, thus Z x e − tiµ ˜ σ ( x, t ) dt − Z x e tiµ ˜ σ ( x, t ) dt = − Z x Z s σ ( ξ ) σ ( s ) e iµs e − iµξ dξds + α ( µ ) , where α ( µ ) = O ( γ ( x, µ )) . Analogously we get the last claim16 emma 4.3. The linear operator T σ ( 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, (4.2) where σ ∈ L p [0 , is bounded in B .Proof. Note that (cid:16) Z x | ( T σ f )( x, t ) | r dt (cid:17) /r = (cid:16) Z x (cid:12)(cid:12)(cid:12) Z x χ ( s − t ) σ ( s ) f ( s, s − t ) ds (cid:12)(cid:12)(cid:12) r dt (cid:17) /r ≤ Z x | σ ( s ) | (cid:16) Z s | f ( s, s − t ) | r dt (cid:17) /r ds ≤ Z x | σ ( s ) | (cid:16) Z s | f ( s, τ ) | r dτ (cid:17) /r ds ≤ Z x | σ ( s ) | ds sup s ∈ [0 , (cid:16) Z s | f ( s, τ ) | r dτ (cid:17) /r ≤ k σ k L k f k B . (4.3)For the proof of continuity we take ǫ and x such that ≤ t ≤ x + ǫ ≤ .Then (cid:13)(cid:13)(cid:13) ( T σ f )( x + ǫ, · ) − ( T σ f )( x, · ) (cid:13)(cid:13)(cid:13) L r [0 , ≤ (cid:16) Z x (cid:12)(cid:12)(cid:12) Z x + ǫx σ ( s ) f ( s, s − t ) ds (cid:12)(cid:12)(cid:12) r dt (cid:17) /r + (cid:16) Z x + ǫx (cid:12)(cid:12)(cid:12) Z x + ǫt σ ( s ) f ( s, s − t ) ds (cid:12)(cid:12)(cid:12) r dt (cid:17) /r . First integral may be estimated as follows (cid:16) Z x (cid:12)(cid:12)(cid:12) Z x + ǫx σ ( s ) f ( s, s − t ) ds (cid:12)(cid:12)(cid:12) r dt (cid:17) /r ≤ Z x + ǫx | σ ( s ) | (cid:16) Z x | f ( s, s − t ) | r dt (cid:17) /r ds ≤ Z x + ǫx | σ ( s ) | (cid:16) Z s | f ( s, τ ) | r dτ (cid:17) /r ds ≤ k f k B Z x + ǫx | σ ( s ) | ds and this expression goes to zero whenever ǫ does.Second integral can be treated in an analogous way, hence the proof is com-pleted Lemma 4.4.
The operators T kj , k, j = 1 , , k = j satisfy the following esti-mate k T nkj f k B ≤ a n n ! k f k B , f ∈ B, n ∈ N , k, j = 1 , , k = j. (4.4) Proof.
Consider the operator T . Note that directly from third line of (4.3) we17et (cid:16) Z x | ( T f )( x, t ) | r dt (cid:17) /r ≤ Z x | σ ( s ) | (cid:16) Z s | ( T σ f )( s, τ ) | r dτ (cid:17) /r ds ≤ Z x | σ ( s ) | Z s | σ ( τ ) | (cid:16) Z τ | f ( τ, ξ ) | r dξ (cid:17) /r dτ ds ≤ k f k B Z x | σ ( s ) | Z s | σ ( τ ) | ds. (4.5)Define η ∈ C [0 , by η ( x ) := Z x | σ ( s ) | (cid:18)Z s | σ ( τ ) | dτ (cid:19) ds, x ∈ [0 , . This function is increasing and bounded by a = k σ k L k σ k L . It suffices toprove that for all ( x, t ) ∈ ∆ and n = 1 , , . . . , (cid:16) Z x | ( T n f )( x, t ) | r dt (cid:17) /r ≤ k f k B n ! η n ( x ) , f ∈ B. (4.6)For n = 1 the estimate (4.6) was shown above. Arguing by induction, supposethat (4.6) holds for some n ∈ N . Then for ( x, t ) ∈ ∆ and f ∈ B from (4.5) wehave (cid:16) Z x | ( T n +112 f )( x, t ) | r dt (cid:17) /r ≤ Z x | σ ( s ) | Z s | σ ( τ ) | (cid:16) Z τ | ( T n f )( τ, ξ ) | r dξ (cid:17) /r dτ ds ≤ k f k B n ! Z x | σ ( s ) | Z s | σ ( τ ) | η n ( τ ) dτ ds ≤ k f k B n ! Z x | σ ( s ) | Z s | σ ( τ ) | dτ η n ( s ) ds = k f k B n ! Z x η n ( s ) dη ( s ) = k f k B ( n + 1)! η n +1 ( x ) . Therefore (4.6) hold true and then after taking supremum over x ∈ [0 , we get(4.4).Next proposition we state below without a proof, since it can be found in [7,Prop. 6.1]. Proposition 4.5. If σ j ∈ L p [0 , , ≤ p < and F ∈ M ( B ) , then Z x e − iµt ( ˜ T F )( x, t ) dt = − Z x e − iµs J ( s ) Z s e iµξ F ( s, ξ ) dξds. (4.7) 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. (4.8)18 emma 4.6. If µ ∈ P d , then there hold the following inequalities (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 ˜ a γ ( µ ) , (4.9) (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 ˜ a γ ( µ ) , (4.10) (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 ) ≤ a + 1) e d (cid:16) γ ( µ ) γ ( x, µ ) + γ ( µ ) (cid:17) ,x ∈ [0 , , (4.11) (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 a n − ( n − γ ( µ ) , n ≥ . (4.12) Proof.
Note 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) (4.13) ≤ e d n k σ k L p (cid:13)(cid:13)(cid:13) Z s e iµξ σ ( ξ ) dξ (cid:13)(cid:13)(cid:13) L q + k σ k L p (cid:13)(cid:13)(cid:13) Z s e iµξ σ ( ξ ) dξ (cid:13)(cid:13)(cid:13) L q o ≤ e d max {k σ k L p , k σ k L p } γ ( µ ) , x ∈ [0 , . We thus proved the estimate (4.9).Next, from (4.7), if µ ∈ P d , x ∈ [0 , and F ∈ M ( B ) , 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, (4.14)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 a (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 , . (4.15)We use (4.15) and (4.13) to 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 a (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 ˜ a γ ( µ ) , thus, the estimate (4.10) holds. 19ue to 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 p k γ ( y, µ ) k L q ≤ a γ ( µ ) , the inequality (4.11) 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) . (4.16)Whereas using (4.8), (4.9), 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 (4.16) follows.The estimate (4.12) will be showed, if we prove 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 ) . (4.17)We proceed by induction. Using (4.14) for F = ˜ T ˜ J and (4.16), 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 a γ ( µ ) ≤ e d (cid:0)R x σ ( s ) γ ( s, µ ) ds (cid:1) e d a γ ( µ ) ≤ e d (cid:0) a γ ( µ ) + e d a γ ( µ ) (cid:1) . n = 2 .Let suppose now that (4.17) holds for some n ≥ . We thus once again use(4.14) to derive (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 , , thus (4.17) holds also for n + 1 , and the proof of (4.17) is completed. References [1] M. M. Malamud A. V. Agibalova and L. L. Oridoroga. On the completenessof general boundary value problems for × first-order systems of ordinarydifferential equations. Methods Funct. Anal. Topology , 18:4–18, 2012.[2] S. Albeverio, R. Hryniv, and Y. Mykytyuk. Inverse spectral problemsfor Dirac operators with summable potentials.
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