Mathematical Notes, vol. 79, no. 4, 2006, pp. 589–593. Translated from Matematicheskie Zametki, vol. 79, no. 4, 2006, pp. 636–640. c Original Russian Text Copyright 2006 by L. L. Oridoroga, S. Hassi.
Completeness and Riesz Basis Property of Systems of Eigenfunctions and Associated Functions of Dirac-Type Operators with Boundary Conditions Depending on the Spectral Parameter L. L. Oridoroga and S. Hassi Received November 10, 2004; in final form, September 12, 2004
Key words: Riesz basis property, eigenfunction, associated function, boundary value problem,
Dirac equation.
1. INTRODUCTION It is well known (see [1]) that the system of eigenfunctions and associated functions of the Sturm-Liouville problem (1) −y + q(x)y = λ2 y with separable boundary conditions y (0) − h0 y(0) = y (1) − h1 y(1) = 0
(2)
is complete in the space L2 [0, 1] for any complex-valued potential q ∈ L1 [0, 1] and any h0 , h1 ∈ C . In the same paper [1], the completeness of the system of eigenfunctions and associated functions is established in the case of an arbitrary (rather than just separable) nondegenerate boundary conditions. The completeness of the system of eigenfunctions and associated functions of the differential equation n−2 (n) qj (x)y = λn y (3) y + j=0
of arbitrary order with separable (nonregular) boundary conditions was announced by Keldysh in the well-known paper [2] and was first proved by Shkalikov in [3]. In [4], Malamud and one of the authors studied the completeness of the system of eigenfunctions and associated functions of boundary problems for systems of differential equations of first order with boundary conditions of general form (independent of the spectral parameter). In [5, 6], Tarapova studied the completeness of the system of eigenfunctions and associated functions of Eq. (1) in the case of nonlinear boundary conditions depending on the spectral parameter. In [7–9], a similar result is obtained for Dirac-type systems with separable boundary conditions (both linear and nonlinear) depending on the spectral parameter. In the recent papers [10, 11], Trooshin (Trushin) and Yamamoto give sufficient conditions for the system of eigenfunctions and associated functions of a Dirac system with separable linear boundary conditions independent of the spectral parameter to form a Riesz basis. 0001-4346/2006/7934-0589
c 2006 Springer Science+Business Media, Inc.
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In the present paper, certain generalizations of the results listed above are obtained. Namely, in Sec. 2 we give sufficient completeness conditions for the system of eigenfunctions and associated functions of a system of n(≥ 2) differential equations of first order with separable linear boundary conditions depending on the spectral parameter. This result generalizes Theorem 2 from [4] and coincides with the latter in the case of boundary conditions independent of λ (i.e., in the case of constant polynomials P0 and P1 ). In addition, for n = 2 and arbitrary P0 (λ) and P1 (λ) , it coincides with the result from [7]. In Sec. 3, this result is refined in the case n = 2 . It is shown that for n = 2 , the system of eigenfunctions and associated functions of a problem with separable boundary conditions depending on the spectral parameter forms a Riesz basis. As a corollary, the result from [10] is obtained (see Corollary 7). 2. COMPLETENESS OF THE SYSTEM OF EIGENFUNCTIONS AND ASSOCIATED FUNCTIONS FOR SYSTEMS OF ARBITRARY ORDER Let b1 < b2 < · · · < bκ < 0 < bκ+1 < · · · < bn . Consider the diagonal selfadjoint n n × n matrix ∗ B = B = diag(b1 , b2 , . . . , bn ) . Consider the boundary problem in the space 1 L2 [0, 1] for the system of differential equations of the form −iBy + Q(x)y = λy ,
y = col{y1 , y2 , . . . , yn },
(4)
with a matrix potential of the form Q(x) = (qij (x))ni,j=1 ,
where
qii ≡ 0,
i ∈ {1, 2, . . . , n},
and qij ∈ L∞ [0, 1] for i, j ∈ {1, 2, . . . , n} . Let P0 (λ) = (P0ij (λ))ni,j=1 and P1 (λ) = (P1ij (λ))ni,j=1 be matrix polynomials such that det(P0 P0∗ + P1 P1∗ ) > 0 for λ ∈ C . Consider Eq. (4) with boundary conditions P0 (λ)y(0) + P1 (λ)y(1) = 0. Let us introduce the following notation: P1ij (λ) Pi+ (λ) = P0ij (λ) P0ij (λ) Pij− (λ) = P1ij (λ)
(5)
for bk < 0 (i.e., for k ≤ κ), for bk > 0 (i.e., for k ≥ κ), for bk < 0 (i.e., for k ≤ κ), for bk > 0 (i.e., for k ≥ κ).
Let Nk be the largest of the degrees of the polynomials P0jk and P1jk (for all 1 ≤ j ≤ n). + − + − Denote by Cjk ( Cjk ) the coefficient of λNk in the polynomial Pjk ( Pjk , respectively). In other + − + − ( Cjk ) is equal to the leading coefficient of the polynomial Pjk ( Pjk , respectively) if words, Cjk the degree of this polynomial is Nk , and is zero otherwise. Let us introduce matrices C + and C − as follows: + n )j ,k=1 , C + := (Cjk
− n C − := (Cjk )j ,k=1 .
Definition 1. A set Φ of eigenvectors and associated vectors is said to satisfy condition (P) whenever the following statement holds: if the set Φ contains an eigenfunction or associated function corresponding to the eigenvalue λk , then, together with it, it contains all the associated functions of higher order corresponding to the same eigenvalue. MATHEMATICAL NOTES
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Theorem 2. Suppose that det C + = 0 and det C − = 0 . Then the system of eigenfunctions and n 2 associated functions of problem (4), (5) is complete in the space 1 L [0, 1] . Moreover, suppose that a set Φ consisting of N := min1≤k≤n Nk eigenfunctions and associated functions satisfies condition (P). Then the system of eigenfunctions associated functions of n and 2 problem (4), (5) without the set Φ is also complete in the space 1 L [0, 1] . The proof of Theorem 2 substantially depends on Theorem 1 from [4] about the asymptotics of solutions of (4). 3. BASIS PROPERTY OF THE SYSTEM OF EIGENFUNCTIONS AND ASSOCIATED FUNCTIONS FOR SECOND-ORDER SYSTEMS In this section, we consider the system of eigenfunctions and associated functions of a system of two first-order differential equations with separable linear boundary conditions depending on the spectral parameter. −1 Namely, let b1 < 0 < b2 , and let B = diag(b−1 1 , b2 ) be a diagonal 2 × 2 matrix. In the space L2 [0, 1] ⊕ L2 [0, 1] , consider the boundary problem for the system of differential equations of the form (6) −iBy + Q(x)y = λy , where
Q(x, t) =
0 q2
q1 0
,
y(x) =
y1 (x) y2 (x)
,
and qj ∈ C 1 [0, 1] . Let us add to (6) separable boundary conditions depending on the spectral parameter
P11 (λ)y1 (0) + P12 (λ)y2 (0) = 0, P21 (λ)y1 (1) + P22 (λ)y2 (1) = 0.
(7)
The following theorem, as well as Theorem 6 below, can be derived from Shkalikov’s general results (see [12]) on the basis property of the system of eigenfunctions and associated functions of ordinary differential equations (and systems of such equations); see also the papers [13] and [14], in which the results from [12] are generalized. However, our proof is based on a version of Bari’s theorem about the preservation of Riesz basis under quadratically small perturbations and does not involve the fairly complex general constructions from the papers [12–14]. In this connection, let us also mention the paper [15] by Gomilko and Radzievskii about the basis property of the system of eigenfunctions of an ordinary functional differential operator of nth order in Lp [0, 1] . The technique of our proof of Theorem 3 is close to the one used in [15]. Theorem 3. Let P11 (λ) and P12 (λ) be coprime polynomials with deg P11 = deg P12 = N0 , and let P21 (λ) and P22 (λ) be coprime polynomials with deg P21 = deg P22 = N1 . Let Φ be a set consisting of N = N0 + N1 eigenfunctions and associated functions such that the system of eigenfunctions and associated functions of problem (6), (7) remains complete in the space L2 [0, 1] ⊕ L2 [0, 1] after the set Φ is removed from it. Then the system of eigenfunctions and associated functions of problem (6), (7) without the set Φ forms a Riesz basis in the space L2 [0, 1] ⊕ L2 [0, 1] . The proof of this theorem is based on the following lemmas, which describe the asymptotic behavior of the eigenvalues and eigenfunctions of the operator (6), (7). The proofs of these lemmas are based on the existence of triangular transformation operators for the system (4), which were constructed by Malamud in [16]. MATHEMATICAL NOTES
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Lemma 4. Let P11 (λ) and P12 (λ) be coprime polynomials with deg P11 = deg P12 = N0 , and let P21 (λ) and P22 (λ) be coprime polynomials with deg P21 = deg P22 = N1 . Let Cij be the leading coefficient of the polynomial Pij (λ) , and set C1 := C11 C21 , C2 := C12 C22 . Denote by Λ a set containing (with account of multiplicity) N = N0 + N1 eigenvalues of problem (6), (7). Then all the other eigenvalues (with account of multiplicity) can be renumbered in such a way that i ln(C1 /C2 ) + 2πn 1 , where n ∈ Z. (8) +O λn = b2 − b1 |n| Lemma 5. Let q1 (λ) and q2 (λ) be continuously differentiable functions. Then the solutions ϕ (x ; λ) := col(ϕ1 (x ; λ), ϕ2 (x ; λ))
and
; λ) := col(ψ1 (x ; λ), ψ2 (x ; λ)) ψ(x
and
ϕ2 (0 ; l) = ψ1 (0 ; l) = 0,
of Eq. (6) with initial conditions ϕ1 (0 ; l) = ψ2 (0 ; l) = 1 satisfy the following estimates: ϕ1 (x ; λ) = exp(b1 λix) + ψ1 (x ; λ) =
1 O(exp(b1 λix)), λ
1 O(exp(b1 λix)), λ 1 ψ2 (x ; λ) = O(exp(b1 λix)) λ
(9)
1 O(exp(b2 λix)), λ 1 ψ2 (x ; λ) = exp(b2 λix) + O(exp(b2 λix)) λ
(10)
1 O(exp(b1 λix)), λ
ϕ2 (x ; λ) =
for λ ∈ C+ and 1 O(exp(b2 λix)), λ 1 ψ1 (x ; λ) = O(exp(b2 λix)), λ
ϕ1 (x ; λ) =
ϕ2 (x ; λ) =
for λ ∈ C− . Theorem 6. Let P11 (λ) and P12 (λ) be coprime polynomials with deg P11 = deg P12 = N . Let P21 and P22 be nonzero constants (i.e., polynomials of zero degree). Suppose that a set Φ consisting of N eigenfunctions and associated functions satisfies condition (P). Then the system of eigenfunctions and associated functions of problem (6), (7) without the set Φ forms a Riesz basis in the space L2 [0, 1] ⊕ L2 [0, 1] . Setting N0 = N1 = 0 in the assumptions of Theorem 6, we arrive at the result obtained in [10]. Corollary 7 [10]. Let h1 and h2 be distinct from 0. Then the system of eigenfunctions and associated functions of problem (6) with boundary conditions y1 (0) + h1 (λ)y2 (0) = 0, y1 (1) + h2 (λ)y2 (1) = 0 forms a Riesz basis in the space L2 [0, 1] ⊕ L2 [0, 1] . ACKNOWLEDGMENTS The authors wish to express their sincere gratitude to A. A. Shkalikov for very useful discussion of the results and for the information about the papers [15] and [14]. MATHEMATICAL NOTES
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REFERENCES 1. V. A. Marchenko, Sturm–Liouville Operators and their Applications [in Russian], Naukova Dumka, Kiev, 1977. 2. M. V. Keldysh, Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 77 (1951), no. 1, 11–14. 3. A. A. Shkalikov, Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 10 (1976), no. 4, 69–80. 4. M. M. Malamud and L. L. Oridoroga, Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 34 (2000), no. 3, 88–90. 5. E. I. Tarapova, Teoriya funktsii, funktsion. analiz i ih prilozh., 31 (1979), 157–160. 6. E. I. Tarapova, Teoriya funktsii, funktsion. analiz i ih prilozh., 33 (1979), 82–87. 7. L. L. Oridoroga, Dopovidi NAN Ukraini (2001), no. 3, 34–39. 8. L. L. Oridoroga, Methods Funct. Anal. Topology, 7 (2001), no. 1, 82–87. 9. S. Hassi and L. L. Oridoroga, Mat. Zametki [Math. Notes], 74 (2003), no. 2, 316–320. 10. I. Trooshin (Trushin) and M. Yamamoto, Appl. Anal., 80 (2001), no. 1–2, 19–51. 11. I. Trooshin (Trushin) and M. Yamamoto, J. Inverse Ill-Posed Problems, 10 (2002), no. 6, 643–658. 12. A. A. Shkalikov, Trudy Semin. I. G. Petrovskogo (1983), no. 9, 190–229. 13. A. A. Shkalikov, in: Functional Analysis and its Applications to Mechanics and Probability [in Russian], Moscow State Univ., Moscow, 1984, pp. 124–128. 14. L. M. Luzhina, Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1988), no. 1, 31–35. 15. A. M. Gomilko and G. V. Radzievskii, Mat. Zametki [Math. Notes], 49 (1991), no. 1, 47–55. 16. M. M. Malamud, Trudy Moskov. Mat. Obshch. [Trans. Moscow Math. Soc.], 60 (1999), 199–258. (L. L. Oridoroga) Donetsk National University, Ukraine E-mail : (L. L. Oridoroga)
[email protected] (S. Hassi) Vaasa University, Finland E-mail : (S. Hassi)
[email protected]
MATHEMATICAL NOTES
Vol. 79
No. 4
2006
