Riesz bases consisting of root functions of 1D Dirac operators

arXiv:1108.4225v1 [math.SP] 22 Aug 2011

RIESZ BASES CONSISTING OF ROOT FUNCTIONS OF 1D DIRAC OPERATORS PLAMEN DJAKOV AND BORIS MITYAGIN Abstract. For one-dimensional Dirac operators       dy 1 0 0 P y1 Ly = i , + vy, v = , y= 0 −1 dx Q 0 y2 subject to periodic or antiperiodic boundary conditions, we give necessary and sufficient conditions which guarantee that the system of root functions contains Riesz bases in L2 ([0, π], C2 ). In particular, if the potential matrix v is skew-symmetric (i.e., Q = −P ), or more generally if Q = tP for some real t 6= 0, then there exists a Riesz basis that consists of root functions of the operator L. 2010 Mathematics Subject Classification. 47E05, 34L40.

1. Introduction We consider one-dimensional Dirac operators of the form       0 P y1 1 0 dy + v(x) y, v = , y= , (1.1) Lbc (v)y = i Q 0 y2 0 −1 dx

with periodic matrix potentials v such that P, Q ∈ L2 ([0, π], C2 ), subject to periodic (P er+ ) or antiperiodic (P er− ) boundary conditions (bc): (1.2)

P er+ : y(π) = y(0);

P er− : y(π) = −y(0).

Our goal is to give necessary and sufficient conditions on potentials v which guarantee that the system of periodic (or antiperiodic) root functions of LP er± (v) contains Riesz bases. The free operators L0P er± = LP er± (0) have discrete spectrum: ( 2Z if bc = P er+ Sp(L0P er± ) = Γ± , where Γ± = 2Z + 1 if bc = P er− and each eigenvalue is of multiplicity 2. The spectra of perturbed operators LP er± (v) = L0P er± + v is also discrete; for n ∈ Γ± with large enough |n| the perturbed operator − + has ”twin” eigenvalues λ± n close to n. In the case where λn 6= λn for large enough |n|, could the corresponding normalized ”twin eigenfunctions” form a Riesz basis? Recently, in the case of Hill operators, many authors focused on this problem (see [1, 2, 5, 6, 8, 10, 11, 12, 15, 16] and the bibliography there). It may happen P. Djakov acknowledges the hospitality of Department of Mathematics and the support of Mathematical Research Institute of The Ohio State University, July - August 2011. B. Mityagin acknowledges the support of the Scientific and Technological Research Council of Turkey and the hospitality of Sabanci University, April - June, 2011. 1

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PLAMEN DJAKOV AND BORIS MITYAGIN

+ that λ− n 6= λn for |n| > N∗ but the system of normalized eigenfunctions fails to give a convergent eigenfunction expansion (see [2, Theorem 71]). In the present paper we consider such a problem in the case of 1D periodic Dirac operators. In [7], we have singled out a class of potentials v which smoothness could − be determined only by the rate of decay of related spectral gaps γn = λ+ n − λn , ± where λn are the eigenvalues of L = L(v) considered on [0, π] with periodic (for even n) or antiperiodic (for odd n) boundary conditions. This class X is determined by the properties of the functionals βn− (v; z) and βn+ (v, z) (see below (2.8) ) to be equivalent in the following sense: there are c, N > 0 such that

c−1 |βn+ (v; zn∗ )| ≤ |βn− (v; zn∗ )| ≤ c|βn+ (v; zn∗ )|,

|n| > N,

− zn∗ = (λ+ n + λn )/2 − n.

Section 3 contains the main results of this paper. We prove that if v ∈ X then the system of root functions of the operator LP er± (v) contains Riesz bases in L2 ([0, π], C2 ). Theorem 3.1, which is analogous to Theorem 1 in [6] (or Theorem 2 in [5]), gives necessary and sufficient conditions for existence of such Riesz bases. Theorem 3.2 is a modification of Theorem 3.1 that is more suitable for application to concrete classes of potentials. Applications of Theorems 3.1 and 3.2 are given in Section 4. In particular, we prove that if the potential matrix v is skew-symmetric (i.e., Q = −P ) then the system of root functions of LP er± (v) contains Riesz bases in L2 ([0, π], C2 ). 2. Preliminaries 1. Let H be a separable Hilbert space, and let (eα , α ∈ I) be an orthonormal basis in H. If A : H → H is an automorphism, then the system

(2.1)

fα = Aeα ,

α ∈ I,

is an unconditional basis in H. Indeed, for each x ∈ H we have ! X X X −1 −1 x = A(A x) = A hx, (A−1 )∗ eα ifα = hx, f˜α ifα , hA x, eα ieα = α

α

α

i.e., (fα ) is a basis, its biorthogonal system is {f˜α = (A−1 )∗ eα , α ∈ I}, and the series converge unconditionally. Moreover, it follows that X (2.2) 0 < c ≤ kfα k ≤ C, m2 kxk2 ≤ |hx, f˜α i|2 kfα k2 ≤ M 2 kxk2 , α

−1

with c = 1/kA k, C = kAk, M = kAk · kA−1 k and m = 1/M. A basis of the form (2.1) is called Riesz basis. One can easily see that the property (2.2) characterizes Riesz bases, i.e., a basis (fα ) is a Riesz bases if and only if (2.2) holds with some constants C ≥ c > 0 and M ≥ m > 0. Another characterization of Riesz bases is given by the following assertion (see [9, Chapter 6, Section 5.3, Theorem 5.2]): If (fα ) is a normalized basis (i.e., kfα k = 1 ∀α), then it is a Riesz basis if and only if it is unconditional. A countable family of bounded projections {Pα : H → H, α ∈ I} is called unconditional basis of projections if Pα Pβ = 0 for α 6= β and X x= Pα (x) ∀x ∈ H, α∈I

where the series converge unconditionally in H.

RIESZ BASES

3

If {Hα , α ∈ I} is a maximal family of mutually orthogonal subspaces of H and Qα is the orthogonal projection on Hα , α ∈ I, then {Qα , α ∈ I} is an unconditional basis of projections. A family of projections {Pα , α ∈ I} is called a Riesz basis of projections if there is a family of orthogonal projections {Qα , α ∈ I} and an isomorphism A : H → H such that Pα = A Qα A−1 ,

(2.3)

α ∈ I.

In view of (2.3), if {Pα } is a Riesz basis of projections, then there are constants a, b > 0 such that X (2.4) akxk2 ≤ kPα xk2 ≤ bkxk2 ∀x ∈ H. α

For a family of projections P = {Pα , α ∈ I} the following properties are equivalent (see [9, Chapter 6]): (i) P is an unconditional basis of projections; (ii) P is a Riesz basis of projections.

Lemma 2.1. Let (Pα , α ∈ I) be a Riesz basis of two-dimensional projections in a Hilbert space H, and let fα , gα ∈ Ran Pα , α ∈ I be linearly independent unit vectors. Then the system {fα , gα , α ∈ Γ} is a Riesz basis if and only if

(2.5)

κ := sup |hfα , gα i| < 1.

Proof. Suppose that the system {fα , gα , α ∈ I} is a Riesz basis in H. Then X x= (fα∗ (x)fα + gα∗ (x)gα ), x ∈ H, α

where tions

fα∗ , gα∗

are the conjugate functionals. By (2.2), the one-dimensional projec-

Pα1 (x) = fα∗ (x)fα , Pα2 (x) = gα∗ (x)gα , α ∈ I, are uniformly bounded. On the other hand, it is easy to see that
−1
−1 , , kPα2 k2 ≥ 1 − |hfα , gα i|2 kPα1 k2 ≥ 1 − |hfα , gα i|2

so (2.5) holds. Conversely, suppose (2.5) holds. Then we have for every α ∈ I

(1 − κ) |fα∗ (x)|2 + |gα∗ (x)|2 ≤ kPα (x)k2 ≤ (1 + κ) |fα∗ (x)|2 + |gα∗ (x)|2 which implies, in view of (2.4), X
a b |fα∗ (x)|2 + |gα∗ (x)|2 ≤ kxk2 ≤ kxk2 . 1+κ 1 − κ α

Therefore, (2.2) holds, which means that the system {fα , gα , α ∈ I} is a Riesz basis in H.  2. We consider the Dirac operator (1.1) with bc = P er± in the domain     y Dom (LP er± (v)) = y = 1 : y1 , y2 are absolutely continuous, y(π) = ±y(0) . y2

Then the operator LP er± (v) is densely defined and closed; its adjoint operator is   0 Q ∗ . (2.6) (LP er± (v)) = LP er± (v ∗ ), v ∗ = P 0

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PLAMEN DJAKOV AND BORIS MITYAGIN

Lemma 2.2. The spectra of the operators LP er± (v) are discrete. There is an N = N (v) such that the union ∪|n|>N Dn of the discs Dn = {z : |z − n| < 1/4} contains all but finitely many of the eigenvalues of LP er+ and LP er− while the remaining finitely many eigenvalues are in the rectangle RN = {z : |Re z|, |Im z| ≤ N + 1/2}. Moreover, for |n| > N the disc Dn contains two (counted with algebraic multi+ plicity) periodic (if n is even) or antiperiodic (if n is odd) eigenvalues λ− n , λn such − + − + − + that Re λn < Re λn or Re λn = Re λn and Im λn ≤ Im λn . See details and more general results about localization of these spectra in [13, 14] and [2, Section 1.6]. Lemma 2.2 allows us to apply the Lyapunov–Schmidt projection method and reduce the eigenvalue equation Ly = λy for λ ∈ Dn to an eigenvalue equation in the two-dimensional space En0 = {L0 Y = nY } (see [2, Section 2.4]). This leads to the following (see in [2] the formulas (2.59)–(2.80) and Lemma 30). Lemma 2.3. (a) For large enough |n|, n ∈ Z, there are functionals αn (v, z) and βn± (v; z), |z| < 1 such that a number λ = n + z, |z| < 1/4, is a periodic (for even n) or antiperiodic (for odd n) eigenvalue of L if and only if z is an eigenvalue of the matrix   αn (v, z) βn− (v; z) (2.7) . βn+ (v; z) αn (v, z) (b) A number λ = n + z ∗ , |z ∗ | < 41 , is a periodic (for even n) or antiperiodic (for odd n) eigenvalue of L of geometric multiplicity 2 if and only if z ∗ is an eigenvalue of the matrix (2.7) of geometric multiplicity 2.

The functionals αn (z; v) and βn± (z; v) are well defined for large enough |n| by explicit expressions in terms of the Fourier coefficients p(m), q(m), m ∈ 2Z of the potential entries P, Q about the system {eimx , m ∈ 2Z} (see [2, Formulas (2.59)– (2.80)]). Here we provide formulas only for βn± (v; z) : ∞ X (2.8) βn± (v; z) = σν± with σ0+ = q(2n), σ0− = p(−2n), ν=0

σν+ =

X

j1 ,...,j2ν 6=n

σν− =

X

j1 ,...,j2ν 6=n

q(n + j1 )p(−j1 − j2 )q(j2 + j3 ) . . . p(−j2ν−1 − j2ν )q(j2ν + n) , (n − j1 + z)(n − j2 + z) . . . (n − j2ν−1 + z)(n − j2ν + z)

p(−n − j1 )q(j1 + j2 )p(−j2 − j3 ) . . . q(j2ν−1 + j2ν )p(−j2ν − n) , (n − j1 + z)(n − j2 + z) . . . (n − j2ν−1 + z)(n − j2ν + z)

where j1 , . . . , j2ν ∈ n + 2Z. Next we summarize some basic properties of αn (z; v) and βn± (z; v). Proposition 2.4. (a) The functions αn (z; v) and βn± (z; v) depend analytically on z for |z| ≤ 1. For |n| ≥ n0 the following estimates hold:  p  (2.9) |αn (v; z)|, |βn± (v; z)| ≤ C E|n| (r) + 1/ |n| , |z| ≤ 1/2; (2.10)

∂αn , (v; z) ∂z

±  p  ∂βn ≤ C E|n| (r) + 1/ |n| , (v; z) ∂z

|z| ≤ 1/4,

where r = (r(m)), r(m) = max{|p(±m)|, q(±m)}, C = C(krk), n0 = n0 (r) and X 2 |r(k)|2 . (Em (r)) = |k|≥m

RIESZ BASES

5

(b) For large enough |n|, the number λ = n + z, z ∈ D = {ζ : |ζ| ≤ 1/4}, is an eigenvalue of LP er± if and only if z ∈ D satisfies the basic equation (2.11)

(z − αn (z; v))2 = βn+ (z; v)βn− (z, v),

(c) For large enough |n|, the equation (2.11) has exactly two roots in D counted with multiplicity. Proof. The assertion (a) is proved in [2, Proposition 35]. Lemma 2.3 implies (b). By (2.9), supD |αn (z)| → 0 and supD |βn± (z)| → 0 as n → ∞. Therefore, (c) follows from the Rouch´e theorem.  − In view of Lemma 2.2, for large enough |n| the numbers zn∗ = (λ+ n + λn )/2 − n are well defined. The following estimate of γn from above follows from (2.9) and (2.10) (see [2, Lemma 40]).

Lemma 2.5. For large enough |n|, (2.12)

− − ∗ + ∗ γn = |λ+ n − λn | ≤ (1 + δn )(|βn (zn )| + |βn (zn )|)

with δn → 0 as |n| → ∞. Remark. Here and sometimes thereafter, we suppress the dependence on v in the notations and write αn (z) and βn± (z). ± 3. In view of the above consideration, there is n0 = n0 (v) such that λ± n , βn (z) and αn (z) are well-defined for |n| > n0 , and Lemmas 2.2, 2.3, 2.5 and Proposition 2.4 hold. Let us set

(2.13)

M± = {n ∈ Γ± :

+ n ∈ Γ± , |n| > n0 , λ− n 6= λn }.

Definition. Let X ± be the class of all Dirac potentials v with the following property: there are constants c ≥ 1 and N ≥ n0 such that (2.14)

1 + |β (v; zn∗ )| ≤ |βn− (v; zn∗ )| ≤ c |βn+ (v; zn∗ )| c n

if n ∈ M± , |n| ≥ N.

Lemma 2.6. If v ∈ X ± and the set M± is infinite, then for n ∈ M± with sufficiently large |n| we have (2.15)

1 ± |β (v; zn∗ )| ≤ |βn± (v; z)| ≤ 2|βn± (v; zn∗ )| 2 n

∀ z ∈ Kn := {z : |z − zn∗ | ≤ γn }.

Proof. By Lemma 2.5, if v ∈ X ± then for n ∈ M± with large enough |n| we have βn± (zn∗ ) 6= 0. In view of (2.10), if z ∈ Kn then for large enough |n| ± βn (z) − βn± (zn∗ ) ≤ εn |z − zn∗ | ≤ εn γn ,  p  where εn = C E|n| (r) + 1/ |n| → 0 as |n| → ∞. By Lemma 2.5, for large enough |n| we have γn ≤ 2 (|βn− (zn∗ )| + |βn+ (zn∗ )|) . Then, for n ∈ M, ±
βn (z) − βn± (zn∗ ) ≤ 2εn |βn− (zn∗ )| + |βn+ (zn∗ )| ≤ 2εn (1 + c) βn± (zn∗ ) which implies, for sufficiently large |n|, [1 − 2εn (1 + c)] βn± (zn∗ ) ≤ βn± (z) ≤ [1 + 2εn (1 + c)] βn± (zn∗ ) . Since εn → 0 as |n| → ∞, (2.15) follows.



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PLAMEN DJAKOV AND BORIS MITYAGIN

Proposition 2.7. Suppose that v ∈ X ± and the corresponding set M± is infinite. Then for n ∈ M± with large enough |n| √

2 c |βn− (v; zn∗ )| + |βn+ (v; zn∗ )| ≤ γn ≤ 2 |βn− (v; zn∗ )| + |βn+ (v; zn∗ )| . (2.16) 1 + 4c

Proof. The estimate of γn from above follows from Lemma 2.5. By Lemma 2.5, for n ∈ M± with large enough |n| we have βn± (zn∗ ) 6= 0. Set tn = |βn+ (zn+ )|/|βn− (zn+ )|,

zn+ = λ+ n − n,

n ∈ M± .

By Lemma 2.6, tn is well defined for large enough |n|. By Lemma 49 in [2], there exists a sequence (δn )n∈Z with δn → 0 as |n| → ∞ such that, for n ∈ M± with large enough |n|,  √ 
2 tn (2.17) |γn | ≥ − δn |βn− (zn∗ )| + |βn+ (zn∗ )| . 1 + tn

In view of (2.15) in Lemma 2.6, for large enough |n| we have 1/(4c) ≤ tn ≤ 4c. Therefore, by (2.17) it follows ! √
2 4c γn ≥ − δn |βn− (zn∗ )| + |βn+ (zn∗ )| , 1 + 4c which implies (since δn → 0 as |n| → ∞) the left inequality in (2.16). This completes the proof.  3. Riesz bases of root functions In view of Lemma 2.2, the Dirac operators LP er± (v) have discrete spectra; for N large enough and n ∈ Γ± with |n| > N the Riesz projections Z Z 1 1 ± −1 ± (z − LP er± ) dz, Pn = (3.1) SN = (z − LP er± )−1 dz 2πi ∂RN 2πi |z−n|= 41 ± are well–defined and dim SN < ∞, dim Pn± = 2. Further we suppress in the notations the dependence on the boundary conditions P er± and write SN , Pn only. By [4, Theorem 3], X kPn − Pn0 k2 < ∞, (3.2) n∈Γ± ,|n|>N

where Pn0 are the Riesz projections of the free operator. Moreover, the Bari–Markus criterion implies (see Theorem 9 in [4]) that the spectral Riesz decompositions X
(3.3) f = SN f + Pn f ∀f ∈ L2 [0, π], C2 n∈Γ± ,|n|>N

converge unconditionally. In other words, {S
N , Pn , n ∈ Γ± , |n| > N } is a Riesz basis of projections in the space L2 [0, π], C2 . Theorem 3.1. (A) If v ∈ X ± , then there exists a Riesz basis in L2 ([0, π], C2 ) which consists of root functions of the operator LP er± (v). (B) If v 6∈ X ± , then the system of root functions of the operator LP er± (v) does not contain Riesz bases.

RIESZ BASES

7

Remark. To avoid any confusion, let us emphasize that in Theorem 3.1 two independent theorems are stacked together: one for the case of periodic boundary conditions P er+ , and another one for the case of antiperiodic boundary conditions P er− . Proof. We consider only the case of periodic boundary conditions bc = P er+ since the proof is the same in the case of antiperiodic boundary conditions bc = P er− . (A) Fix v ∈ X + , and let N = N (v) > n0 (v) be chosen so large that Lemma 2.6, Proposition 2.7 and (3.1)–(3.3) holds for |n| > N. If n 6∈ M+ then λ∗n = n + zn∗ is a double eigenvalue. In this case we choose f (n), g(n) ∈ Ran(Pn ) so that (3.4)

kf (n)k = kg(n)k = 1,

LP er+ (v)f (n) = λ∗n f (n),

hf (n), g(n)i = 0.

+ If n ∈ M+ then λ− n and λn are simple eigenvalues. Now we choose corresponding eigenvectors f (n), g(n) ∈ Ran(Pn ) so that

− (3.5) kf (n)k = kg(n)k = 1, LP er+ (v)f (n) = λ+ n f (n), LP er + (v)g(n) = λn g(n).

Let H be the closed linear span of the system Φ = {f (n), g(n) : n ∈ Γ+ , |n| > N }. By (3.3), L2 ([0, π], C2 ) = H ⊕ Ran(SN ). Since dim SN < ∞, the theorem will be proved if we show that the system Φ is a Riesz basis in the space H. By (3.3), the system of two-dimensional projections {Pn : n ∈ Γ+ , |n| > N } is Riesz basis of projections in H. By Lemma 2.1, the system Φ is a Riesz basis in H if and only if sup |hf (n), g(n)i| < 1. n∈Γ+ ,|n|>N

By (3.4), we need to consider only indices n ∈ M+ . Next we show that (3.6)

sup |hf (n), g(n)i| < 1. M+

By Lemma 2.6 the quotient ηn (z) = βn− (z)/βn+ (z) is a well defined analytic function on a neighborhood of the disc Kn = {z : |z − zn∗ | ≤ γn }. Moreover, in view of (2.14) and (2.15), we have (3.7)

1 ≤ |ηn (z)| ≤ 4c 4c

for n ∈ M+ , z ∈ Kn .

Since ηn (z) does not vanish in Kn , there is an appropriate branch Log of log z (which depend on n) defined on a neighborhood of ηn (Kn ). We set Log (ηn (z)) = log |ηn (z)| + iϕn (z); then ηn (z) = βn− (z)/βn+ (z) = |ηn (z)|eiϕn (z) ,

(3.8) so the square root borhood of Kn by (3.9)

q

βn− (z)/βn+ (z) is a well defined analytic function on a neighq p i βn− (z)/βn+ (z) = |ηn (z)|e 2 ϕn (z) .

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PLAMEN DJAKOV AND BORIS MITYAGIN

Now the basic equation (2.11) splits into the following two equations q + + (3.10) z = ζn (z) := αn (z) + βn (z) βn− (z)/βn+ (z), q z = ζn− (z) := αn (z) − βn+ (z) βn− (z)/βn+ (z). (3.11)

For large enough |n|, each of the equations (3.10) and (3.11) has exactly one root in the disc Kn . Indeed, in view of (2.10), sup dζn± /dz → 0 as n → ∞. |z|≤1/2

Therefore, for large enough |n| each of the functions ζn± is a contraction on the disc Kn , which implies that each of the equations (3.10) and (3.11) has at most one root in the disc Kn . On the other hand, Lemma 2.2 implies that for large enough |n| the basic equation (2.11) has exactly two simple roots in Kn , so each of the equations (3.10) and (3.11) has exactly one root in the disc Kn . For large enough |n|, let z1 (n) (respectively z2 (n)) be the only root of the equation (3.10) (respectively (3.11)) in the disc Kn . Of course, we have − + + either (i) z1 (n) = λ− n −n, z2 (n) = λn −n or (ii) z1 (n) = λn −n, z2 (n) = λn −n.

Further we assume that (i) takes place; the case (ii) may be treated in the same way, and in both cases we have − |z1 (n) − z2 (n)| = γn = |λ+ n − λn |.

(3.12) We set (3.13)

f 0 (n) = Pn0 f (n),

g 0 (n) = Pn0 g(n).

From (3.2) it follows that kPn − Pn0 k → 0. Therefore,

kf (n) − f 0 (n)k = k(Pn − Pn0 )f (n)k ≤ kPn − Pn0 k → 0,

kg(n) − g 0 (n)k → 0,

so |hf (n) − f 0 (n), g(n) − g 0 (n)i| → 0. Since kf (n)k2 = kf 0 (n)k2 + kf (n) − f 0 (n)k2 and hf (n), g(n)i = hf 0 (n), g 0 (n)i + hf (n) − f 0 (n), g(n) − g 0 (n)i, we obtain (3.14)

kf 0 (n)k, kg 0 (n)k → 1,

lim sup |hf (n), g(n)i| = lim sup |hf 0 (n), g 0 (n)i|. n→∞

By Lemma 2.3, f 0 (n) is an eigenvector of the matrix

n→∞



 αn (z1 ) βn− (z1 ) correβn+ (z1 ) αn (z1 )

sponding to its eigenvalue z1 = z1 (n), i.e.,   αn (z1 ) − z1 βn− (z1 ) f 0 (n) = 0. βn+ (z1 ) αn (z1 ) − z1  T n (z1 ) Therefore, f 0 (n) is proportional to the vector z1β−α , 1 . Taking into account + n (z1 ) (3.8), (3.9) and (3.10) we obtain p  i kf 0 (n)k |ηn (z1 )|e 2 ϕ(z1 ) 0 (3.15) f (n) = p . 1 1 + |ηn (z1 )| In an analogous way, from (3.8), (3.9) and (3.11) it follows   p i kg 0 (n)k − |ηn (z2 )|e 2 ϕ(z2 ) . (3.16) g 0 (n) = p 1 1 + |ηn (z2 )|

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9

Now, (3.15) and (3.16) imply (3.17) where

p p 1 − |ηn (z1 )| |ηn (z2 )| eiψn p , hf (n), g (n)i = kf (n)kkg (n)k p 1 + |ηn (z1 )| 1 + |ηn (z2 )| 0

0

0

ψn =

0

1 [ϕn (z1 (n)) − ϕn (z2 (n)]. 2

Next we explain that (3.18)

ψn → 0

as n → ∞.

Since ϕn = Im (Log ηn ) we obtain, taking into account (3.12), that d |ϕn (z1 (n)) − ϕn (z2 (n)| ≤ sup (Log ηn ) · γn , dz [z1 ,z2 ]

where [z1 , z2 ] denotes the segment with end points z1 = z1 (n) and z2 = z2 (n). By (2.10) in Proposition 2.4 and (2.15) in Lemma 2.6 we estimate d 1 dβn− 1 dβn+ (Log ηn ) = − (z) − + (z), dz βn (z) dz βn (z) dz as follows:

where εn

z ∈ [z1 , z2 ],

d εn εn (Log ηn ) ≤ |β − (z ∗ )| + |β + (z ∗ )| dz n n n n   1 = C E|n| (r) + √ → 0 as n → ∞. Therefore, (2.14) and (2.16) |n|

imply that |ϕn (z1 (n)) − ϕn (z2 (n)| ≤ 4(1 + c) · εn → 0, i.e., (3.18) holds. From (3.17) it follows (3.19)

|hf 0 (n), g 0 (n)i|2 = kf 0 (n)k2 kg 0 (n)k2 · Πn ,

with (3.20)

p 1 + |ηn (z1 )||ηn (z2 )| − 2 |ηn (z1 )||ηn (z2 )| cos ψn Πn = . (1 + |ηn (z1 )|) (1 + |ηn (z2 )|)

Now (3.18) implies cos ψn > 0 for large enough n, so taking into account that kf 0 (n)k, kg 0 (n)k ≤ 1, we obtain by (3.7) |hf 0 (n), g 0 (n)i|2 ≤ Πn ≤ with

1 + |ηn (z1 )||ηn (z2 )| ≤δ 0 : d−1 |βn± (0)| ≤ |βn± (z)| ≤ d |βn± (0)| ∀z ∈ D = {z : |z| < 1/4}.

Theorem 3.2. Suppose bc = P er+ (or bc = P er− ), and v is a Dirac potential such that (3.30) and (3.31) hold for n ∈ Γ+ (respectively n ∈ Γ− ). Then (a) the system of root functions of LP er+ (v) (respectively LP er− (v)) is complete and contains at most finitely many linearly independent associated functions; (b) the system of root functions of LP er+ (v) (respectively LP er− (v)) contains Riesz bases if and only if (3.32)

0 < lim inf + n∈Γ

|βn− (0)| , |βn+ (0)|

lim sup n∈Γ+

|βn− (0)| 0, then there is a symmetric potential v˜ such that LP er± (v) is similar to the self-adjoint operator LP er± (˜ v ), so its spectrum Sp (LP er± (v)) ⊂ R. (b) If t < 0, then there is a skew-symmetric potential v˜ such that LP er± (v) is similar to LP er± (˜ v ). Moreover, there is an N = N (v) such that for |n| > N either + − ± + (i) λ− and λ n n are simple eigenvalues and λn = λn , Im λn 6= 0 or − (ii) λ+ n = λn is a real eigenvalue of algebraic and geometric multiplicity 2. (c) For large enough |n| βn+ (zn∗ , v) = t · βn− (zn∗ , v),

(4.2)

which implies Xt ⊂ X + ∪ X − . (d) The system of root functions of LP er+ (v) (or LP er− (v)) contains Riesz bases. Proof. For every c 6=  0, the Dirac to the Dirac operator  operator LP er± (v)  is similar  c 0 0 cP . Indeed, if C = , then a simple calculation LP er± (vc ) with vc = 1 0 1 0 cQ shows that CLP er± (v) = LP er± (vc )C. p If v ∈ Xt we set v˜ = vc with c = |t|. Then 1c Q = ct P = √t P = ±cP . |t|

Therefore, v˜ is symmetric or skew-symmetric, respectively, for t > 0 and t < 0. (b) By (2.6), (LP er± (v))∗ = LP er± (v ∗ ) with     0 tP 0 Q ∗ = 1 = vt , v = 0 P 0 tQ

so the operator LP er± (v) is similar to its adjoint operator. Therefore, if λ ∈ Sp (LP er± (v)) , then λ ∈ Sp (LP er± (v)) as well. On the other hand by Lemma 2.2, there is an N = N (v) such that for |n| > N the disc Dn = {z : |z − n| < 1/4} contains exactly two (counted with algebraic multiplicity) periodic (for even n) or antiperiodic (for odd n) eigenvalues of the operator LP er± . Therefore, if λ ∈ Dn with Im λ 6= 0 is an eigenvalue of LP er± then λ ∈ Dn is also an eigenvalue of LP er± and λ 6= λ, so λ and λ are simple, i.e., (i) holds.   w1 is a corresponding eigenvector, Suppose λ ∈ Dn is a real eigenvalue. If w 2      w2 w2 w2 = λL , i.e., is also then passing to conjugates we obtain L −w1 −w 1  1   −w w2 w1 = 0, so , an eigenvector corresponding to the eigenvalue λ. But −w1 w2 these vector-functions are linearly independent. Hence (ii) holds. (c) By (i) and (ii) it follows that 1 − (λ + λ+ n ) − n is real for |n| > N. 2 n In view of (2.8), this implies that (4.2) holds. (d) In view of (4.2), we have v ∈ X, so the claim follows from Theorem 3.1.  zn∗ =

Example 4.2. If a, b, A, B are non-zero complex numbers and   0 P (4.3) v = with P (x) = ae2ix + be−2ix , Q(x) = Ae2ix + Be−2ix , Q 0

14

PLAMEN DJAKOV AND BORIS MITYAGIN

then the system of root functions of LP er+ (v) (or LP er− (v)) contains at most finitely many linearly independent associated functions. Moreover, the system of root functions of LP er+ (v) contains Riesz bases always, while the system of root functions of LP er− (v) contains Riesz bases if and only if |aA| = |bB|. Let us mention that if bc = P er+ then it is easy to see by (2.8) that βn± (z) = 0 whenever defined, so the claim follows from Theorem 3.1. If bc = P er− , then the result follows from Theorem 3.2 and the asymptotics   −2  p  n−1 n+1 n−1 ! 1 + O(1/ |n| , (4.4) βn+ (0) = A 2 a 2 4−n+1 2 (4.5)

βn− (0)

=b

n+1 2

B

n−1 2

4

−n+1



 −2  p  n−1 ! 1 + O(1/ |n| . 2

Proofs of (4.4), (4.5) and similar asymptotics, related to other trigonometric polynomial potentials and implying Riesz basis existence or non-existence, will be given elsewhere (see similar results for the Hill-Schr¨odinger operator in [5, 6]). References [1] N. Dernek and O. A. Veliev, On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operator. Israel J. Math. 145 (2005), 113–123. [2] P. Djakov and B. Mityagin, Instability zones of periodic 1D Schr¨ odinger and Dirac operators, in Russian, Uspekhi Mat. Nauk 61 (2006), 77–182; Enlish translation: Russian Math. Surveys 61 (2006), 663–766. [3] Djakov, P., Mityagin, B.: Asymptotics of instability zones of the Hill operator with a two term potential. J. Funct. Anal. 242 (2007), 157–194. [4] P. Djakov and B. Mityagin, Bari-Markus property for Riesz projections of 1D periodic Dirac operators, Math. Nachr. 3 (2010), 443-462. [5] P. Djakov and B. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Doklady Math. 83 (2011), 5–7. [6] P. Djakov and B. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, arXiv:1007.3234, to be published in Mathematische Annalen, online: 23 November 2010. [7] P. Djakov and B. Mityagin, 1D Dirac operators with special periodic potentials, arXiv:1007.3234. [8] F. Gesztesy and V. Tkachenko, A Schauder and Riesz basis criterion for non-self-adjoint Schr¨ odinger operators with periodic and anti-periodic boundary conditions, arXiv 1104.4846 (Apr 26 and June 4, 2011). [9] I. C. Gohberg and M. G. Krein, Introduction to the theory of linear non-self-adjoint operators, vol. 18 (Translation of Mathematical Monographs). Providence, Rhode Island, American Mathematical Society, 1969. [10] A. Makin, On spectral decompositions corresponding to non-self-adjoint Sturm-Liouville operators, Dokl. Math. 73 (2006), 15–18. [11] A. S. Makin, Convergence of expansions in the root functions of periodic boundary value problems, Dokl. Math. 73 (2006), 71-76. [12] A. Makin, On the basis property of systems of root functions of regular boundary value problems for the Sturm-Liouville operator, Differ. Eq. 42 (2006), 1717–1728. [13] B. Mityagin, Convergence of expansions in eigenfunctions of the Dirac operator. (Russian) Dokl. Akad. Nauk 393 (2003), 456–459. [14] B. Mityagin, Spectral expansions of one-dimensional periodic Dirac operators. Dyn. Partial Differ. Equ. 1 (2004), 125–191. [15] A. A. Shkalikov and O. A. Veliev, On the Riesz basis property of eigen- and associated functions of periodic and anti-periodic Sturm-Liouville problems, Math. Notes 85 (2009), 647–660.

RIESZ BASES

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[16] O. A. Veliev, On the nonself-adjoint ordinary differential operators with periodic boundary conditions, Israel J. Math. 176 (2010), 195–207. Sabanci University, Orhanli, 34956 Tuzla, Istanbul, Turkey E-mail address: [email protected] Department of Mathematics, The Ohio State University, 231 West 18th Ave, Columbus, OH 43210, USA E-mail address: [email protected]