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BARI–MARKUS PROPERTY FOR RIESZ PROJECTIONS OF 1D PERIODIC DIRAC OPERATORS 9 PLAMEN DJAKOVANDBORIS MITYAGIN 0 0 Abstract. The Dirac operators 2 n Ly=i 1 0 dy +v(x)y, y= y1 , x∈[0,π], a „0 −1«dx „y2« J with L2-potentials 7 v(x)= 0 P(x) , P,Q∈L2([0,π]), „Q(x) 0 « ] P considered on [0,π] with periodic, antiperiodic or Dirichlet boundary S conditions (bc), havediscrete spectra, and theRiesz projections . h 1 1 at SN = 2πiZ|z|=N−21(z−Lbc)−1dz, Pn= 2πiZ|z−n|=14(z−Lbc)−1dz m are well–defined for |n|≥N if N is sufficiently large. It is proved that [ kPn−Pn0k2 <∞, 1 |nX|>N 6v where Pn0, n∈Z, are theRiesz projections of thefree operator. Then, by the Bari–Markus criterion, the spectral Riesz decomposi- 5 8 tions 0 f =SNf + Pnf, ∀f ∈L2; . |nX|>N 1 converge unconditionally in L2. 0 9 0 : v 1. Introduction i X Thequestionforunconditionalconvergenceofthespectraldecompositions r a is one of the central problems in Spectral Theory of Differential Operators [2, 3, 18, 24, 25, 21]. In the case of ordinary differential operators on a finite interval, say I = [0,π], dmy m−2 dky (1.1) ℓ(y)= + q (x) , q Hk(I), dxm k dxk k ∈ k=0 X with strongly regular boundary conditions (bc) the eigenfunction decomposition (1.2) f(x) = c (f)u (x), ℓ(u )= λ u , u (bc), k k k k k k ∈ k X 2000 Mathematics Subject Classification. 34L40(primary),47B06, 47E05(secondary). 1 2 PLAMENDJAKOVANDBORISMITYAGIN converge unconditionally for every f L2(I) (see [20, 15, 3]). ∈ If (bc) are regular but not strictly regular the system of root functions (eigenfunctions andassociated functions)in general is not abasis inL2.But if therootfunctions arecombined properlyin disjointgroupsB , B =N, n n ∪ then the series (1.3) f(x)= P f, P f = c (f)u (x), n n k k Xn kX∈Bn converges unconditionally in L2 (see [27, 28]). Let us be more specific in the case of operators of second order (1.4) ℓ(y)= y′′+q(x)y, 0 x π. ≤ ≤ Then, Dirichlet bc = Dir : y(0) = y(π) = 0 is strictly regular; however, Periodic bc = Per+ : y(0) = y(π), y′(0) = y′(π) and Antiperiodic bc = Per− : y(0) = y(π), y′(0) = y′(π) are regular, but not strictly regular. − − Analysis – even if it becomes more difficult and technical – could be extended to singular potentials q H−1. A. Savchuk and A. Shkalikov ∈ showed ([26], Theorems 2.7 and 2.8) that for both Dirichlet bc or (properly understood) Periodic or Antiperiodic bc, the spectral decomposition (1.3) converges unconditionally. An alternative proof of this result is given in [10]. For Dirac operators (2.1) the results on unconditional convergence are sparse and not complete so far [28, 16, 17, 29, 30, 12]. The case of separate boundary conditions, at least for smooth potential v, has been studied in detail in [16]. For periodic (or antiperiodic) bc B.Mityagin proved unconditional convergence of the series (1.3) with dimP = 2, n N(v), for potentials v Hb, b > 1/2 – see Theorem 8.8 n | | ≥ ∈ [23] for a precise statement. Ourtechniquesfrom[10]toanalyzetheresolvents (λ L )−1 ofHilloper- bc − ators with the weakest (in Sobolev scale) assumption v H−1 on ”smooth- ∈ ness” of the potential are adjusted and extended in the present paper to Dirac operators with potentials in L2. We prove (see Theorem 3 for a pre- cisestatement) thatifv L2 andbc = Per±,Dir thesequenceofdeviations ∈ P P0 is in ℓ2.Then,theBari–Markus criterion (see [1,19]or [11], Ch.6, k n− nk Sect.5.3, Theorem 5.2)) shows that the spectral decomposition (1.5) f = S f + P f, f L2, N n ∀ ∈ |n|>N X where, for n N(v), | |≥ 2 bc = Per± (1.6) dimP = , n (1 bc = Dir converge unconditionally. This is Theorem 9, the main result of the present paper. BARI–MARKUS PROPERTY 3 Further analysis requires thorough discussion of the algebraic structure of regular and strictly regular bc for Dirac operators. Then we can claim a general statement which is an analogue of (1.5)–(1.6), or Theorem 9, with bc = Dir in case of strictly regular boundary conditions, and bc = Per± in case of regular but not strictly regular boundary conditions. We will give all the details in another paper. 2. Preliminary results Consider the Dirac operator on I = [0,π] 1 0 dy (2.1) Ly = i +v(x)y, 0 1 dx (cid:18) − (cid:19) where 0 P(x) y (2.2) v(x) = , y = 1 , Q(x) 0 y 2 (cid:18) (cid:19) (cid:18) (cid:19) and v is an L2–potential, i.e., P,Q L2(I). ∈ f We equip the space H0 of L2(I)–vector functions F = 1 with the f 2 (cid:18) (cid:19) scalar product 1 π F,G = f (x)g (x)+f (x)g (x) dx. 1 1 2 2 h i π Z0 (cid:16) (cid:17) Consider the following boundary conditions (bc) : (a) periodic Per+ : y(0) = y(π), i.e., y (0) = y (π) and y (0) = y (π); 1 1 2 2 (b) anti-periodic Per− : y(0) = y(π), i.e., y (0) = y (π) and y (0) = 1 1 2 − − y (π); 2 − (c) Dirichlet Dir : y (0) = y (0), y (π) = y (π). 1 2 1 2 The corresponding closed operator with a domain f (2.3) ∆ = f (W2(I))2 : f = 1 (bc) bc ∈ 1 f2 ∈ (cid:26) (cid:18) (cid:19) (cid:27) will be denoted by L , or respectively, by L and L . If v = 0, i.e., bc Per± dir P 0,Q 0, we write L0 (or simply L0), or L0 ,L0 respectively. Of ≡ ≡ bc Per± Dir course, it is easy to describe the spectra and eigenfunctions for L0 . bc (a) Sp(L0 ) = n even = 2Z; each number n 2Z is a double eigen- Per+ { } ∈ value, and the corresponding eigenspace is (2.4) E0 = Span e1,e2 , n 2Z, n { n n} ∈ where e−inx 0 (2.5) e1(x) = , e2(x) = ; n 0 n einx (cid:18) (cid:19) (cid:18) (cid:19) (b) Sp(L0 ) = n odd = 2Z+1; thecorrespondingeigenspaces E0 are given by (2.P4e)r−and ({2.5) bu}t with n 2Z+1; n ∈ 4 PLAMENDJAKOVANDBORISMITYAGIN (c) Sp(L0 ) = n Z ; each eigenvalue n is simple. The corresponding Dir { ∈ } normalized eigenfunction is 1 (2.6) g (x)= e1 +e2 , n Z, n √2 n n ∈ (cid:0) (cid:1) so the corresponding (one-dimensional) eigenspace is (2.7) G0 = Span g . n { n} We study the spectral properties of the operators L and L by Per± Dir using their Fourier representations with respect to the eigenvectors of the corresponding free operators given above in (2.4)–(2.7). Let (2.8) P(x) = p(m)eimx, Q(x) = q(m)eimx, m∈2Z m∈2Z X X and (2.9) P(x) = p (m)eimx, Q(x) = q (m)eimx, 1 1 m∈1+2Z m∈1+2Z X X be, respectively, the Fourier expansions of the functions P and Q about the systems eimx, m 2Z and eimx, m 1+2Z . { ∈ } { ∈ } Then (2.10) v 2 = p(m)2+ q(m)2 = p (m)2+ q (m)2 . 1 1 k k | | | | | | | | m∈2Z m∈1+2Z X (cid:0) (cid:1) X (cid:0) (cid:1) Inits Fourierrepresentation, theoperatorL0 isdiagonal, andV isdefined byitsaction onvectors e1 ande2,withn 2Zforbc = Per+ andn 1+2Z n n ∈ ∈ for bc= Per−. In view of (2.2) and (2.8), we have (2.11) Ve1 = q(k+n)e2, Ve2 = p( k n)e1, n k n − − k k∈n+2Z k∈n+2Z X X so, the matrix representation of V is 0 V12 (2.12) V , (V12) = p( k n), (V21) = q(k+n). ∼ V21 0 kn − − kn (cid:18) (cid:19) In the case of Dirichlet boundary conditions the operator L0 is diagonal as well. The matrix representation of V given by the following lemma. Lemma 1. Let (gn)n∈Z be the orthogonal normalized basis of eigenfunctions of L0 in the case of Dirichlet boundary conditions. Then (2.13) V := Vg ,g = W(k+n), k,n Z, kn n k h i ∈ with (p( m)+q(m))/2 m even (2.14) W(m)= − . ((p1( m)+q1(m))/2 m odd − BARI–MARKUS PROPERTY 5 Theprooffollows from a directcomputation of Vg ,g .Let usmention, n k h i that the sequences p (m) and q (m) in (2.14) are Hilbert transforms of p(n) 1 1 and q(n) (see [6], Lemma 2 in Section 1.3) but we do not need this fact. In the following only the relation (2.10) is essential. In view of (2.4)–(2.7) the operator R0 = (λ L0)−1 is well defined, respectively, for λ 2Z if bc = Per+, λ λ1+2Z−if bc = Per−, and λ Z if 6∈ 6∈ 6∈ bc= Dir. The operator R0 is diagonal, and we have λ 1 1 (2.15) R0e1 = e1, R0e2 = e2 for bc= Per±, λ n λ n n λ n λ n n − − and 1 (2.16) R0g = g for bc= Dir. λ n λ n n − The standard perturbation type formulae for the resolvent R = (λ λ − L0 V)−1 are − ∞ (2.17) R =(1 R0V)−1R0 = (R0V)kR0, λ − λ λ λ λ k=0 X and ∞ (2.18) R = R0(1 VR0)−1 = R0(VR0)k. λ λ − λ λ λ k=0 X The simplest conditions that guarantee convergence of the series (2.17) or (2.18) in ℓ2 are R0V < 1, respectively, VR0 < 1. k λ k k λk In the case of Dirac operators there are no such good estimates but there are good estimates for the norms of (R0V)2 and (VR0)2 (see [5] and [6], λ λ Section 1.2, for more comments). But now we are going to suggest another approach that is borrowed from the study of Hill operators with periodic singular potentials (see [8, 9, 10]). Notice, that one can write (2.17) or (2.18) as ∞ (2.19) R = R0 +R0VR0 + = K2+ K (K VK )mK , λ λ λ λ ··· λ λ λ λ λ m=1 X provided (2.20) (K )2 = R0. λ λ In view of (2.15) and (2.16), we define an operator K = K with the λ property (2.20) by 1 1 (2.21) K e1 = e1, K e2 = e2 for bc= Per±, λ n √λ n n λ n √λ n n − − and 1 (2.22) K g = g for bc= Dir, λ n n √λ n − 6 PLAMENDJAKOVANDBORISMITYAGIN where √z = √reiϕ/2 if z =reiϕ, π ϕ < π. − ≤ Then R is well–defined if λ (2.23) K VK < 1. λ λ ℓ2→ℓ2 k k In view of (2.11) and (2.21), for periodic or anti–periodic boundary conditions bc = Per±, we have (2.24) q(k+n) p( k n) (K VK )e1 = e2, (K VK )e2 = − − e1, λ λ n (λ k)1/2(λ n)1/2 k λ λ n (λ k)1/2(λ n)1/2 k k − − k − − X X so, the Hilbert–Schmidt norm of the operator K VK is given by λ λ q(k+m)2 p( k m)2 (2.25) K VK 2 = | | + | − − | , k λ λkHS λ k λ m λ k λ m k,m | − || − | k,m | − || − | X X where k,m 2Z for bc= Per+ and k,m 1+2Z for bc= Per−. ∈ ∈ In an analogous way (2.13), (2.14) and (2.22) imply, for Dirichlet boundary conditions bc= Dir, W(k+n) (2.26) (K VK )g = g , k,n Z, λ λ n (λ k)1/2(λ n)1/2 k ∈ k − − X and therefore, we have W(k+m)2 (2.27) K VK 2 = | | , k,m Z. k λ λkHS λ k λ m ∈ k,m | − || − | X For convenience, we set (2.28) r(m)= max(p(m), p( m))+max(q(m), q( m)), m 2Z, | | | − | | | | − | ∈ if bc = Per±, and (2.29) r(m)= W(m) m Z, | | ∈ ifbc = Dir.NowwedefineoperatorsV¯ andK¯ whichdominate,respectively, λ V and K , as follows: λ (2.30) V¯e1 = r(k+n)e2, V¯e2 = r(k+n)e1 for bc= Per±, n k n k k∈n+2Z k∈n+2Z X X (2.31) V¯g = r(k+n)g for bc = Dir, n k k∈Z X and 1 1 (2.32) K¯ e1 = e1, K¯ e2 = e2 for bc = Per±, λ n n λ n n λ n λ n | − | | − | p 1 p (2.33) K¯ g = g for bc= Dir. λ n n λ n | − | p BARI–MARKUS PROPERTY 7 Since the matrix elements of the operator K VK do not exceed, by λ λ absolute value, the matrix elements of K¯ V¯K¯ , we estimate from above the λ λ Hilbert–SchmidtnormoftheoperatorK VK byoneandthesameformula: λ λ r(i+k)2 (2.34) K VK 2 K¯ V¯K¯ = | | , k λ λkHS ≤ k λ λkHS λ i λ k i,k | − || − | X where i,k 2Z if bc = Per+ and i,k 1+2Z if bc = Per−, or i,k Z ∈ ∈ ∈ if bc = Dir. Next we estimate the Hilbert–Schmidt norm of the operator K¯ V¯K¯ for λ C = λ : λ n = 1/2 . λ λ n For each ℓ2–∈sequenc{e x =| (x−(j)|)j∈Z an}d m N we set ∈ 1/2 (2.35) (x) = x(j)2 . m E  | |  |j|≥m X   Lemma 2. In the above notations, if n = 0, then 6 (2.36) r(i+k)2 r 2 K¯ V¯K¯ 2 = | | C k k +( (r))2 , λ C , k λ λkHS λ i λ k ≤ n E|n| ! ∈ n Xi,k | − || − | | | where C is an absolute constant. p Remark: For convenience, here and thereafter we denote by C any absolute constant. Proof. Since (2.37) 2λ i n i if i = n, λ C = λ : λ n = 1/2 , n | − | ≥ | − | 6 ∈ { | − | } the sum in (2.36) does not exceed r(n+k)2 r(n+i)2 r(i+k)2 4r(2n)2 +4 | | +4 | | +4 | | . | | n k n i n i n k k6=n | − | i6=n | − | i,k6=n | − || − | X X X In view of (4.3) and (4.4) in Lemma 7, each of the above sums does not exceed the right-hand side of (2.36), which completes the proof. (cid:3) Corollary: There is N N such that ∈ (2.38) K VK 1/2 for λ C , n > N. λ λ n k k≤ ∈ | | 3. Core results By our Theorem 18 in [6] (about spectra localization), for sufficiently large n , say n > N, the operator L has exactly two (counted with Per± | | | | their algebraic multiplicity) periodic (for even n) or antiperiodic (for odd n) eigenvalues inside the disc with a center n of radius 1/2. The operator L Dir has, for all sufficiently large n , one eigenvalue in every such disc. | | 8 PLAMENDJAKOVANDBORISMITYAGIN Let P and P0 be the Riesz projections corresponding to L and L0, i.e., n n 1 1 P = (λ L)−1dλ, P0 = (λ L)−1dλ, n 2πi − n 2πi − ZCn ZCn where C = λ : λ n = 1/2 . n { | − | } Theorem 3. Suppose L and L0 are, respectively, the Dirac operator (2.1) with an L2 potential v and the free Dirac operator, subject to periodic, antiperiodic or Dirichlet boundary conditions bc = Per± or Dir. Then, there is N N such that for n > N the Riesz projections P and P0 corresponding ∈ | | n n to L and L0 are well defined and we have (3.1) P P0 2 < . k n − nk ∞ |n|>N X Proof. Now we present the proof of the theorem up to a few technical in- equalities. They will be proved later in Section 4, Lemmas 7 and 8. 1. Let us notice that the operator–valued function K is analytic in λ C R . But (2.19), (3.2) below and all formulas of this section – which + \ are essentially variations of (2.19) – always have even powers of K , and λ K2 = R0 is analytic onC Sp(L0).Certainly, thisjustifiestheuseofCauchy λ λ \ formula or Cauchy theorem when warranted. In view of (2.38), the corollary after the proof of Lemma 2, if n is suffi- | | ciently large then the series in (2.19) converges. Therefore, ∞ 1 (3.2) P P0 = K (K VK )s+1K dλ. n− n 2πi λ λ λ λ ZCn s=0 X Remark. We are going to prove (3.1) by estimating the Hilbert–Schmidt norms P P0 which dominate P P0 . Of course, these norms are k n− nkHS k n− nk equivalent as long as the dimensions dim(P P0) are uniformly bounded n − n because for any finite dimensional operator T we have T T (dimT)1/2 T HS k k ≤ k k ≤ k k but in the context of this paper for all projections dimP , dimP0 2. n n ≤ 2. If bc = Dir, then, by (2.6), P P0 2 = (P P0)g ,g 2. k n − nkHS |h n − n m ki| m,k∈Z X By (3.2), we get ∞ (P P0)g ,g = I (s,k,m), h n − n m ki n s=0 X where 1 I (s,k,m) = K (K VK )s+1K g ,g dλ. n λ λ λ λ m k 2πi h i ZCn BARI–MARKUS PROPERTY 9 Therefore, ∞ P P0 2 I (s,k,m) I (t,k,m). k n − nkHS ≤ | n |·| n | |n|>N s,t=0|n|>Nm,k∈Z X X X X Now, the Cauchy inequality implies ∞ (3.3) P P0 2 (A(s))1/2(A(t))1/2, k n − nkHS ≤ |n|>N s,t=0 X X where (3.4) A(s) = I (s,k,m)2. n | | |n|>Nm,k∈Z X X Notice that A(s) depends on N but this dependence is suppressed in the notation. From the matrix representation of the operators K and V we get λ (3.5) W(k+j )W(j +j ) W(j +m) K (K VK )s+1K e ,e = 1 1 2 ··· s , λ λ λ λ m k h i (λ k)(λ j ) (λ j )(λ m) 1 s j1X,...,js − − ··· − − and therefore, 1 W(k+j )W(j +j ) W(j +m) 1 1 2 s (3.6) I (s,k,m) = ··· dλ. n 2πi (λ k)(λ j ) (λ j )(λ m) ZCnj1X,...js − − 1 ··· − s − In view of (2.29), we have W(k+j )W(j +j ) W(j +m) 1 1 2 s (3.7) ··· B(λ,k,j ,...,j ,m), 1 s (λ k)(λ j ) (λ j )(λ m) ≤ (cid:12) − − 1 ··· − s − (cid:12) (cid:12) (cid:12) where (cid:12) (cid:12) (cid:12) (cid:12) (3.8) r(k+j )r(j +j ) r(j +j )r(j +m) 1 1 2 s−1 s s B(λ,k,j ,...,j ,m)= ··· , s > 0, 1 s λ k λ j λ j λ m 1 s | − || − |···| − || − | and r(m+k) (3.9) B(λ,k,m) = λ k λ m | − || − | inthecasewhens = 0andtherearenoj-indices. Moreover, by(2.29),(2.31) and (2.33), we have (3.10) B(λ,k,j ,...,j ,m)= K¯ (K¯ V¯K¯ )s+1K¯ e ,e . 1 s λ λ λ z m k h i j1X,...,js Lemma 4. In the above notations, we have (3.11) A(s) B (s)+B (s)+B (s)+B (s), 1 2 3 4 ≤ 10 PLAMENDJAKOVANDBORISMITYAGIN where 2 (3.12) B (s) = sup B(λ,n,j ,...,j ,n) ; 1 1 s   |nX|>Nλ∈Cn j1X,...,js   2 (3.13) B (s) = sup B(λ,k,j ,...,j ,n) ; 2 1 s   |nX|>NkX6=nλ∈Cn j1X,...,js   2 (3.14) B (s) = sup B(λ,n,j ,...,j ,m) ; 3 1 s   |nX|>NmX6=nλ∈Cn j1X,...,js   2 ∗ (3.15) B (s) = sup B(λ,k,j ,...,j ,m) , s 1, 4 1 s   ≥ |nX|>NmX,k6=nλ∈Cn j1X,...,js   where the symbol over the sum in the parentheses means that at least one ∗ of the indices j ,...,j is equal to n. 1 s Proof. Indeed, in view of (3.4), we have A(s) A (s)+A (s)+A (s)+A (s), 1 2 3 4 ≤ where A (s) = I (s,n,n)2, A (s)= I (s,k,n)2 1 n 2 n | | | | |n|>N |n|>Nk6=n X X X A (s)= I (s,n,m)2, A (s)= I (s,k,m)2. 3 n 4 n | | | | |n|>Nm6=n |n|>Nm,k6=n X X X X By (3.6)–(3.9) we get immediately that A (s) B (s), ν = 1,2,3. ν ν ≤ On the other hand, by the Cauchy formula, W(k+j )W(j +j ) W(j +m) 1 1 2 s ··· dλ =0 if k,j ,...,j ,m = n. 1 s (λ k)(λ j ) (λ j )(λ m) 6 ZCn − − 1 ··· − s − Therefore, removing from the sum in (3.6) the terms with zero integrals, and estimating from above the remaining sum, we get ∗ I (s,k,m) sup B(λ,k,j ,...,j ,m) , m,k = n. n 1 s | | ≤   6 λ∈Cn j1X,...,js   From here it follows that A (s) B (s), which completes the proof. (cid:3) 4 4 ≤