Dirichlet-Neumann bracketing for a class of banded Toeplitz matrices
aa r X i v : . [ m a t h . SP ] D ec Dirichlet-Neumann bracketing for a class of bandedToeplitz matrices
Martin Gebert A BSTRACT . We consider boundary conditions of self-adjoint banded Toeplitzmatrices. We ask if boundary conditions exist for banded self-adjoint Toeplitzmatrices which satisfy operator inequalities of Dirichlet-Neumann bracketingtype. For a special class of banded Toeplitz matrices including integer powersof the discrete Laplacian we find such boundary conditions. Moreover, for thisclass we give a lower bound on the spectral gap above the lowest eigenvalue.
1. Introduction and result
In this note we are concerned with self-adjoint banded Toeplitz matrices. Let T : = (0, 2 π ] and L ∈ N . We consider symbols of the form f : T → R , f ( x ) = N X k =− N a k e − ikx (1.1)for some N ∈ N , a k ∈ C with a k = a − k ∈ C for k = − N , ..., N . These give rise to self-adjoint banded Toeplitz matrices given by the sequence ..., 0, a − N , ..., a , ..., a N , 0, ... and T f , L is the corresponding L × L Toeplitz matrix T f , L = a a · · · a N .. . .. . a − N · · · a · · · a N . .. . . . a − N · · · a − a . (1.2)Throughout we assume that the matrix size L is bigger than the band width N + .Moreover, T f stands for the so-called Laurent or bi-infinite Toeplitz matrix T f : ℓ ( Z ) → ℓ ( Z ), ( T f b ) n : = X m ∈ Z a m − n b m (1.3)where b = ¡ b m ¢ m ∈ Z ∈ ℓ ( Z ) and a k = π Z π d x f ( x ) e − ikx , k ∈ Z . We write ( T f ) [ a , b ] for the restriction of T f to ℓ ([ a , b ]) ⊂ ℓ ( Z ) for a , b ∈ Z with a < b . Then T f , b − a + is the same matrix as ( T f ) [ a , b ] and we use both notations interchangeably.For further reading about banded Toeplitz matrices we refer to [BG05]. We brake T f , L into the direct sum of two Toeplitz matrices T f , L ⊕ T f , L = a · · · a N . .. . . . ... a − N · · · a · · · · · · · · · · · · a · · · a N ... . . . .. . a − N · · · a (1.4)where L , L ∈ N with L + L = L and we assume for convenience that L , L Ê N + . It is clear that the difference T f , L − T f , L ⊕ T f , L is of no definite sign andtherefore no operator inequality between the two operators T f , L and T f , L ⊕ T f , L holds.We are interested in adding boundary conditions to T f , L and T f , L which over-come this lack of monotonicity. For a banded Toeplitz matrix with band size N + boundary conditions refer to adding Hermitian N × N matrices at the corner of therespective boundary, i.e. a boundary condition ⋆ is given by a Hermitian N × N matrix B ⋆ and T ⋆ ,0 f , L : = T f , L + µ B ⋆
00 0 ¶ and T ⋆ f , L : = T f , L + µ e B ⋆ ¶ (1.5)where e B ⋆ is the reflection of B ⋆ along the anti-diagonal, i.e. e B ⋆ : = U ∗ B ⋆ U with U : C N → C N , ( U x ) k : = x N − k + for x = ( x , ..., x N ) ∈ C N . The superscript in theabove indicates simple boundary condition at the respective endpoint which refersto no N × N matrix added. If simple boundary conditions are imposed at bothendpoints we drop the superscripts and note T f , L = T f , L .Our goal is to find boundary conditions N and D which give rise to a chain ofoperator inequalities of the form T N f , L ⊕ T N ,0 f , L É T f , L É T D f , L ⊕ T D ,0 f , L (1.6)subject to the constraint inf T f É T N , N f , R É T D , D f , R É sup T f (1.7)where R ∈ { L , L } . Inequality (1.6) is easily satisfied for boundary conditions givenby large multiples of the N × N identity however the non-trivial constraint is (1.7)which ensures that the spectra of the restricted operators are subsets of the spectrumof the corresponding infinite-volume operator (1.3). We address the question: Given a banded self-adjoint Toeplitz matrix, do boundary conditions in thesense of (1.5) exist such that inequalities (1.6) and (1.7) hold for all L , L ∈ N with L = L + L and L , L greater than the band width? Throughout we mainly focus on the boundary condition N and later on findboundary conditions N for a special class of banded Topelitz matrices which sat-isfy the respective inequalities in (1.6) and (1.7). We don’t know if the answer tothe above question remains yes for general banded self-adjoint Toeplitz matrices.A chain of inequalities of the form (1.6) and (1.7) is referred to as Dirichlet-Neumann bracketing. This stems from the following: For the continuous negative IRICHLET-NEUMANN BRACKETING FOR TOEPLITZ MATRICES 3
Laplace operator Dirichlet and Neumann boundary conditions naturally satisfy theoperator inequality (1.6), see e.g. [RS78, Sec. XIII]. Inspired by the continuousdefinition, this was later extended to the discrete Laplacian as well [Kir08, Sec.5.2]. In both cases an inequality of the form (1.6) is by now a standard tool inmathematical physics and was, for example, used in the proof of Lifshitz tails forrandom Schrödinger operators [Sim85, Kir08, KM07] and Weyl asymptotics forcontinuum Schrödinger operators [RS78, Sec. XIII].It might be tempting to think the natural Neumann boundary condition for T f , L satisfies the first inequality in (1.6). This boundary condition, which we denote bythe superscript N , is given by the Toeplitz-plus-Hankel matrix T N ,0 f , L = T f , L + a − · · · a − N · · · ... ... a − N ... . .. , (1.8)see e.g. [NCT99]. Here we abuse notation a little as the superscript N for Neu-mann boundary condition has nothing to do with the subscript N indicating theband width of the matrix. Except in the case of a self-adjoint -diagonal Toeplitzmatrix this boundary condition does not satisfy T NL , f ⊕ T N ,0 L , f É T L , f . To see this,we consider the square of the negative discrete Laplacian on ℓ ( Z ) . Throughout,the negative discrete Laplacian − ∆ is the -diagonal Toeplitz matrix given by therows ( · · · , 0, −
1, 2, −
1, 0, · · · ) and therefore ( − ∆ ) = ∆ is -diagonal and given by ( · · · , 0, 1, −
4, 6, −
4, 1, 0, · · · ) . In that case a computation shows that ¡ ∆ ¢ L − ¡ ∆ ¢ NL ⊕ ¡ ∆ ¢ N ,0 L = .. . ... − − − − −
10 1 − ... . .. (1.9)and everywhere else. This matrix is not of definite sign and therefore the firstinequality from the left in (1.6) does not hold.In this note we introduce what we call modified Neumann boundary conditions N which satisfy the first inequality in (1.6) and (1.7) for Toeplitz matrices givenby symbols of the form f E , ··· , E n , α , ··· , α n ( x ) : = f α E ( x ) · · · f α n E n ( x ) = n Y i = ¡ − x − E i ) ¢ α i (1.10)for x ∈ T and some distinct E , ..., E n ∈ T and α , ..., α n ∈ N . In the above, we haveset f α E ( x ) : = ¡ − x − E ) ¢ α . Note that the minimum of f E , ··· , E n , α , ··· , α n is andit is attained at the points E , ..., E n .We remark that T f E is a -diagonal Laurent matrix given by rows ( · · · , 0, e iE , 2, e − iE , 0, · · · ) which is unitarily equivalent to the discrete Laplacian − ∆ M. GEBERT and we set − ∆ E : = T f E . Using this notation, we can write T f E ··· , En , α ··· , α n = n Y i = ¡ − ∆ E i ¢ α i (1.11)which is a banded Laurent matrix with band width N + where N = P ni = α i .The main theorem regarding Dirichlet-Neumann bracketing for T f E ··· , En , α ··· , α n isthe following: Theorem 1.1.
Let n ∈ N , E , .., E n ∈ T be distinct and α , ..., α n ∈ N . Let g = f E , ··· , E n , α , ··· , α n be of the form (1.10) and N = P ni = α i . Then there exist boundaryconditions which we call modified Neumann and Dirichlet boundary conditions, N and D , such that T N g , L ⊕ T N ,0 g , L É T g , L É T D g , L ⊕ T D ,0 g , L (1.12) and = inf T g É T N , N g , L ⊕ T N , N g , L É T N g , L ⊕ T N ,0 g , L (1.13) for all L , L ∈ N with L + L = L and L , L Ê N + . The boundary conditions N and D are given explicitly in Definition below. Remarks 1.2. (i) For band width greater than the boundary conditions N and D differ from the Neumann boundary N condition mentioned in (1.8) and theDirichlet boundary condition used in e.g. [NCT99] which coincides with what wecall simple boundary condition.(ii) It would be desirable to have the inequality T D f , L ⊕ T D ,0 f , L É sup T f as wellbut our modified Dirichlet boundary condition D defined in Definition 2.2 does notsatisfy this. We obtain D by a general principle that any inequality T N g , L ⊕ T N ,0 g , L É T g , L induces modified Dirichlet boundary condition such that T g , L É T D g , L ⊕ T D ,0 g , L and vice versa, see Lemma 4.1.(iii) The theorem holds for any integer power ( m ∈ N ) of the discrete Laplacianas the symbol of ( − ∆ ) m : ℓ ( Z ) → ℓ ( Z ) is g ( x ) = f m ( x ) = ¡ − x ) ¢ m , x ∈ T (1.14)and is of the form (1.10). In that case n = , E = and α = m .Considering only symbols (1.10) seems very restrictive. But, for example, The-orem 1.1 gives Dirichlet-Neumann bracketing for a rather large class of -diagonalreal-valued Toeplitz matrices: Corollary 1.3 ( -diagonal real-valued Toeplitz matrices) . Let h : T → R be thesymbol h ( x ) = a e − i x + a e − i x + a + a e i x + a e i x (1.15) where a , a , a ∈ R with a > and − É a a É . Then there exist modified Neu-mann and Dirichlet boundary conditions, N and D , for the Toeplitz matrix T h . L such that T N h , L ⊕ T N ,0 h , L É T h , L É T D h , L ⊕ T D ,0 h , L (1.16) and inf T h É T N , N h , L ⊕ T N , N h , L É T N h , L ⊕ T N ,0 h , L (1.17) IRICHLET-NEUMANN BRACKETING FOR TOEPLITZ MATRICES 5 for all L , L ∈ N with L + L = L and L , L Ê . The upcoming paper [GRM] will heavily rely on the established Dirichlet-Neumann bracketing to prove Lifshitz tails of the integrated density of states forself-adjoint Toeplitz matrices with random diagonal perturbations. Fractional pow-ers of Toeplitz matrices of the form (1.11) serve there as model operators. This is acontinuation of our study of Lifshitz tails of randomly perturbed fractional Lapla-cians in [GRM20]. Generally, Dirichlet-Neumann bracketing is a common tool inproving Lifshitz tails, see e.g. [Kir08, Sec. 6]. Another main ingredient and of inde-pendent interest is a lower bound on the spectral gap above the ground state energyof Toeplitz matrices with modified Neumann boundary condition. We prove here:
Proposition 1.4 (Spectral gap) . Let n ∈ N , E , .., E n ∈ T be distinct and α , ..., α n ∈ N . Let g = f E , ··· , E n , α , ··· , α n be of the form (1.10) and N = P ni = α i . We denote by λ L É ... É λ LL the eigenvalues of T N , N g , L counting multiplicities and ordered increas-ingly. Then λ Lk = for k =
1, ..., N and there exists C > such that for all L Ê N + λ LN + Ê CL α max , (1.18) where α max : = max © α i : i =
1, ..., n ª . In the case of T N , Ng . L , i.e the Neumann boundary conditons defined in (1.8), thelatter proposition follows rather directly from the explicit diagonalization of T N , Ng . L ,see [NCT99]. For the modified Neumann boundary conditions T N , N g , L it is morecomplicated as an explicit diagonalization of T N , N g , L is not known.
2. Definition of boundary conditions N and D The boundary conditions in Theorem 1.1 rely on a representation of self-adjointToeplitz matrices T f E ··· , En , α ··· , α n as a sum of rank-one operators. To see this wewrite for E ∈ T − ∆ E = D ∗ E D E (2.1)where D E : = T h E : ℓ ( Z ) → ℓ ( Z ) is the Laurent matrix given by the symbol h E : T → C , h E ( x ) = − e − iE e − i x , i.e. D E = .. . . . . e − iE e − iE e − iE .. . .. . . (2.2)Using this decomposition and (1.11), we write T f E ··· , En , α ··· , α n = n Y i = ¡ D ∗ E i D E i ¢ α i = ³ n Y i = D α i E i ´ ∗ ³ n Y i = D α i E i ´ (2.3) M. GEBERT where we used that all Laurent matrices commute. We denote by ( δ k ) k ∈ Z the stan-dard basis of ℓ ( Z ) . Inserting the identity = P k ∈ Z | δ k 〉〈 δ k | in the above, we obtain T f E ··· , En , α ··· , α n = X k ∈ Z ¯¯¯ n Y i = D α i E i δ k ED n Y i = D α i E i δ k ¯¯¯ (2.4)where the above series converge strongly. For k ∈ Z we define the vector ψ gk : = n Y i = D α i E i δ k = U k n Y i = D α i E i δ (2.5)whose support satisfies supp ψ gk = [ k , k + N ] ⊂ Z where supp ϕ = { n ∈ Z : ϕ ( n ) for ϕ ∈ ℓ ( Z ) and N = P ni = α i . In the above U k : ℓ ( Z ) → ℓ ( Z ) , ( U k x ) n = x n − k , isthe right shift by k ∈ Z . Summarizing the above computation, we have proved thefollowing: Proposition 2.1.
Let n ∈ N , E , .., E n ∈ T be distinct and α , ..., α n ∈ N . Let g = f E , ··· , E n , α , ··· , α n be of the form (1.10) . Then T g = X k ∈ Z | ψ gk 〉〈 ψ gk | (2.6) with ψ gk ∈ ℓ ( Z ) given by (2.5) . Now given Proposition 2.1 it is straight forward to define the boundary condi-tions N and D in the following way: Definition 2.2 (Boundary conditions N and D ) . Let n ∈ N , E , .., E n ∈ T be distinctand α , ..., α n ∈ N . Let g = f E , ··· , E n , α , ··· , α n be of the form (1.10), N = P ni = α i and ψ gk , k ∈ Z , be given in Proposition 2.1.For a ∈ Z ∪ { −∞ } and b ∈ Z with b − a > N + we define the restriction of T g to [ a , b ] ⊂ Z with simple boundary conditions at a and(i) boundary condition N at b ∈ Z by ¡ T g ¢ N [ a , b ] : = ³ X k ∈ Z :[ k , k + N ] ⊂ ( −∞ , b ] | ψ gk 〉〈 ψ gk | ´ [ a , b ] . (2.7)To be precise, for a = −∞ the respective intervals are open at a .(ii) boundary condition D at b ∈ Z by ( T g ) D [ a , b ] : = T g ) [ a , b ] − ¡ T g ¢ N [ a , b ] . (2.8)Accordingly, we define ¡ T g ¢ N / D ,0[ a , b ] by reflection along the anti-diagonal. In partic-ular,(iii) boundary conditions N at both a , b ∈ Z are given by ¡ T g ¢ N , N [ a , b ] : = X k ∈ Z :[ k , k + N ] ⊂ [ a , b ] | ψ gk 〉〈 ψ gk | . (2.9)(iv) boundary conditions D at both a , b ∈ Z by ¡ T g ¢ D , D [ a , b ] : = T g ) [ a , b ] − ¡ T g ¢ N , N [ a , b ] . (2.10) IRICHLET-NEUMANN BRACKETING FOR TOEPLITZ MATRICES 7
Remarks 2.3. (i) From the definition of the boundary conditions N and D one notes that only the respective N × N corner of ¡ T g ¢ [ a , b ] at the boundary ischanged. More precisely, ¡ T g ¢ N / D [ a , b ] = ¡ T g ¢ [ a , b ] + µ e B N / D ¶ (2.11)with e B N = − X k ∈ Z : b + ∈ [ k , k + N ] P | ψ gk 〉〈 ψ gk | P É (2.12)and e B D = X k ∈ Z : b + ∈ [ k , k + N ] P | ψ gk 〉〈 ψ gk | P Ê (2.13)where P is the projection onto the N -dimensional space ℓ ([ b − N + b ]) . Therefore N and D are boundary conditions in the sense of (1.5).(ii) For functions g as in Theorem 1.1, the latter directly implies ¡ T g ¢ N [ a , b ] É ¡ T g ¢ [ a , b ] É ¡ T g ¢ D [ a , b ] . (2.14)
3. An exampleExample 3.1.
Let E ∈ T . We consider the symbol g ( x ) = f E ,1,1 ( x ) = ¡ − x ) ¢¡ − x − E ) ¢ , x ∈ T . (3.1)The function g satisfies g Ê and its minimal value is and attained at x = and x = E . In the case E = we have T g = ( − ∆ ) which was also discussed inthe introduction, see (1.9). A short computation shows that T g is the -diagonalToeplitz matrix T g = . .. . .. . .. e − iE − − e − iE + e − iE + e iE − − e iE e iE . .. . .. . .. . (3.2)To define the Neumann boundary condition, we write using Proposition 2.1 T g = X k ∈ Z | ψ gk 〉〈 ψ gk | (3.3)where for k ∈ Z we have supp ψ gk = [ k , k + . Moreover | ψ gk 〉〈 ψ gk | = . .. ... − − e iE e iE − − e − iE + e − iE + e iE − − e iE e − iE − − e − iE ... . .. (3.4) M. GEBERT and everywhere else. Then the boundary conditions N and D from Definition 2.2are of the form ¡ T g ¢ N ,0[ a , b ] = ¡ T g ¢ [ a , b ] − + e iE + e − iE − − e iE · · ·− − e − iE ... . . . (3.5)and ¡ T g ¢ D ,0[ a , b ] = ¡ T g ¢ [ a , b ] + + e iE + e − iE − − e iE · · ·− − e − iE ... . . . (3.6)where the latter two matrices are everywhere else. Here one clearly sees that theboundary conditions N and D consist of adding or subtracting a sign-definite × matrix in the respective corner of ¡ T g ¢ [ a , b ] . This is consistent with our definition ofboundary conditions in the introduction.
4. Proof of Theorem 1.1 and Corollary 1.3 P ROOF OF T HEOREM L , L Ê N + and L + L = L . The chain of in-equalities = inf T g É T N , N g , L ⊕ T N , N g , L É T N g , L ⊕ T N ,0 g , L É T g , L (4.1)follows directly from the definition of the boundary condition N as we drop in thedefinition of N non-negative rank-one projections from T g , L . For the upper boundin the last inequality of (1.16) we note that T g , L = µ T g , L P T g , L P ⊥ P ⊥ T g , L P T g , L ¶ Ê Ã T N g , L T N ,0 g , L ! (4.2)interpreted as an operator on ℓ ([1, L ]) ⊕ ℓ ([ L + L ]) and P stands here for pro-jection onto ℓ ([1, L ]) ⊕ {0} and P ⊥ = − P . Now Lemma 4.1 below gives the resultas the definition of the modified Dirichlet boundary condition in (2.8) is preciselyof the form (4.5). (cid:3) In the next lemma we show that any boundary condition satisfying the firstinequality in (1.16) naturally induces a boundary condition satisfying the secondinequality in (1.16).
Lemma 4.1.
Let H and H be two possibly infinite-dimensional Hilbert spaces.Let A : H ⊕ H → H ⊕ H be a bounded operator A = µ A A A A ¶ . (4.3) Assume there exist A N : H → H and A N : H → H such that µ A A A A ¶ Ê µ A N A N ¶ . (4.4) Then µ A A A A ¶ É µ A − A N
00 2 A − A N ¶ . (4.5) IRICHLET-NEUMANN BRACKETING FOR TOEPLITZ MATRICES 9 P ROOF . We conjugate inequality (4.6) by the unitary U = µ − ¶ . Hence (4.6) isequivalent to U ∗ AU Ê U ∗ µ A N A N ¶ U = µ A N A N ¶ . (4.6)We note that U ∗ AU = µ A − A − A A ¶ (4.7)which together with (4.6) gives µ A
00 2 A ¶ − µ A − A − A A ¶ É µ A
00 2 A ¶ − µ A N A N ¶ (4.8)which is the result. (cid:3) P ROOF OF C OROLLARY b ∈ T . We compute for x ∈ T (2 − x − b ))(2 − x + b )) = (2 − e i x e − ib − e − i x e ib )(2 − e i x e ib − e − i x e − ib ) = e − i x − b ) e − i x + + b ) − b ) e i x + e i x = : w b ( x ). (4.9)Let h : T → R be of the form as described in Corollary 1.3 h ( x ) = a e − i x + a e − i x + a + a e i x + a e i x (4.10)where a , a , a ∈ R with a > and − É a a É . We rewrite h ( x ) = a ¡ e − i x + a a e − i x + a a + a a e i x + e i x ¢ . (4.11)As we assumed − É a a É there exists b ∈ T such that b ) = a a and henceusing the definition of w b in (4.9) we obtain h ( x ) = a ¡ (2 − x − b ))(2 − x + b )) ¢ + a − − b ) = a w b ( x ) + c (4.12)with c : = a − − b ) . Theorem 1.1 implies there exists boundary conditions N and D such that = inf T w b É T N , N w b , L ⊕ T N , N w b , L É T N w b , L ⊕ T N ,0 w b , L É T w b , L É T D w b , L ⊕ T D ,0 w b , L (4.13)for all L , L ∈ N with L + L = L and L , L Ê N + . Multiplying w b with a Ê and adding c will not change the chain of operator inequalities (4.13) and the resultfollows. (cid:3)
5. Proof of Proposition 1.4 P ROOF OF P ROPOSITION g and N as in the assumptions and L ∈ N with L Ê N + . We first prove that λ Lk = for all k =
1, ..., N . To do so, we consider the N vectors ϕ j i E i = ¡ k j i e iE i k ¢ k = L = ¡ j i e iE i , · · · , L j i e iE i L ¢ T ∈ C L = ℓ ([1, L ]) (5.1) where i =
1, ..., n and j i =
0, ..., α i − . A computation shows that for all k ∈
1, ..., L − N ¡ D α i E i ϕ j i E i ¢ k = (5.2)where we see D α i E i here as an operator D α i E i : ℓ ([1, L ]) → ℓ ([1, L ]) . Therefore by thedefinition of ψ gk in (2.5) we obtain for k ∈
1, ..., L − N 〈 ψ gk , ϕ j i E i 〉 = (5.3)for all i =
1, ..., n and j i =
0, ..., α i − . Recalling the definition of T N , N g , L in (2.9),we obtain from the previous identity T N , N g , L ϕ j i E i = X m ∈ Z :[ k , k + N ] ⊂ [1, L ] | ψ gk 〉〈 ψ gk | ϕ j i E i 〉 = (5.4)for i =
1, ..., n and j i =
0, ..., α i − . Lemma 5.2 shows that the N vectors in (5.1)are linearly independent and therefore span a N dimensional space which implies λ Lk = for k =
1, ..., N .Next we prove the lower bound on λ LN + = λ LN + ¡ T N , N g , L ¢ , where we use thenotation λ Lk ( · ) if we want to emphasize to underlying operator. We consider firstthe L × L restriction of T g with periodic boundary conditions T per g , L : = a · · · a N a − N · · · a − . . . .. . . .. ..... . .. . a − a − N · · · a · · · a N a .. . .. .... . . . . .. . .. a · · · a N − a − N · · · a . (5.5)For k =
1, ..., L we define the vector ψ ( k ) = ¡ ψ ( k )1 , .., ψ ( k ) L ¢ T ∈ C L ψ ( k ) m : = p L e π k ( m − L i , m =
1, ..., L . (5.6)A computation shows for k =
1, ..., L that T per g , L ψ ( k ) = g ³ π kL ´ ψ ( k ) . (5.7)Therefore, the family of vectors ¡ ψ ( k ) ¢ k = L form an ONB of eigenvectors of T per g , L corresponding to the eigenvalues g ³ π kL ´ , k =
1, ..., L .Using the definition of T N , N g , L in (2.9), we observe that T per g , L − T N , N g , L = L X k = L − N + | ψ gk 〉〈 ψ gk | (5.8)where for k = L − N +
1, ..., L ψ gk = ¡ c L − k + , · · · , c N , 0, · · · , 0, c , · · · , c L − k ¢ T ∈ C L (5.9) IRICHLET-NEUMANN BRACKETING FOR TOEPLITZ MATRICES 11 with c k : = ( ψ g ) k for k =
0, ..., N . Therefore, the difference in (5.8) is rank N . Fromthe first part of the proof we know that λ Lj ( T N , N g , L ) = for j =
1, ..., N . Now themin-max principle implies the lower bound λ LN + ¡ T N , N g , L ¢ Ê λ L ¡ T per g , L ¢ = min k = L g ³ π kL ´ (5.10)and the last equality follows from (5.7). Next we define for E ∈ T the unitary U E : C L → C L , ( U E b ) m = e − iEm b m for b ∈ C L and m =
1, ..., L . Then, by the definition of ψ gk the following identity holds U E T N , N g , L U ∗ E = T N , N g E , L (5.11)where g E ( x ) = g ( x − E ) , x ∈ T , and we extended g here periodically such that g ( x − E ) makes sense for any x ∈ T and E ∈ T . As the spectrum does not changeunder conjugation by a unitary, we obtain λ LN + ¡ T N , N g , L ¢ = λ LN + ¡ T N , N g E , L ¢ (5.12)for all E ∈ T and using the lower bound (5.10) we end up with λ LN + ¡ T N , N g , L ¢ = max E ∈ T λ LN + ¡ T N , N g E , L ¢ Ê max E ∈ T min k = L g E ³ π kL ´ . (5.13)Given the distinct minima E , ..., E n ∈ T of the function g , Lemma 5.1 below pro-vides a constant C > such that for all L > N + there exists ˜ E ∈ T such that C L É min i = n dist ¡ E i , ³ π kL − ˜ E ´ mod π : k =
1, ..., L ¢ É π L . (5.14)We note that C > in the above is independent of L and only depends on n . Since E , ..., E n are the minima of the function g , we obtain with the ˜ E ∈ T found above,inequality (5.14) and Taylor’s theorem the lower bound(5.13) Ê min k = L g ³ π kL − ˜ E ´ Ê C L α max (5.15)for some C > depending on g but independently of L which is the assertion. (cid:3) Lemma 5.1.
Let E , ..., E n ∈ T be n ∈ N distinct points and set E ( n ) : = © E i , i =
1, ..., n ª . Then there exists e E ∈ T such that dist( S e EL , E ( n ) ) Ê π n L (5.16) where S e EL : = n³ π kL − e E ´ mod π : k =
1, ..., L o (5.17) and dist( A , B ) = min © | a − b | : a ∈ A , b ∈ B ª for A , B ⊂ R . P ROOF . We prove the lemma by induction on n ∈ N .For n = let E (1) = © E ª . Then we choose e E = − E + π L and therefore (5.16) istrue. Assume the result is true for n − distinct points E , ..., E n − and let E n be apoint distinct from the others. By assumption there exists e E such that dist( S e EL , E ( n − ) Ê π n − L . (5.18)If dist ¡ S e EL , E n ¢ Ê π n − L we are done. If this is not the case we obtain by adding orsubtracting π n L to e E that there exists ˆ E ∈ T such that dist ¡ S ˆ EL , E n ¢ Ê π n L . (5.19)Since | e E − ˆ E | É π n L and dist( S ˜ EL , E ( n − ) Ê π n − L , we obtain dist( S ˆ EL , E ( n − ) Ê π n L (5.20)which is the assertion together with (5.19). (cid:3) Lemma 5.2.
Let n ∈ N , E , .., E n ∈ T be distinct, α , ..., α n ∈ N and N = P ni = α i .Moreover, let L ∈ N with L Ê N . The N vectors ϕ j i E i = ¡ k j i e iE i k ¢ k = L = ¡ j i e iE i , · · · , L j i e iE i L ¢ T ∈ C L (5.21) where i =
1, ..., n and j i =
0, ..., α i − are linearly independent. P ROOF . Let i ∈ {1, ..., n } and j i ∈ {0, ..., α i − . We introduce the short-hand notation z i : = e iE i and define the truncation of ϕ j i i to C N ˆ ϕ j i i : = ¡ z i , 2 j i z i , ..., N j i z Ni ¢ T ∈ C N . (5.22)This is just the truncation of ϕ j i i to the first N rows. Now det ¡ ˆ ϕ , ..., ˆ ϕ α − , ˆ ϕ , · · · , ˆ ϕ α n − n ¢ is a confluent Vandermonde determinant whichcan be computed explicitly and evaluates to ¯¯ det ¡ ˆ ϕ , ..., ˆ ϕ α − , ˆ ϕ , · · · , ˆ ϕ α n − n ¢¯¯ = n Y i = ( α i − Y É i < j É n | z i − z j | α i α j , (5.23)see e.g [HG80, Thm. 1]. Since z i z j for all i j , we obtain that the latter deter-minant is non-zero. Therefore, the N vectors ˆ ϕ , ..., ˆ ϕ α n − n are linearly independent.This implies that the vectors © ϕ j i i : i =
1, ..., n , j i =
0, ..., α i − ª are linearly indepen-dent as well. (cid:3) Acknowledgements
The author thanks Constanza Rojas-Molina for many interesting and enjoyablediscussions on the subject and Peter Müller for helpful comments on an earlierversion of the paper.
IRICHLET-NEUMANN BRACKETING FOR TOEPLITZ MATRICES 13
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