arXiv:1503.01761v1 [math-ph] 5 Mar 2015 ∗ J-rnbe1 NSIsiu ore M 52 rnbe 38402 Grenoble, 5582, UMR Fourier Institut CNRS 1, UJF-Grenoble ooopi ucinlcluu o n otatosdeve contractions cnu for calculus functional holomorphic ecnrsrc teto ocnrcinoeaosta ea short). we that for as operators (cnu contraction properties, to their attention of restrict can some we study and normal, necessary not h w oin onie hspoeuewrseulywell equally works procedure This [J1]. coincide. notions two the aigtelmtΛ limit the taking fterno prtrrsrce obxsΛ boxes to restricted operator random th the considering of in consists definition appealing physically intermpoie h soitddniyo tt measu state of density operat random associated the the is of provides function theorem functional the tion the of element function, matrix a diagonal Given assoc r measure the the on operator. is functions self-adjoint continuous It supported Ki]. we compactly CL, a on [CFKS, acting with textbooks the operators, self-adjoint e.g. see random n mathematical of important properties an is tral measure states of density The Introduction 1 tran Borel the to related is circle t unit Moreover, the on restrictions. betw functional volume DOS discrepancy finite the r their despite integral and an measure, operators admit limiting shown the is restric of volume functional terms F finite DOS the the restrictions. vol of limit, volume infinite spectrum volume finite the sure the of almost measure of the counting functions be of fi to volume suitable proven with unit is repres contractions functional random integral ra For DOS natural for disk. the enjoy functional unit to the state shown on of and is density contra functional ergodic the DOS random with The with analogy associated in is space, which disk unit the steteei ocniuu ucinlcluu nti fra this in calculus functional continuous no is there the As ervstteecntutosi h rmwr fbuddr bounded of framework the in constructions these revisit We edfiealna ucinl h O ucinl nsae o spaces on functional, DOS the functional, linear a define We est fSae o admContractions Random for States of Density → Z d ne roiiyasmtos h ii xssams sur almost exists limit the assumptions, ergodicity Under . li Joye Alain Abstract ⊂ Z d ysial onaycniin,adin and conditions, boundary suitable by ∗ rc e ntvlm ffunctions of volume unit per trace e sm r opeeynnunitary non completely are ssume eitga ersnaino the of representation integral he enda h xetto fa of expectation the as defined oe naHrysaeo the of space Hardy a on loped ldfie hsclsignificance, physical defined ll e o prtr defined operators For re. fr ftelmtn measure. limiting the of sform a xs eae oterandom the to related axis, eal e h pcr fnnnormal non of spectra the een ecie eo.B rescaling, By below. described o ntr prtr,see.g. see operators, unitary for nly ncs h normalised the case in inally, France , in ovre nteinfinite the in converges tions to ntesuyo h spec- the of study the in otion r n h is representa- Riesz the and or, dmsl-don operators. self-adjoint ndom ae oapstv functional positive a to iated to prtr naHilbert a on operators ction ievlm approximations, volume nite peetto nteds in disk the on epresentation nain nteui circle unit the on entations m ii ftetaeper trace the of limit ume ooopi ucin on functions holomorphic f eok ersr othe to resort we mework, no prtr htare that operators andom l and ely Z d a , disk, as recalled in Section 2. This allows us to define a density of state functional (DOS functional for short) on H∞(D) in analogy with that defined for self-adjoint operators, see Definition 3.1. We show in Proposition 3.2 that the DOS functional possesses a natural integral representation on the unit circle T by a function ϕ L1(T) H1(D), and that ∈ \ 0 when restricted to the disc algebra A(D) = H (D) C(D¯), it further admits an integral ∞ ∪ representation on the disk by a complex harmonic function mϕ, as Proposition 3.5 shows. Restrictions of random contractions to finite volume boxes Λ Zd are considered in ⊂ Section 4, under suitable ergodicity assumptions. The infinite volume limit of the trace per unit volume of functions of random contractions is shown to coincide with the DOS functional defined via the full operator in Propositions 4.3 and 4.4. We make use of this alternative construction of the DOS functional on A(D) to show that a priori estimates of the spectral radius of the finite volume restrictions imply more structure on the integral representations on T and D by means of ϕ and mϕ: these functions are shown to be defined by the complex conjugate of a function ψ that is holomorphic in a neighbourhood of the unit disk, see Theorem 4.6 and Corollary 4.8. Then, we consider the situation where the normalised counting measures on the spectra of finite volume restrictions of the random contractions converge weakly , in the infinite volume limit. Theorem 4.9 states that, despite the fact that finite volume restrictions of non normal operators have spectra that generally differ significantly from the full operator [D1, D2, GoKh2, TE], the limiting measure provides us with yet an alternative integral representation of the DOS functional on the unit disk. Moreover, the holomorphic function ψ is directly related to the Borel transform of the limiting measure. Some examples are worked out in Section 5 to illustrate the various features of the DOS functional. We start with random contractions defined as multiples of random unitary operators. Then we consider the non self-adjoint Anderson model (NSA model), whose finite volume restrictions have eigenvalue distributions that give rise to limiting measure, see e.g. [GoKh1, GoKh2, D2]. Finally, the DOS functional computed for certain non unitary unitary band matrices, whose spectral properties are studied in [HJ], and whose finite volume restrictions display similar features as those of the NSA model.

2 Functional Calculus for CNU Contractions

We recall from [SFBK] the main properties of the functional calculus developed for con- tractions. Let T be a contraction on a separable Hilbert space . Consider the unique decompo- H sition T = T T on = , (1) 0 ⊕ 1 H H0 ⊕H1 n n N where 0 = ψ T ψ = ψ = T ∗ ψ ,n , 1 = 0 and Tj = T j , H { ∈H|k k k k k k ∈ } H H⊖H |H j = 0, 1, such that T0 is unitary, and T1 is completely non unitary (cnu). The analysis of random unitary operators is by now well known, so we restrict attention to cnu contractions and therefore assume that T = T1 in the following. We recall here some basic facts from harmonic analysis. In the following, D denotes the open disk, its boundary is ∂D that we will also identify with the torus T. The set of holomorphic functions on an open set S C ⊂ is denoted by Hol(S).

2 The Hardy class Hp(D) consists in holomorphic functions on D such that

1 it p sup0

Theorem 2.1 Assume T is a cnu contraction on a separable Hilbert space . Then, for D n D H any u H∞( ), s.t. u(z)= n 0 cnz for z , u(T ) is defined by the strong limit ∈ ≥ ∈ P n n u(T )= s- lim ur(T ), where ur(T )= cnr T , (3) r 1− → n 0 X≥ which exists. The map H (D) u u(T ) ( ) is an algebra homomorphism which ∞ ∋ 7→ ∈ B H further satisfies :

I if u(z) 1 a) u(T )= ≡ T if u(z)= z  b) u(T ) u k k ≤ k k∞ c) u (T ) u(T ) in norm, resp. strong, resp. weak sense if n → u u uniformly on D, resp. boundedly a.e. on T, resp. boundedly on D n → d) u(T )∗ =u ˜(T ∗). (4)

D T Remarks 2.2 i) By the identification of H∞( ) and L+∞( ), this functional calculus can be viewed as a homomorphism L (T) f f(T ) ( ), with +∞ ∋ → ∈B H I if f(t) = 1 a.e. f(T )= T if f(t)= eit a.e.  f(T ) f . (5) k k ≤ k k∞ ii) Conditions a), c) in the strong sense and (5) make this functional calculus maximal and unique, see [SFBK]. iii) If u A(D), u (T ) u(T ) 0, as n . ∈ k n − k→ →∞ iv) If T = T T , as in (1), with unitary part T having purely absolutely continuous 0 ⊕ 1 0 spectrum, then Theorem 2.1 holds with c) in the strong sense. See [SFBK], Theorem 2.3.

3 3 DOS Functional

We deal here with random contractions T defined on a separable Hilbert space with ω H ergodic properties we express as follows. We first assume some regularity assumptions. Let the probability space (Ω, , P), where Zd R N P F Ω is identified with X , X , d , and = k Zd dµ, where dµ is a probability { } ⊂ ∈ ⊗ ∈ distribution on X and is the σ-algebra generated by the cylinders. We assume that F Ω ω T ( ) is measurable i.e. ∋ 7→ ω ∈B H ϕ, ψ , Ω ω ϕ T (ω)ψ is measurable. (6) ∀ ∈H ∋ 7→ h | i Ergodicity is expressed in the following framework. We consider = l2(Zd) and consider H for j Zd the shift operator S defined on Ω by ∈ j S (ω) = ω , k Zd, where ω X, (7) j k k+j ∈ k ∈ P so that the measure is ergodic under the set of commuting translations Sj j Zd . Let { } ∈ ϕj j Zd denote the canonical basis of and Vj be the unitary operator defined by { } ∈ H d Vjϕk = ϕk j, k Z . (8) − ∀ ∈ d We further assume the existence of a periodic lattice Γ Z spanned by γi i 1,2,...,d , ⊂ { } ∈{ } γ Zd and the corresponding primitive cell B = d x γ , 0 x < 1 Zd so that for i ∈ { i=1 i i ≤ i }∩ any j Zd, there exist a unique b B and a unique g Γ with j = b + g. ∈ ∈ P ∈ The random contractions we consider are ergodic in the following sense:

1 T = V T V − , g Γ. (9) Sg (ω) g ω g ∀ ∈ Definition 3.1 The DOS functional L : H (D) C is defined for all f H (D) by ∞ → ∈ ∞ 1 L(f)= E( ϕ f(T )ϕ ), (10) B h b| ω bi b B | | X∈ where B denotes the cardinal of B and T is a measurable cnu random contraction. | | ω We first note that

Proposition 3.2 The map L : H (D) C is a bounded and admits the following integral ∞ → representation: there exists ϕ L1(T) H1(D), such that for all f H (D) ∈ \ 0 ∈ ∞ dt L(f)= f(eit)ϕ(eit) . (11) T 2π Z 1 D Remarks 3.3 0) The set H0 ( ) should be understood as the set of boundary values of these functions. i) The ergodicity assumption (9) plays no role here.

4 ii) Such functionals on H∞(D) are called weakly continuous, see e.g. [G], Section V. iii) The representation (11) says that for any G H1(D) ∈ 0 it it dt it it dt f(e )(ϕ + G)(e ) = f(e )ϕ(e ) , f H∞(D), (12) T 2π T 2π ∀ ∈ Z Z see [G], Theorem 5.2. It shows that the relevant information is carried by the negative Fourier coefficients only. iv) In case ϕ Lp(T) L1(T), with p > 1, we can actually represent L by a unique ∈ ⊂ ϕ Lp obtained from ϕ by substracting the contribution from the sum over positive Fourier − ∈ coefficients. We give conditions for this to hold in Theorem 4.6 below. v) Making the dependence on T of the functional L explicit in the notation, we have for all f H (D), ∈ ∞ LT (f)= LT ∗ (f˜). (13) Proof: Linearity and the bound L(f) f stem directly from the properties of the | | ≤ k k∞ functional calculus recalled above. Then one makes use of the following equivalence, see [G], Theorem 5.3: L is a weakly continuous functionals on H∞(D) iff L is continuous under bounded pointwise convergence; i.e. if fn H∞(D), fn M and fn(z) f(z), for ∈ k k∞ ≤ → all z D, then L(f ) L(f). For such a sequence f , we have by point c) of Thm. 2.1 ∈ n → n that ϕ f (T )ϕ ϕ f(T )ϕ for all k Zd, and ϕ f (T )ϕ M, uniformly h k| n ω ki → h k| ω ki ∈ |h k| n ω ki| ≤ in n,k,ω. Hence by Lebesgue dominated convergence, we also have E( ϕ f (T )ϕ ) h b| n ω bi → E( ϕ f(T )ϕ ), for any b B, which yields L(f ) L(f), for B finite. h b| ω bi ∈ n → | | Remarks 3.4 i) The bound L(f) f is saturated: L(1) = 1. | | ≤ k k∞ ii) For any j N, the function z zj H (D) and, denoting the Fourier coefficients of ∈ 7→ ∈ ∞ ϕ by ϕˆ(k) k Z, { } ∈ 1 L( j)= eijtϕ(eit) =ϕ ˆ( j), with ϕˆ(0) = 1. (14) · T 2π − Z Then we observe that for functions in A(D) H (D), we get an alternative represen- ⊂ ∞ tation of L( ) on the disk. · Let us denote by P [ϕ](reit) = P [ϕ](x,y) the harmonic function in D given by the Poisson integral of ϕ L1(T), with the usual abuse of notation. Due to the fact that it ∈ it 1 T limr 1− P [ϕ](re ) = ϕ(e ) almost everywhere and in L ( ) norm, we can approximate L(f)→ by an integral over smooth functions: for any f A(D), and any 0

L(f) = f(x + iy)mϕ(x,y)dxdy, where D Z 1 m (x,y) = (∂ + i∂ ) (x iy)P [ϕ](x,y) . (17) ϕ 2π x y { − } Proof: Consider the approximation (16). By Stokes theorem applied to f(z)P [ϕ](z) ∈ C∞(D), we have dt dz f(reit)P [ϕ](reit) = f(z)P [ϕ](z)¯z T 2π D r22iπ Z Zr∂ ∂ dz¯ dz ∂ dxdy = f(z)P [ϕ](z)¯z ∧ = f(z) P [ϕ](z)¯z . (18) D ∂z¯ { } r22iπ D ∂z¯ { } r2π Zr Zr Thanks to (16), we can take the limit r 1 which yields → − 1 ∂ 1 ∂ m (z)= P [ϕ](z)¯z = P [ϕ](z)+ P [ϕ](z) z¯ . (19) ϕ π ∂z¯ { } π ∂z¯     Using the fact that P [ϕ] is harmonic, one finally gets ∂ 1 ∂ ∂2 m (z)= P [ϕ](z), and m (z) = 0. (20) ∂z ϕ π ∂z ∂z∂z¯ ϕ

Remark 3.6 In keeping with the fact that ϕ ϕ + G, where G H1(T) does not change 7→ ∈ 0 the representation, one checks that

mϕ+G(z)= mϕ(z)+ G(z)/π, where f(z)G(z)dxdy = 0. (21) D Z The integral representations of L( ) discussed so far are intrinsic. There are of course · many alternative integral representations in the disk: since A(D) C(D¯), we can extend ⊂ L to Lˆ : C(D¯) C, by means of Hahn-Banach Theorem, with L = Lˆ . Given Lˆ, → k k k k since D¯ is compact, the Riesz Representation Theorem asserts the existence of a unique complex Borel measure dµ on D such that Lˆ(f) = f(x,y)dµ(x,y) for all f C(D¯) and D¯ ∈ Lˆ = µ (D). Thus, k k | | R Lemma 3.7 Let Tω be a measurable random contraction. There exists a complex Borel measure dµ on D¯ such that µ (D¯) = 1 such that | | L(f)= f(x + iy)dµ(x,y), f A(D). (22) D¯ ∀ ∈ Z Remark 3.8 The measure dµ uniquely determined by the extension Lˆ of L to C(D¯). The example discussed in Section 5.4 illustrates the fact that there may be infinitey many such integral representations of L.

6 4 Finite Volume Approximations

Let Λ Zd be given by (2n + 1)d symmetric translates of B along Γ of the form ⊂ d Λ= B + niγi B + gm, (23) Z ≡ ni∈ i=1 m=1, ,(2n+1)d −n≤[ni≤n X ···[ with Λ = (2n + 1)d B , and let | | | |

= span ϕ , j Λ , C = , (24) HΛ { j ∈ } HΛ H⊖HΛ together with the corresponding orthogonal projections onto these subspaces PΛ, PΛC .

Dropping ω from the notation for now, let assume that the cnu contraction T can be written as C T = T Λ T Λ + F Λ, (25) ⊕ Λ ΛC Λ where T and T are defined on and C and F is a trace class operator on , and HΛ HΛ H furthermore T Λ is a cnu contraction. Such decompositions can be obtained for example by setting

Λ T = T Λ + boundary conditions (26) |H ΛC T = T C + boundary conditions, (27) |HΛ for suitable boundary conditions at ∂Λ, see below. For any f H (D), the operators ∈ ∞ f(T ),f(T Λ) are well defined by functional calculus and we consider two random functionals on H∞(D) given by 1 1 L (f)= tr(f(T Λ)), L˜ (f)= tr(P f(T )P ). (28) Λ Λ Λ Λ Λ Λ | | | | From the bound A rank(A) A on the trace norm , we deduce that for all k k1 ≤ k k k · k1 f H (D) ∈ ∞ LΛ(f) f , L˜Λ(f) f , (29) k k ≤ k k∞ k k ≤ k k∞ so that all arguments of the proof of Proposition 3.2 apply. Hence LΛ and L˜Λ can be 1 i i written in the form (11) with corresponding random L (T) functions ϕΛ(e ·) andϕ ˜Λ(e ·). More precisely we have

Lemma 4.1 Let λ , j = 1, ..., Λ , be the eigenvalues of the cnu contraction T Λ, repeated ac- j | | cording to their algebraic multiplicities. Then, f H (D), L (f)= 1 f(eit)ϕ (eit)dt ∀ ∈ ∞ Λ 2π T Λ where R Λ | | it 1 1 ϕΛ(e ) = it . (30) Λ 1 λje− | | Xj=1 −

7 Proof: The finite dimensional contraction T Λ being cnu, σ(T ) D. Hence, for any Λ ⊂ f H (D), and any ρ< 1 large enough ∈ ∞ Λ Λ 1 | | 1 | | 1 f(z) LΛ(f) = f(λj)= dz Λ Λ 2iπ ρ∂D z λj | | Xj=1 | | Xj=1 Z − Λ it 1 | | ρ fρ(e ) = it dt. (31) Λ 2π T ρ λje− | | Xj=1 Z − For each j, since f H (D), Lebesgue dominated convergence implies that the limit ρ 1 ∈ ∞ → exists, which yields the result.

Remarks 4.2 i) An application of Stokes theorem shows that

Λ 1 | | 1 f(z) LΛ(f)= 2 dxdy, (32) Λ π D (1 λjz¯) | | Xj=1 Z − an expression of the fact that 1 (1 λ z¯) 2 is the Bergman reproducing kernel. π − j − ii) Introducing the normalised counting measure of σ(T Λ), dmΛ on the closed unit disc D¯ by Λ 1 | | dmΛ(x,y)= δ(x λ )δ(y λ ), λ σ(T Λ), (33) Λ −ℜ j −ℑ j j ∈ | | Xj=1 where the eigenvalues are repeated according to their algebraic multiplicities, we can extend LΛ to C(D¯) by Λ LˆΛ(f)= f(x,y)dm (x,y), f C(D¯). (34) D¯ ∀ ∈ Z However, L˜ (f) does not make sense for f C(D¯). Λ ∈ 4.1 Infinite Volume Limit Restoring the variable ω in the notation for a moment, we show that the random functionals LΛ,ω(f) and L˜Λ,ω(f) converge almost surely to the deterministic DOS functional L(f) as Λ Zd, when f A(D) H (D). By Λ Zd or Λ , we mean n in definition → ∈ ⊂ ∞ → | |→∞ →∞ (23). This is done along the same lines as in the unitary case under ergodicity assumption, see [J1], for example.

We start by a deterministic statement:

Proposition 4.3 Assume T and T Λ given by (25) are cnu contractions. C If P (T T Λ T Λ ) = o( Λ ) as Λ , then , for all f A(D), k Λ − ⊕ k1 | | | |→∞ ∈

lim LΛ(f) L˜Λ(f) = 0. (35) Λ − | |→∞

8 Proof: We need to show that

1 Λ lim tr(f(T ) tr(PΛf(T )PΛ) = 0. (36) Λ Λ − | |→∞ | |  Λ Λ As f(T )= PΛf(T )PΛ, cyclicity of the trace yields

tr(f(T Λ) P f(T )P ) = tr((f(T Λ) f(T ))P ) = tr(P (f(T Λ) f(T ))). (37) − Λ Λ − Λ Λ − On the other hand, since A(D) consists in uniformly continuous functions on D¯, we have

N j f(z)= ajz + RN (z), where sup RN (z) = RN 0 as N . (38) z D¯ | | k k∞ → →∞ Xj=0 ∈ Hence, together with the bound A rank(A) A , we get the uniform estimate k k1 ≤ k k

1 Λ rank(PΛ) Λ tr(RN (T ) PΛRN (T )PΛ) RN (T ) RN (T ) 2 RN . (39) Λ | − |≤ Λ k − k≤ k k∞ | | | | Therefore, we can focus on f(z)= zj, j N. With ∈ j 1 − j Λ ΛC j k Λ ΛC Λ ΛC j k 1 (T (T T ) )P = T (T T T )(T T ) − − P (40) − ⊕ Λ − ⊕ ⊕ Λ Xk=0 j 1 − k Λ ΛC Λ j k 1 = T (T T T )P (T O) − − (41) − ⊕ Λ ⊕ Xk=0 we get for all j N, using T and T Λ are contractions and cyclicity of the trace, ≤ 1 C j C tr(P (T j (T Λ T Λ )j)) P (T T Λ T Λ ) . (42) Λ | Λ − ⊕ |≤ Λ k Λ − ⊕ k1 | | | | C This estimate, the assumption P (T T Λ T Λ ) = o( Λ ) and (39) end the proof. k Λ − ⊕ k1 | | We finally turn to the infinite volume limit of L˜ (f), for f A(D), under the ergodicity Λ,ω ∈ assumption (9) on the way randomness enters the contraction Tω.

Proposition 4.4 Assume T is ergodic in the sense of (9). For all f A(D), ω ∈

lim L˜Λ,ω(f)= L(f), almost surely. (43) Λ | |→∞ Proof: By construction of Λ, see (23), for any H : Zd C, → (2n+1)d d H(k)= H(b + g ), where g = n γ Γ. (44) m m i i ∈ k Λ b B m=1 i=1 X∈ X∈ X X

9 Thus, by the ergodicity assumption,

(2n+1)d 1 1 L˜ (f) = ϕ f(T )ϕ = ϕ V ∗ f(T )V ϕ Λ,ω Λ h k| ω ki Λ h b+gm | gm Sgm (ω) gm b+gm i k Λ b B m=1 | | X∈ | | X∈ X (2n+1)d 1 1 = ϕ f(T )ϕ = ϕ f(T n1 nd )ϕ (45). b Sgm (ω) b b Sγ Sγ (ω) b Λ h | i Λ h | 1 ··· d i b B m=1 b B ni n | | X∈ X | | X∈ | X|≤ Thanks to Birkhoff Theorem, we have on a set Ω Ω of measure one, and for all b B, f ⊂ ∈ 1 lim ϕ f(T n1 nd )ϕ = E( ϕ f(T )ϕ ). (46) d b Sγ Sγ (ω) b b ω b n (1 + 2n) h | 1 ··· d i h | i →∞ ni n | X|≤ Since A(D) is separable, the statement above is true for a dense countable set of functions

fm m N on m NΩfm =Ω0 Ω, a set of measure one. Since B < , we infer { } ∈ ∩ ∈ ⊂ | | ∞ 1 1 lim ϕk f(Tω)ϕk = E( ϕb f(Tω)ϕb ) a.s. (47) Λ Λ h | i B h | i | |→∞ k Λ b B | | X∈ X∈ | | which proves the statement.

Remarks 4.5 i) We consider A(D) only, a separable space, since H∞(D) is not. 1 Λ | | ii) As Λ k=1 ϕk f(Tω)ϕk f , Lebesgue dominated convergence implies that | | h | i ≤ k k∞

P Λ 1 | | lim E(L˜Λ,ω(f)) = lim E( ϕk f(Tω)ϕk )= L(f). (48) Λ Λ Λ h | i | |→∞ | |→∞ | | Xk=1 iii) Under the assumptions of Propositions 4.3 and 4.4

lim LΛ,ω(f)= L(f), almost surely. (49) Λ | |→∞ iv) The result also holds if T = T T with purely absolutely continuous unitary part T . 0 ⊕ 1 0 4.2 Spr(T Λ) < 1

We show that if an a priori uniform estimate on the spectral radius of TΛ holds, we deduce anti-analyticity of the integral representation ϕ of L(f).

Let us drop the dependence on ω Ω in the notation. The form LΛ is represented by it ∈ it integration against ϕΛ(e ), see (30), which can be written with z = e as Λ 1 | | 1 ϕΛ(z)= ψΛ(z), where ψΛ(z)= . (50) Λ 1 λ¯jz | | Xj=1 − As TΛ is cnu, ψΛ is holomorphic in D and we have the absolutely converging power series k ∞ tr T ψ (z)= Λ∗ zk, z D. (51) Λ Λ ∀ ∈ Xk=0 | |

10 Theorem 4.6 Let Tω satisfy the hypotheses of Propositions 4.3 and 4.4. Assume there exists r< 1 such that Spr(T ) r, Λ Zd and ω Ω. (52) Λ ≤ ∀ ∈ ∀ ∈ Then, there exists ψ Hol(D/r) such that Proposition 3.2 holds with ϕ(eit)= ψ(eit): ∈ dt L(f)= f(eit)ψ(eit) . (53) T 2π Z

Remarks 4.7 i) We can satisfy the hypothesis by properly rescaling the operator Tω. ii) The property Spr(T ) r < 1 Λ does not imply Spr(T ) < 1. Indeed, finite volume Λ ≤ ∀ approximations of non normal operators typically miss important parts of σ(T ), see Section 5.3 and (86) in particular.

Proof: If (52) holds, ψ is holomorphic in the larger disc D/r for all Λ Zd, ω Ω and Λ ∈ ∈ 1 ψ (z) , z D/r. (54) | Λ |≤ 1 r z ∀ ∈ − | | In particular, the family ψ of holomorphic functions on D/r is uniformly bounded { Λ}Λ on each compact subset of D/r. Hence, by Montel Theorem, see e.g. [R], Theorem 14.6, ψ is a normal family. Therefore, for each fixed ω Ω , the set of measure one on which { Λ}Λ ∈ 0 Proposition 4.4 holds, there exists a subsequence ψΛk k N which converges uniformly on { } ∈ each compact subsets of D/r to a function ψ(z) which is holomorphic on D/r. In particular, for all f A(D), ∈

it it dt it it dt lim LΛk (f) = lim f(e )ψΛk (e ) = f(e )ψ(e ) , (55) k k T 2π T 2π →∞ →∞ Z Z where ψ(z) is analytic in a neighbourhood of D. By Remark 4.5 iii), we get that ϕ L1(T) ∈ which represents L(f) is given by ϕ(t)= ψ(eit), and ψ is independent of ω. Consequently,

Corollary 4.8 With ψ = b zn Hol(D/r), and for all f(z)= a zn A(D), n=0 n ∈ n=0 n ∈

P ∞ P L(f)= ψ f 2 T = ψˆ fˆ 2 N = ¯bnan. (56) h | iL+( ) h | il ( ) n=0 X The integral representation (17) reads 1 ∂ m (z)= m (z)= ψ(z)z . (57) ϕ ψ π ∂z¯ n o 4.3 Representations of L via finite volume approximations Λ Let us consider now the random counting measure dmω (33) and assume that it admits a weak limit, almost surely: for all ω Ω with P(Ω ) = 1, and for all f C(D¯), ∈ 0 0 ∈ ˆ Λ lim LΛ,ω(f) = lim f(x,y)dmω (x,y)= f(x,y)dmω(x,y). (58) Λ Λ D¯ D¯ | |→∞ | |→∞ Z Z 11 Then, for any ω Ω , dm 0 provides another representation of L on A(D), since, ∈ 0 ω ≥ specialising to f A(D), we get from (49) ∈

L(f)= f(x + iy)dmω(x,y). (59) D¯ Z

Λ Theorem 4.9 Let Tω satisfy the assumptions of Propositions 4.3 and 4.4 and dmω in (33) converge weakly to dmω, almost surely. Then L admits the following representation

L(f)= f(x + iy)dm¯ (x,y), f A(D), (60) D¯ ∀ ∈ Z where dm¯ 0 is given by E(dm ). ≥ ω Further assume Spr(T Λ) r< 1, for all ω Ω and Λ. Then, Theorem 4.6 holds with ω ≤ ∈ dm¯ (x,y) ψ(z)= , z D/r, (61) D¯ 1 z(x iy) ∀ ∈ Z − − and Proposition 3.5 holds with

1 dm¯ (x,y) mψ¯(z)= . (62) π D¯ (1 z¯(x + iy))2 Z − Remark 4.10 If the weak limit of the normalised counting measure of the finite volume spectrum exists, see e.g. [GoKh2] for such cases, it provides another representation of the DOS functional. In that sense, the spectrum of the finite volume restrictions acquire a global meaning, in spite of the fact that it can be very different from the spectrum of the infinite volume operator. In particular, the support of the limiting measure can be disjoint from the spectrum of the operator, see Section 5.4 for such an example.

Proof: The first statement is a consequence of (59) and Fubini’s Theorem. The assump- tion on Spr (T Λ) implies supp dm¯ z r . Then, Corollary 4.8 applied to f(z) = zk ω ⊂ {| | ≤ } yields the coefficients of the power expansion of the holomorphic function

ψ(z)= b zk, z D/r. (63) k ∈ k 0 X≥ We have k k bk = (x iy) dm¯ (x,y), with bk r , k N. (64) D¯ − | |≤ ∈ Z Thus, exchanging integration and summation, we get expression (61) for z D/r. The last ∈ statement follows from Corollary 4.8.

Remark 4.11 Thanks to Remarks 2.2 iv), and 4.5 iv), all results of Section 4 hold if T writes as T = T T , see (1), with a unitary part T that is purely absolutely continuous. 0 ⊕ 1 0

12 5 Special Cases

We take a closer look at various particular cases allowing us to get further informations on the integral representation ϕ.

5.1 The Normal Case

A first special case of interest occurs when Tω is normal, i.e., when there exists orthogonal projection valued measures dEω(x + iy) (x+iy) σ(Tω ) such that in the weak sense, { } ∈

Tω = (x + iy)dEω(x + iy). (65) Zσ(Tω ) In such a case, we have a continuous functional calculus: for any f C(D¯) ∈

f(Tω)= f(x + iy)dEω(x + iy). (66) Zσ(Tω ) Hence, assuming that T is normal for any ω Ω, Definition 3.1 gives rise to a positive ω ∈ functional on C(D¯), so that by Riesz-Markov Theorem

L(f)= f(x + iy)dm(x,y), where dm is a non-negative Borel measure on D¯. (67) D¯ Z In this favourable framework, we have

C Lemma 5.1 Let T , T Λ be contractions such that P (T T Λ T Λ ) = o( Λ ), uniformly ω ω k Λ − ω ⊕ ω k1 | | in ω Ω. Further assume T and T Λ are normal and the ergodicity assumption (9) holds. ∈ ω Then, as Λ , →∞ dmΛ dm a.s., in the weak- sense. (68) ω → ∗ Proof: The same arguments using Stone Weierstrass and Birkhoff theorems together with the separability of C(D¯) prove the result as in the previous section.

Applying Propositions 3.2 and 3.5 to the normal case, we get for any f A(D) ∈

f(x + iy)mϕ(x,y)dxdy = f(x + iy)dm(x,y), (69) D¯ D¯ Z Z where dm(x,y) is the non negative usual density of states, and mϕ(x,y) is harmonic and in general complex valued. This special case makes explicit the lack of uniqueness in the representation of analytic functionals.

Remark 5.2 If, moreover, there exists 0

13 5.2 Multiple of Unitary Operators Consider now a special normal case where the statement above allows us to make the link between the DOS functional and the density of state measure of a random unitary operator 2 Uω more explicit. Let 0

Tω = rUω. (71) Λ We assume that there exist finite volume approximations such that Uω is a finite dimen- sional unitary matrix; see e.g. [J1] for examples of this situation with Λ = n + 1,n . Λ {− } Consequently, we have the trivial bounds Spr (Tω ) = Spr(Tω) = r, r < 1. Stressing the r dependence in the notation, the DOS functional L(r)( ) on A(D) is represented by · dt L(r)(f)= ψ(r)(eit)f(eit) , where ψ(r)(z) Hol(D/r). (72) T 2π ∈ Z The density of states measure dk for ergodic unitary operators is a normalised positive reg- ular Borel measure on T characterised as in Section 5.1 by Definition 3.1 with f continuous on the circle via Riesz-Markov Theorem, [J1]:

2 1 it E( ϕk f(Uω)ϕk )= f(e )dk(t), f C(∂D). (73) 2 h | i T ∀ ∈ Xk=1 Z The explicit link is provided by

Proposition 5.3 Let Tω = rUω, with 0

14 Proof: Let f A(D). From (72) and (73), we get ∈ dt ψ(r)(eit)f(eit) = f(reit)dk(t). (78) T 2π T Z Z (r) n D The coefficients of the expansion ψ (z) = n N bnz , for z /r are given by bn = ∈ ∈ rnkˆ(n),n N, where kˆ(n)= dk(t)e itn. Hence, with sign(0) = 0, ∈ T − P R (r) it n ˆ int (r) it nˆ int 2 ψ (e ) 1 = n Z r| |k(n)e ψ (e )= r k(n)e ℜ (r) it− ∈ n int (79) ⇒ 2 ψ (e ) = i Z sign(n)r kˆ(n)e . n N  Pn | | X∈ ℑ − ∈ P The last two series coincide with the real and imaginary parts of dk (P ( )+ iQ ( ))(eit), ∗ r · r · where Pr(t) and Qr(t), are the Poisson and conjugate Poisson kernels given by

1+ reit 1 r2 + i2r sin(t) P (t)+ iQ (t)= = − , 0

The NSA model Hω(g), is a one parameter deformation of the one dimensional random Anderson model of solid state physics defined as follows in the canonical basis of l2(Z):

g g Hω(g)ϕj = e− ϕj 1 + e ϕj+1 + ωjϕj, j Z. (81) − ∀ ∈

We assume that g 0 and ωj j Z are i.i.d. real valued random variables distributed ≥ { } ∈ according to a measure dµ supported on a compact interval [ B,B]. Let E be the ellipse − g g+iθ (g+iθ) E = e + e− θ [0, 2π] , (82) g { | ∈ } which coincides with the spectrum of H0(g), the NSA model in absence of random potential. It is proven in [D2] that, provided B eg + eg, ≥ σ(H (g)) = E + [ B,B], almost surely. (83) ω g − Λ 2 Moreover, consider Hω (g) the finite volume restriction of Hω(g) to l (Λ), where Λ = n,n Λ {− Λ } with Dirichlet boundary conditions. Then, for g > 0, the matrix Hω (g) is similar to Hω (0) kg with similarity transform given by W ϕk = e ϕk, so that its spectrum is real. The matrix Λ Hω (0) is the finite volume restriction of the Anderson model Hω(0), with σ(H (0)) = E + [ B,B], almost surely. (84) ω 0 − 15 To cast these considerations in our framework, we assume B eg + e g and set ≥ − g g Λ Λ g g Tω(g)= Hω(g)/(e + e− + B), resp. Tω (g)= Hω (g)/(e + e− + B), (85) on l2(Z), resp. l2(Λ) with Dirichlet boundary conditions. By construction, both operators are contractions. Moreover

Lemma 5.5 The matrix T Λ(g) is cnu for Λ large enough, and all ω Ω, whereas the ω | | ∈ operator Tω(g)) is cnu almost surely. Also, for g> 0,

Λ 2+ B Spr(Tω(g)) = 1, a.s. and lim Spr(Tω (g)) = < 1, ω Ω. (86) Λ eg + e g + B ∀ ∈ →∞ − Proof: Statements (86) are consequences of (83), (84), and properties of finite volume Λ approximations of self adjoint operators. Thus Tω (g) is cnu for Λ large enough. Given | | 2 (83), Tω(g) is cnu if 1 are not eigenvalues of Tω(g). Suppose ϕ l (Z) are normalized ± ± ∈ and satisfy Tω(g)ϕ = ϕ . Then, using the fact that Tω∗(g) is a contraction, and Cauchy- ± ± ± Schwarz inequality, we get Tω∗(g)ϕ = ϕ . Hence 1 are eigenvalues with eigenvectors ± ± ± ± ϕ for the self-adjoint Anderson type operator (Tω(g)+ Tω∗(g))/2. But this is known to ± happen with zero probability only.

Stressing the g dependence in the notation, we denote the DOS functional on A(D) by L(g)( ). The foregoing and Theorem 4.9 immediately show that · Λ Λ Lemma 5.6 Let dmω (g) be the normalised counting measure of σ(Tω (g))). Then, for any g> 0, dmΛ(g) dk , a.s., in the weak sense, (87) ω → g Λ where dk 0 is the density of states of the rescaled self-adjoint Anderson model Hω (0) . g ≥ eg+e−g+B Hence, for any f A(D), and any g 0, ∈ ≥

(g) x + iy L (f)= f g g dk(x), (88) [ 2+B,2+B] e + e− + B Z −   where dk 0 is the density of states of HΛ(0). ≥ ω

Remark 5.7 The support of dkg is a subset of the almost sure spectrum of Tω(g).

The estimates provided in Lemma 5.5 show that L(g)( ) admits an integral representation · on T in term of a holomorphic function ψ(g)(z), see Theorem 4.6. Moreover,

g g (g) Lemma 5.8 Set s(g) = e + e− + B. The function ψ is real analytic and admits a converging power series in z < s(g)/(2 + B) . Moreover, ψ(g) admits an analytic {| | } continuation on C [s(g)/(2 + B), [ ] , s(g)/(2 + B)] given by \ { ∞ ∪ −∞ − } dk(x) ψ(g)(z)= . (89) [ (2+B),2+B] 1 zx/s(g) Z − −

16 Remark 5.9 With the Borel transform Fk of dk given by dk(x) Fk(z)= , z C [ 2+ B, 2+ B], (90) [ 2+B,2+B] x z ∀ ∈ \ − Z − − we have for z = 0, 6 ψ(g)(z)= F (s(g)/z) s(g)/z. (91) − k Proof: It is a special case of Theorem 4.9 where the coefficients of the power expansion of ψ(g) are given by

n n [ 2+B,2+B] x dk(x) (2 + B) − N bn = bn = g g n , , bn n n . (92) R (e + e− + B) | |≤ s(g) ∀ ∈

5.4 Non Unitary Band Matrices We further illustrate both the discrepancy between limiting measure and spectrum of the full operator, and the multiplicity of representations of the DOS functional by measures on D¯ by considering random contractions with a band matrix representation.

2 Let Tω be defined on l (Z) by its matrix representation in the canonical basis given as . .. eiω2j−1 γ eiω2j−1 δ  0 0  0 0 eiω2j+1 γ eiω2j+1 δ T =   , (93) ω  eiω2j+2 α eiω2j+2 β 0 0     0 0     .   eiω2j+4 α eiω2j+4 β ..     where ωj j Z are T-valued iid random variables. The deterministic coefficients α,β,γ,δ { } ∈ are assumed to give rise to a non unitary matrix

α β C = s.t. C C2 1. (94) 0 γ δ k 0k ≤  

iωj Actually, Tω = DωT , where Dω is a diagonal unitary random operator with elements e , and T is a deterministic operator whose representation has the form (93), where all ωj’s are equal to zero. Such random ergodic operators arise in the analysis of certain random quantum walks and some of their spectral properties are analysed in [HJ]. In particular, it 2 is shown there that Tω is a cnu contraction on l (Z) if and only if

det C < 1, α < 1, and δ < 1. (95) | 0| | | | | Moreover, T = 1, and Spr(T ) may take the value 1, depending on the parameters. k ωk ω We first note the following simple general result:

17 Lemma 5.10 Let Tω defined by (93) with (94), and (95), and assume the random phases eiωj are uniformly distributed. Then L(f)= f(0), f A(D). (96) ∀ ∈ Proof: One first observes that for any n N , any k Z, ϕ T nϕ = eim(k)ωk τ(ˆω(k)), ∈ ∗ ∈ h k| ω ki where τ(ˆω(k)) is a random variable independent of ω , and m(k) N . This is a conse- k ∈ ∗ quence of the fact that no cancellation of the random phases can occur due to the shape (93) of the matrix representation of T . Hence, E( ϕ T nϕ ) = 0, for all k Z and n N . ω h k| ω ki ∈ ∈ ∗ Thus, approximating any f A(D) by a polynomial, we get the result. ∈

Remark 5.11 We get from (96) that the integral representations of L( ) on T and D are · given by ϕ 1 and m 1 respectively. ≡ ϕ ≡ π The DOS functional (96) can be extended to C(D¯) as integration against a Dirac measure at the origin, but not only. For example, using in polar coordinates, defining for any 0 <ρ< 1

(ρ) 1 (ρ) Lˆ (f)= f(r cos(θ), r sin(θ))δρ(r)dθ = f(x,y)dµ (x,y), (97) 2π D¯ D¯ Z Z we get a one parameter family of extensions of L corresponding to integration against positive measures on D, dµ(ρ) with support of the circle of radius ρ, centered at the origin, such that Lˆ(ρ)(f)= f(0) if f A(D). The support of dµ(ρ) may or may not belong to the ∈ spectrum of Tω: the examples treated in [HJ] show that for det C0 = 0 and uniform i.i.d. phases ωj, the spectrum Tω is given by the origin and ring centered at the origin whose radiuses depend on the parameters. Note finally that any radial probability measure with smooth density µ(r) provides us with an extension Lˆ( ) on C(D¯) of the form · Lˆ(f)= f(r cos(θ), r sin(θ))µ(r)rdrdθ, (98) D¯ Z which agrees with L( ) on A(D), thanks to the mean value property of harmonic functions. · 1 0 We now specify the values of the parameters to C = , 0

Remark 5.12 The support of the limiting measure dm is disjoint from σ(Tω) in this case:

supp dm σ(T )= √g∂D g∂D ∂D = . (104) ∩ ω ∩ { ∪ } ∅ References

[BHJ] O. Bourget, J. S. Howland and A. Joye, Spectral Analysis of Unitary Band Matri- ces.Commun. Math. Phys. 234, (2003), 191-227. [CL] R. Carmona, J. Lacroix, Spectral theory of random Schrodinger Operators, Birkhauser, 1990. [CFKS] H.L. Cycon , R.G. Froese, W. Kirsch, B. Simon, Schr¨odinger Operators, Springer Verlag, 1987. [D1] E. B. Davies, Spectral theory of pseudo-ergodic operators. Comm. Math. Phys. 216 (2001), 687-704. [D2] E. B. Davies, Spectral properties of random non-self-adjoint matrices and operators. Proc. R. Soc. Lond. A. 457 (2001) 191-206. [G] M. Garnett, Bounded Analytic Functions, Graduate Texts in lathematics 236, Springer, 2007. [GoKh1] I. Y. Goldsheid and B. A. Khoruzhenko, Distribution of eigenvalues in non- Hermitian Anderson models. Phys. Rev. Letters, 80(13) (1998), 2897. [GoKh2] I. Y. Goldsheid and B. A. Khoruzhenko, Eigenvalue curves of asymmetric tridi- agonal random matrices. Electronic Journal of Probability, 5, 1-28, (2000). [HJ] E. Hamza and A. Joye, Spectral Properties of Non-Unitary Band Matrices. Ann. Henri Poincar´e, to appear.

19 [HJS] E. Hamza, A. Joye and G. Stolz, Dynamical Localization for Unitary Anderson Models. Math. Phys., Anal. Geom. 12 (2009), 381-444. [J1] A. Joye, Density of States and Thouless Formula for Random Unitary Band Matrices. Ann. H. Poincar´e. 5 (2004), 347-379. [Ki] W. Kirsch, An invitation to random Schr¨odinger operators, Random Schr¨odinger Op- erators, Panorama et Synth`ese, Soc. Math. France, Vol. 25, 1-119, 2008 [R] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1976. [SFBK] B. Sz.-Nagy, C. Foias, H. Berkovici, L. K´erchy: Harmonic Analysis of Operators in Hilbert Spaces. Springer (2010). [TE] L. Trefethen, M. Embree, Spectra and Pseudospectra: The Behaviour of Nonnormal Matrices and Operators, PUP, (2005).

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