Trace formulas for time periodic complex Hamiltonians on lattice
aa r X i v : . [ m a t h - ph ] J a n TRACE FORMULAS FOR TIME PERIODIC COMPLEX HAMILTONIANSON LATTICE
EVGENY, L. KOROTYAEV
Abstract.
We consider time periodic Hamiltonians with complex potentials on the latticeand determine trace formulas. As a corollary we estimate eigenvalues of the quasienergyoperator in terms of the norm of potentials. Introduction and main results
Introduction.
We discuss scattering and trace formulas for the Schr¨odinger equationon the lattice Z d : ddt u ( t ) = − ih ( t ) u ( t ) , h ( t ) = ∆ + V ( t ) , (1.1)where h ( t ) is the Hamiltonian, τ -periodic in time t and ∆ is the discrete Laplacian given by (cid:0) ∆ f (cid:1) x = 12 X | x − y | =1 ( f x − f y ) , f = ( f x ) x ∈ Z d ∈ ℓ ( Z d ) , x = ( x j ) d ∈ Z d . (1.2)It is known that the spectrum of the Laplacian ∆ is absolutely continuous and satisfies σ (∆) = σ ac (∆) = [0 , d ] . Here V ( t ) is τ -periodic in time potential: ( V ( t ) f ) x = V x ( t ) f x , for all ( t, x ) ∈ R × Z d . Introducethe space ℓ p ( Z d ) , p > f = ( f x ) x ∈ Z d equipped with the norm given by k f k p = k f k ℓ p ( Z d ) = (cid:0) X x ∈ Z d | f x | p (cid:1) p , p ∈ [1 , ∞ ) , and let k f k ∞ = k f k ℓ ∞ ( Z d ) = sup x ∈ Z d | f x | . For a Banach space B we write ℓ r ( B ) , r > B -valued sequences with p th power summable norms, and L r ( T τ , B ) for the B -valued L r -space. In the case f ( · ) ∈ L r ( T τ , ℓ p ( Z d ) we define the norm k f k r,p by k f k rp,r = Z T τ k f ( t ) k rℓ p ( Z d ) dt, p, r > . (1.3)Note that k f k p,r k f k q,r for all p > q >
1. We assume that potentials can be complex-valuedand satisfy
Condition V.
Let d > . The function V ( t ) is τ -periodic, and satisfies k V k p, < ∞ , ( p < if d = 31 p < if d > . (1.4)We discuss trace formulas for operators with complex potentials. Recall that, in general,a trace formula is an identity connecting the integral of the potential and various sums of Date : January 12, 2021.2020
Mathematics Subject Classification.
Key words and phrases. trace formula, time-periodic potentials. eigenvalues and integrals of coefficients of S-matrix of the Schr¨odinger operator (or otherspectral characteristics). We shortly describe results about multidimensional trace formulas: • Real potentials. The first result was obtained by Buslaev [6], see also [12], [38] andreferences therein. Trace formulas for Stark operators and magnetic Schr¨odinger operatorswere discussed in [31], [30]. The trace formulas for Schr¨odinger operators on the lattice Z d with real decaying potentials were determined by Isozaki–Korotyaev [15]. • Complex potentials. Unfortunately, we know only few papers about the trace formulas forSchr¨odinger operators with complex-valued potentials decaying at infinity. Trace formulas forSchr¨odinger operators with complex decaying potentials were determined by Korotyaev [25]in the continuous case and in the discrete case by Korotyaev and Laptev [28], Korotyaev [27]and for the specific case Im V R d is more complicated [24]. We do notknow any results about it.For Hilbert space H and T τ = R / ( τ Z ) we introduce the space ˜ H = L ( T τ , H ) of functions f → f ( t ) that are τ –periodic in time with values in H equipped with the norm k f k H = 1 τ Z τ k f ( t ) k H dt. The space ˜ H can be realized as ℓ ( H ) via the Fourier transform Φ : ˜ H → ℓ ( H ) defined by f → Φ f = ( f n ) n ∈ Z , f n = (Φ f ) n = 1 √ τ Z τ e − intω f ( t ) dt, ω = 2 πτ , f ∈ ˜ H . Let ∂ = − i ∂∂t be the self-adjoint operator in L ( T τ ). We also denote ∂ = − i ∂∂t the corre-sponding operator in e H with the natural domain D = D ( ∂ ). We use the notation h A ( t ) i to indicate multiplication by A ( t ) on the space e H . Introduce the operators e h o and e h on e H = L ( T τ , ℓ ( Z d )) by e h o = ∂ + ∆ , e h = e h o + h V ( t ) i . It is known that the spectrum σ (∆) = [0 , d ]. Then the spectrum of e h o has the form σ ( e h o ) = σ ac ( e h o ) = [ n ∈ Z σ (∆ + ωn ) = [ n ∈ Z [ ωn, ωn + 2 d ] . (1.5)Note that if ω > d , then the spectrum of e h o has the band structure with the bands σ (∆ + ωn ) = [ ωn, ωn + 2 d ] separated by gaps. Let B and B be the trace and the Hilbert-Schmidtclass equipped with the norm k · k B and k · k B , respectively. Introduce the free resolvent R o ( λ ) = ( e h o − λ ) − , λ ∈ C ± . Below we show that if V satisfies Condition V, then V R o ( λ ) ∈ B , ∀ λ ∈ C ± . (1.6)This yields D ( e h ) = D ( e h o ) and σ ess ( e h ) = σ ess ( e h o ) . (1.7)Thus the operator e h has only discrete spectrum in C ± . Define the perturbed resolvent R ( λ ) =( e h − λ ) − for all λ ∈ C ± \ σ disc ( e h ). We have the very useful identity h e itω i R ( λ ) h e − itω i = R ( λ + ω ) , ∀ λ ∈ C ± . (1.8) RACE FORMULAS FOR TIME PERIODIC COMPLEX HAMILTONIANS 3
It means that the spectrum of e h (and e h o ) is ω –periodic. Thus it is sufficient to study eigen-values of e h in a strip Re λ ∈ [0 , ω ). We consider the case of the half strip Λ ⊂ C + definedby Λ = [0 , ω ) × i R + ⊂ C + , ω = πτ . The proof for the lower half strip Λ is similar. The operator e h has N ∞ eigenvalues { λ j , j = 1 , ...., N } in the strip Λ. Here and below each eigenvalue is counted according to itsalgebraic multiplicity. We have similar consideration for the case C − .1.2. Main results.
We assume that a potential V satisfy Condition V. We define a operator-valued function F (below we show that F ( λ ) ∈ B ) and the regularized determinant D by F ( λ ) = | V | R ( λ ) | V | e i arg V , λ ∈ C ± , (1.9)and D ( λ ) = det h ( I + F ) e − F i ( λ ) , λ ∈ C ± . (1.10) D is the basic function to study trace formulas. We describe main properties of D . Theorem 1.1.
Let V satisfy Condition V and a constant C ∗ be defined by (3.3).i) Then the operator-valued function F : C ± → B ( L ( T τ , ℓ ( Z d ))) , defined by (1.9) is analyticand H¨older continuous up to the boundary. Moreover, it satisfies: k F ( λ ) k B C • k V k p, ∀ λ ∈ C ± , C • = 1 + (1 + τ dπ ) (cid:0) C g + τ − (cid:1) + C ∗ τ , (1.11) where C g = π + π √ and the constant C ∗ is defined by (3.3).ii) The modified determinant D is analytic in C ± , H¨older up to the boundary and satisfies D ( λ + ω ) = D ( λ ) ∀ λ ∈ C ± , (1.12) D ( λ ) = 1 + O (1) ν as ν := | Im λ | → ∞ , (1.13)sup λ ∈ C ± |D ( λ ) | e C • k V k p, . (1.14) Moreover, if λ ∈ C \ R is an eigenvalue of the operator e h , then ν (1 − e − τν ) k V k , . (1.15) Remark.
1) For complex potentials we discuss eigenvalues of e h only in the domain [0 , ω ] × (0 , iν o ] , ν o = 2 k V k , , since the operator e h does not have zeros in the domain {| Im λ | > ν o } .2) If the operator V is bounded, then eigenvalues of e h belong to the strip {| Im λ | < k Im V k} .We define the disc D r ⊂ C with the radius r > D r = { z ∈ C : | z | < r } , and abbreviate D = D . We define the Hardy space in a disk D . We say a function F belongs the Hardyspace H = H ∞ ( D ) if F is analytic in D and satisfies k F k H := sup λ ∈ D | F ( z ) | < ∞ . For λ ∈ Λ we define the new spectral variable z ∈ D by z = e iτλ ∈ D , λ ( z ) = − iτ ln z ∈ Λ = [0 , ω ] × R + ⊂ C + . EVGENY, L. KOROTYAEV
The function z = e iτλ , λ ∈ Λ is a conformal mapping from the strip Λ onto the unit disk D .Theorem 1.1 shows that a function ψ ( z ) := D ( λ ( z )) , z ∈ D , belongs to the Hardy space H ∞ ( D ).Define operators J , J on L ( T τ ) and operators F , F on e H by( J f )( t ) = i Z t f ( s ) ds, ( J f )( t ) = i Z τ f ( s ) ds, (1.16)and F = J e V , F = J e V , e V ( t ) = e it ∆ V ( t ) e − it ∆ , R = ( I + F ) − . (1.17)Note that the operator I + F is invertible. Theorem 1.2.
Let a potential V satisfy Condition V and the constant C • be defined by (1.11).Then the function ψ ( z ) := D ( λ ( z )) belongs to H ∞ ( D ) and is H¨older up to the boundary andsatisfies k ψ k H ∞ ( D ) e C • k V k p, . (1.18) The zeros { z j } Nj =1 of ψ in D satisfy P Nj =1 (1 − | z j | ) < ∞ . Moreover, the function log ψ ( z ) isanalytic in D r o for some r o > and has the Taylor series : log ψ ( z ) = ψ z + ψ z + ψ z + ......, as | z | < r o , (1.19) where the coefficients ψ n are given by ψ = − Tr F R e τ F , ψ = − Tr F R e τ F + 12 Tr F ( R e τ F ) , .... (1.20) and e τ = e − iτ ∆ and F , F are defined by (1.17) and R = (1 + F ) − . Remark.
1) We transform the analytic problem from the domain C + to the disk D . Theenergy periodic property (1.12) of the determinant D ( λ + ω ) = D ( λ ) and asymptotics (1.13)are crucial here.2) It is unusual that due to (1.19) the determinant log D ( λ ) = e − τ Im λ O (1) as Im λ → ∞ .Recall that the operator e h has N ∞ eigenvalues { λ j , j = 1 , ...., N } in the domain Λ. Eachpoint z j = z ( λ j ) ∈ D is a zero of ψ ( z ). For the function ψ we define the Blaschke product B ( z ) , z ∈ D by: B = 1 if N = 0 and B ( z ) = N Y j =1 | z j | z j ( z j − z )(1 − z j z ) , z j = e iτλ j if N > . (1.21)It is well known that the Blaschke product B ( z ) , z ∈ D given by (1.21) converges absolutelyfor {| z | < } and satisfies B ∈ H ∞ ( D ) with k B k H ∞
1, since ψ ∈ H ∞ [10]. The Blaschkeproduct B has the standard Taylor series at z = 0:log B ( z ) = B − B z − B z − ... as z → , (1.22)where B = log B (0) < B n = n P Nj =1 (cid:16) z nj − z nj (cid:17) , n >
1. In particular we have B = log B (0) = − τ X Im λ j < . (1.23) RACE FORMULAS FOR TIME PERIODIC COMPLEX HAMILTONIANS 5
We describe the canonical representation of the determinant ψ ( z ) , z ∈ D . Corollary 1.3.
Let a potential V satisfy Condition V. Then the determinant ψ has a canonicalfactorization for all | z | < given by ψ ( z ) = B ( z ) e Ψ( z ) , Ψ( z ) = 12 π Z π e it + ze it − z dµ ( t ) ,dµ ( t ) = ln | ψ ( e it ) | dt − d m ( t ) , (1.24) where ln | ψ ( e it ) | ∈ L ( T ) and m > is some singular measure on [0 , π ] , such that supp m ⊂{ t ∈ [0 , π ] : ψ ( e it ) = 0 } . Moreover, Ψ has the Taylor series at z = 0 in some disk {| z | < r } : Ψ( z ) = µ ( T )2 π + µ z + µ z + µ z + µ z + ..., (1.25) where µ ( T ) = Z π dµ ( t ) = Z π log | ψ ( e it ) | dt − m ( T ) , µ n = 1 π Z π e − int dµ ( t ) , n ∈ N . We present our main result about trace formulas.
Theorem 1.4.
Let V satisfy Condition V. Then the following trace formula holds true: − iτ z Tr (cid:18) R ( λ ) − R o ( λ ) + R o ( λ ) V R o ( λ ) (cid:19) = X (1 − | z j | )( z − z j )(1 − z j z ) + 1 π Z π e it dµ ( t )( e it − z ) , (1.26) m ( T )2 π + τ X λ ∈ Λ Im λ j = 12 π Z π log | ψ ( e it ) | dt > , (1.27) B n = ψ n + µ n , n = 1 , , , .... (1.28) where λ ( z ) = iτ ln z, z ∈ D and the measure dµ ( t ) = log | ψ ( e it ) | dt − d m ( t ) , and B n are givenby (1.22), and in particular, B = N X j =1 (cid:18) z j − z j (cid:19) = ψ + 1 π Z T e − it dµ ( t ) , (1.29) Remark.
1) The measure dµ ( t ) in (1.26) is some analog of the spectral shift function forcomplex potentials.2) The trace formula (1.27) has the term m ( R ) which is absent for real potentials. There isan open problem: when this term is absent (or there exists) for specific complex potentials.3) Consider the ψ ( z ) = D ( λ ( z )) , z ∈ D . If z = e it , then λ = ln ziτ = tτ ∈ [0 , ω ]. Then weobtain Z π log | ψ ( e it ) | dt = τ Z ω log |D ( λ + i | dλ. Moreover, we can do the same with all integrals in Theorem 1.4.
Corollary 1.5.
Let a potential V satisfy Condition V. Then the following estimate hold true: m ( T )2 π + τ X λ ∈ Λ Im λ j C • k V k p, , (1.30) where the constant C • is defined by (1.11). EVGENY, L. KOROTYAEV
Remark.
1) The measure dµ ( t ) in (1.26) is some analog of the spectral shift function forcomplex potentials.2) If a potential V does not depend on time then there are estimates of complex eigenvaluesin terms of potentials, see [28], [27]. In the continuous case there are a lot of results about it,see, e.g., [9], [8] and references therein.For time-periodic Hamiltonians many papers have been devoted to scattering mainly foroperators h ( t ) = − ∆ + V ( t, x ) on R d , d >
1, and to the spectral analysis of the correspondingmonodromy operator. Zel’dovich [45] and Howland [13] reduced the problem with a time-dependent Hamiltonian to a problem with a time-independent Hamiltonian by introducing anadditional time coordinate. Completeness of the wave operators for e h, e h o was established byYajima [43]. In [13], [19] it was shown that e h has no singular continuous spectrum. Moreover,Korotyaev [19] proved in that the total number of embedded eigenvalues on the interval [0 , ω ],counting multiplicity, is finite. The case of Schrodinger operators with time-periodic electricand homogeneous magnetic field was discussed in [22], [23], [44], see also recent papers [1],[2], [3], [17], [35]. Moreover, scattering for three body systems was considered in [20], see also[36].Now we discuss stationary case of discrete multidimensional Schr¨odinger operators on thecubic lattice Z d , d >
2, when potentials are real and do not depend on time. For Schr¨odingeroperators with decaying potentials on the lattice Z d , Boutet de Monvel and Sahbani [5] usedMourre’s method to prove completeness of the wave operators, absence of singular continu-ous spectrum and local finiteness of eigenvalues away from threshold energies. Isozaki andKorotyaev [15] studied the direct and the inverse scattering problem as well as trace formu-las. Korotyaev and Moller [29] discussed the spectral theory for potentials V ∈ ℓ p , p > H on theinterval (0 , d ) is absent, see also [4]. An upper bound on the number of discrete eigenvaluesin terms of some norm of potentials was given by Korotyaev and Sloushch [32], Rozenblumand Solomyak[40]. For closely related problems, we mention that Parra and Richard [37] re-proved the results from [5] for periodic graphs. Finally, scattering on periodic metric graphsassociated with Z d was considered by Korotyaev and Saburova [32].2. Regularized determinants
Preliminary analysis.
We present the concepts and facts needed below. Let H be acomplex separable Hilbert space. The class of bounded and compact operators in H we denoteby B ( H ) and B ∞ ( H ) respectively. Let B ( H ) and B ( H ) be the trace and the Hilbert-Schmidtclass equipped with the norm k · k B and k · k B , respectively. If it is evident which H is meant,we shall write simply B , B ∞ ,... We recall some well-known facts about determinants from [11]. • Let
A, B ∈ B and
AB, BA ∈ B . ThenTr AB = Tr BA, (2.1)det( I + AB ) = det( I + BA ) . (2.2) • If A, B ∈ B , then | det( I + A ) | e k A k B , | det( I + A ) − det( I + B ) | k A − B k B e k A k B + k B k B . (2.3) RACE FORMULAS FOR TIME PERIODIC COMPLEX HAMILTONIANS 7
We define the modified determinant det ( I + A ) bydet ( I + A ) = det (cid:18) ( I + A ) e − A (cid:19) , (2.4)and I + A is invertible if and only if det ( A ) = 0, see Chapter IV in [11]). • If A ∈ B , then (see [11]) | det ( I + A ) | e k A k B . (2.5) • Let
A, B ∈ B and
AB, BA ∈ B . Thendet ( I + AB ) = det ( I + BA ) . (2.6) • Suppose a function A ( · ) : Ω → B is analytic for a domain Ω ⊂ C , and the operator( I + A ( z )) − is bounded for any z ∈ Ω. Then the function f ( z ) = det( I + A ( z )) satisfies f ′ ( z ) = f ( z ) Tr (cid:0) I + A ( z ) (cid:1) − A ′ ( z ) ∀ z ∈ Ω . (2.7)In order to investigate the determinant D ( λ ) we need a following lemma. Lemma 2.1.
Let operators
A, B ∈ B act on some Hilbert space H . Then e A e B e − A − B − I ∈ B , (2.8)det (cid:18) e A e − A − B e B (cid:19) = det (cid:18) e B e A e − A − B (cid:19) = 1 , (2.9)det (cid:18) ( I + A + B ) e − A − B (cid:19) = det (cid:18) e − A ( I + A + B ) e − B (cid:19) . (2.10) If in addition I + A is invertible and R = ( I + A ) − , then det (cid:18) ( I + A + B ) e − A − B (cid:19) = det (cid:18) ( I + A ) e − A (cid:19) det (cid:18) ( I + R B ) e − B (cid:19) , (2.11) Proof.
Using the Tailor series e z = 1 + z + z + ... at z = A, B, − A − B we obtain e A e B e − A − B = ( I + A + A )( I + B + B )( I − A − B + C ) = I + G, where A , B , C , G are some trace operators, which yields (2.8). We show (2.9). Define thetrace class valued function F ( t ) − I and the determinant D ( t ) by F ( t ) = e tA e tB e − t ( A + B ) , D ( t ) = det F ( t )for t ∈ R . From (2.7) we obtain the derivative D ′ ( t ) = D ( t ) Tr( F ′ ( t ) F ( t ) − ) . Using this formula, we get: D ′ ( t ) = D ( t ) Tr( F ′ ( t ) F ( t ) − )= D ( t ) Tr (cid:18) e tA e tB (cid:0) e − tB Ae tB − A (cid:1) e − t ( A + B ) (cid:19) e t ( A + B ) e − tB e − tA = D ( t ) Tr (cid:18)(cid:0) e − tB Ae tB − A (cid:1)(cid:19) = 0 , EVGENY, L. KOROTYAEV which yields (2.9). We show (2.10). Using (2.9) we obtaindet (cid:18) ( I + A + B ) e − A − B (cid:19) = det (cid:18) ( I + A + B ) e − B e − A · e A e B e − A − B (cid:19) det (cid:18) ( I + A + B ) e − B e − A (cid:19) det (cid:18) e A e B e − A − B (cid:19) = det (cid:18) e − A ( I + A + B ) e − B (cid:19) = det (cid:18) ( I + A ) e − A (cid:19) det (cid:18) ( I + R B ) e − B (cid:19) , which yields (2.10). If in addition I + A is invertible, then (2.10) givesdet (cid:18) ( I + A + B ) e − A − B (cid:19) = det (cid:18) ( I + A ) e − A (cid:19) det (cid:18) ( I + R B ) e − B (cid:19) , which yields (2.11).Let e B = L ( T τ , B ) be the space of function v : T τ → v ( t ) ∈ B for some Banach space B , which are measurable and satisfy R τ k v ( t ) k B dt < ∞ . Recall that operators J , J act on L ( T τ ) and are given by( J f )( t ) = i Z t f ( s ) ds, ( J f )( t ) = i Z τ f ( s ) ds, (2.12)Note that the operator I + J is invertible. Lemma 2.2. i) The operator ∂ = − i ∂∂t acting on L ( T τ ) has the resolvent given by (( ∂ − λ ) − f )( t ) = Z t ie iλ ( t − s ) f ( s ) ds + iz − z Z τ e iλ ( t − s ) f ( s ) ds, z = e iτλ , (2.13) where f ∈ L ( T τ ) and λ ∈ C \ σ ( ∂ ) .ii) Let an operator function V ∈ L ( T τ , B ( H )) . Then operators J V and J V on e H belongto e B and satisfy k J V k e B τ Z τ k V ( t ) k B dt τ sup t ∈ [0 ,τ ] k V ( t ) k B , k J V k e B Z τ k V ( t ) k B ( τ − t ) dt τ t ∈ [0 ,τ ] k V ( t ) k B . (2.14) Proof. i) We have ( ∂ − λ ) u = f , where u ′ , f ∈ L ( T τ ). Then using u ( τ ) = u (0) we obtain u ′ − iλu = if, ( e − iλt u ) ′ = ie − iλt f, e − iλt u ( t ) = u (0) + Z t ie − iλs f ( s ) dse − iλτ u ( τ ) = u (0) + Z τ ie − iλs f ( s ) ds, ( e − iλτ − u (0) = Z τ ie − iλs f ( s ) ds, which yields (2.13).ii) The Gilbert-Schmidt norms of J j V, j = 1 , k J V k e B = Z τ ds Z τ k V ( t ) k B dt = τ Z τ k V ( t ) k B dt τ sup t ∈ [0 ,τ ] k V ( t ) k B , k J V k e B = Z τ dt Z t k V ( s ) k B ds = Z τ k V ( s ) k B ( τ − s ) ds τ t ∈ [0 ,τ ] k V ( t ) k B . RACE FORMULAS FOR TIME PERIODIC COMPLEX HAMILTONIANS 9
Proposition 2.3.
Let h o be a bounded self-adjoint operator on the separable Hilbert space H .i) The resolvent R o ( λ ) = ( ∂ + h o − λ ) − , λ ∈ C \ R on f ∈ L ( T τ ) × H has the form given by R o ( λ ) f ( t ) = ie itϕ Z τ (cid:18) t − s + e iτϕ − e iτϕ (cid:19) e − isϕ f ( s ) ds,ϕ = λ − h o , e itϕ = za, z = e iτλ , a = e − iτh o , (2.15) where t = 1 , t > and t = 0 , t < .ii) Let, in addition, an operator function V ∈ L ( T τ , B ( H )) and let c = R τ k V ( s ) k B ds . Thenoperator R o ( λ ) V on e H belong to e B and satisfies k R o ( λ ) V k e B cν (1 − e − ντ ) , ν := Im λ > . (2.16) Moreover, if λ ∈ C + is an eigenvalue of the operator e h o + V , then ν (1 − e − ντ ) c. (2.17) Proof.
The statement i) follows from Lemma 2.2.ii) Using (2.15) we present R o in the form R o = R + XR , where R ( λ ) f ( t ) = Z t ie iλ ( t − s ) ϕ f ( s ) ds, R ( λ ) f ( t ) = Z τ e iλ ( t − s ) ϕ f ( s ) ds, X = e iτϕ − e iτϕ . Consider the case ν = Im λ >
0, the proof for ν < R V is k R V k e B = Z τ e − ν ( t − s ) dt Z t k V ( s ) k B ds = Z τ e νs k V ( s ) k B ds Z τs e − νt dt = 12 ν Z τ k V ( s ) k B (1 − e − ν ( τ − s ) ) ds c ν , (2.18)and the Gilbert-Schmidt norm of R V is k R V k e B = Z τ e − νt dt Z τ e νs k V ( s ) k B ds = 1 − e − ντ ν Z τ e νs k V ( s ) k B ds e ντ − ν c. Then the estimate k X k e − ντ − e − ντ = e ντ − gives k XR ( λ ) V k e B ( e ντ − e ντ − c ν = ( e ντ + 1)( e ντ − c ν , (2.19)since ( e ντ +1)( e ντ − = 1 + e ντ − ντ . Then we obtain k R o V k e B k R V k e B + 2 k R V k e B cν e ντ ( e ντ −
1) = 2 cν (1 − e − ντ ) . (2.20)If k R o ( λ ) V k e B <
1, then the operator I + R o ( λ ) V has an inverse. Thus from (2.20) we havethat if cν (1 − e − ντ ) <
1, then λ is not an eigenvalue of the operator H o + V . Then if λ is aneigenvalue of the operator H o + V , then 2 c > ν (1 − e − τν ) . Determinants.
The operator ∂ = − i ∂∂t on L ( T τ ) has the spectrum σ ( ∂ ) = { ω Z } ,where ω = πτ . Lemma 2.4.
Let h o be a bounded self-adjoint operator on the separable Hilbert space H . Letan operator-valued function V ∈ L ( T τ , B ( H )) and R o ( λ ) = ( ∂ + h o − λ ) − , λ ∈ C ± . Theni) Operators R o ( λ ) V and V R o ( λ ) ∈ B ( e H ) for any λ ∈ C ± ; the modified determinant D ( λ ) =det h ( I + V R o ) e − V R o i ( λ ) is well defined, analytic in C ± and satisfies D ( λ ) = 1 + O (1 /ν ) as ν := Im λ → ±∞ , λ ∈ C ± . (2.21) ii) The modified determinant D ( λ ) satisfies D ( λ + ω ) = D ( λ ) ∀ λ ∈ C ± , (2.22) D ′ ( λ ) D ( λ ) = − Tr (cid:18) ( R o ( λ ) V ) R ( λ ) (cid:19) , (2.23)log D ( λ ) = − ∞ X n =2 ( − n n Tr ( R o ( λ ) V ) n , (2.24) where the traces T n := Tr( V R o ( λ )) n , n > satisfy | T n ( λ ) | (cid:18) ν Z τ k V ( s ) k B ds (cid:19) n ∀ ν > /τ. (2.25) Remark.
Due to (2.21) we take the branch of log D so that log D ( λ ) = o (1) as | Im λ | → ∞ . Proof . i) Lemma 2.3 gives that
V R o ( λ ) ∈ B ( e H ) for any λ ∈ C ± . We show that thedeterminant D is well defined. The Taylor series for the entire function e − E and the estimate(2.16) give at A ( λ ) = V R ( λ )[( I + A ) e − A ] = ( I + A )(1 − A + A O (1)) = 1 − A + A O (1) = I + A O (1) . Moreover, this asymptotics and (2.16), (2.3) imply (2.21).ii) The identities (1.7) and (2.6) yield (2.22). Take | Im λ | > r for r > R ( λ ) = R ( λ ) + ∞ X n =1 ( − n (cid:18) R ( λ ) V (cid:19) n R ( λ ) , (2.26)where the right-hand side is uniformly convergent on { λ ∈ C : | Im λ | > r } . Using (2.7) and(2.1), we have the following for λ ∈ Λ: D ′ ( λ ) = −D ( λ ) Tr (cid:18) e A ( λ ) ( I + A ( λ )) − A ( λ ) A ′ ( λ ) e − A ( λ ) (cid:19) = −D ( λ ) Tr( I + A ( λ )) − A ( λ ) A ′ ( λ ) = −D ( λ ) Tr V R ( λ ) V R ( λ )= −D ( λ ) Tr (cid:18) R ( λ ) V R ( λ ) V R ( λ ) (cid:19) = −D ( λ ) Tr (cid:18) ( R ( λ ) V ) R ( λ ) (cid:19) , (2.27)since R ( λ ) V R ( λ ) = R ( λ ) V R ( λ ), which yields (2.23). Thus (2.26) gives(log D ( λ )) ′ = − Tr ∞ X n =0 ( − n (cid:18) R ( λ ) V (cid:19) n +2 R ( λ ) . (2.28) RACE FORMULAS FOR TIME PERIODIC COMPLEX HAMILTONIANS 11
Then integrating we obtain (2.24) since we have the identity ddλ (cid:18) Tr (cid:18) V R ( λ ) (cid:19) n (cid:19) = n Tr (cid:18) V R ( λ ) (cid:19) n R ( λ ) . iii) From ντ > ντ | Tr T n | = | Tr A n ( λ ) | k V R o ( λ ) k n e B (cid:18) ν Z τ k V ( s ) k B ds (cid:19) n . We are ready to prove the main theorem of this section. Here we transform the presentationof the modified determinant D in the forms (2.29), (2.31) convenient for us. Via this presen-tation we determine the asymptotics (2.33) of D ( λ ) in terms of z = e iτλ as Im λ → + ∞ . Infact, we determine the Taylor expansion (2.33) of ψ ( z ) in some disk {| z | < r } Theorem 2.5.
Let h o be a bounded self-adjoint operator on the separable Hilbert space H .Let an operator function V ∈ L ( T τ , B ( H )) . Theni) The modified determinant D ( λ ) = det h ( I + R o V ) e − R o V i ( λ ) is analytic in Λ and thefunction ψ ( z ) = D ( λ ( z )) is analytic in D and has the following form ψ ( z ) = det h ( I + F ) e − F i ( z ) , z = e iτλ ∈ D , (2.29) where the operator F ( z ) acts on e H and is given by F ( z ) = F + γ ( z ) F , γ ( z ) = za − za , a = e − iτh o ,F = J e V , F = J e V , e V ( t ) = e ith o V ( t ) e − ith o , (2.30) where F , F ∈ B ( e H ) . Moreover, the operator I + F is invertible and if R = ( I + F ) − ,then ψ ( z ) = det h ( I + R γF ) e − γF i ( z ) , | z | < . (2.31) iii) The function log ψ ( z ) is analytic in D r for some r > and has the following form log ψ ( z ) = ∞ X n =1 ( − n n Tr F ( R γ ( z ) F ) n . (2.32) Moreover, it has the following Taylor series log ψ ( z ) = ψ z + ψ z + ψ z + ......, as | z | < r, (2.33) where ψ = − Tr F R aF , ψ = − Tr F R a F + 12 Tr F ( R aF ) , .... (2.34) Proof.
Due to Lemma 2.4 an operator R o ( λ ) V ∈ B ( e H ) for any λ ∈ G . Recall that z = e iτλ ,where λ ∈ Λ and 11 t = 1 , t > t = 0 , t <
0. Due to (2.15) the operator ( ∂ + h o − λ ) − V, λ ∈ C ± on e H = L ( T τ , H ) has the form given by( R o ( λ ) V f )( t ) = ie itϕ Z τ (cid:18) t − s + γ ( z ) (cid:19) e − isϕ V ( s ) f ( s ) ds = bF ( z ) b − f,ϕ = λ − h o , e iτϕ = za, z = e iτλ , a = e − iτh o , (2.35) where f ∈ e H and b = e itϕ is a multiplication operator e itϕ f ( t ) in e H . Then from (2.6) weobtain for ψ ( z ) = D ( λ ( z )): ψ ( z ) = det h ( I + R o V ) e − R o V i ( λ ( z )) = det h ( I + F ) e − F i ( z ) . We have the decomposition F = F + γF , where due to (2.14) the operators F , F ∈ B ( e H )and the operator F is invertible, as the Volterra operator. Thus due to Lemma 2.1 we obtain ψ ( z ) = det h ( I + F ) e − F i D ( z ) , D ( z ) := det h ( I + R γ ( z ) F ) e − γ ( z ) F i . (2.36)Recall that the operator e V ( t ) = e ith o V ( t ) e − ith o , where V ( t ) ∈ B ( H ). Due to (2.14) theoperator F ∈ B ( e H ) and if z → k γ ( z ) F k B | z | − | z | k F k B = O ( | z | ) k F k B , which yields D ( z ) →
1. From (2.21) we have ψ ( z ) →
1. From (2.36) we obtain ψ ( z ) = det h ( I + F ) e − F i D ( z ) → , D ( z ) → , which yields det h ( I + F ) e − F i = 1 and we get (2.31).We show (2.32). Estimates in (2.14) yield k F k B C , k F k B C , kRk C o , (2.37)for some constants C , C , C o . Let A = γF and let D ( t ) = det h ( I + R tA ) e − tA i , t ∈ R . From(2.7), (2.37) and R − I = − F R we obtain D ′ ( t ) /D ( t ) = Tr (cid:20)(cid:18) R A − A (cid:19) e − tA (cid:21) e tA ( I + R tA ) − = Tr( R − I ) A ( I + R tA ) − = − Tr F R A ( I + R tA ) − = Tr F X n > ( −R A ) n +1 t n (2.38)and the integration yields (2.32) for z small enough.Let ζ = za, a = e − iτh o . We have γ ( z ) = ζ − ζ = ζ + ζ + ζ + ζ + ..., and then γ ( z ) F = ( ζ + ζ + ζ + ζ + ... ) F , (2.39)From (2.7), (2.37) and (2.32) we obtain for the term with n = 1 and n = 2: − Tr F R γ ( z ) F = − Tr F R ( za + z a + ... ) F , Tr F ( R F ( z )) = Tr F (cid:18) R ( za + z a + ... ) F (cid:19) = z Tr F ( R aF ) + O ( z ) . (2.40)Collecting asymptotics from (2.40) we obtain (2.34). RACE FORMULAS FOR TIME PERIODIC COMPLEX HAMILTONIANS 13 Proof of main theorems
Laplacian on the lattice.
We need results about the resolvent r o ( λ ) = (∆ − λ ) − on ℓ ( Z d ) from [29]: Theorem 3.1.
Let d > . Let u, v ∈ ℓ p ( Z d ) with p < ( if d = 3 d d +1 if d > . Then theoperator-valued function f : C \ [0 , d ] → B , defined by f ( λ ) := u (∆ − λ ) − v (3.1) is analytic and H¨older continuous up to the boundary and satisfies for all λ ∈ C \ [0 , d ] : k f ( λ ) k B C ∗ k u k p k v k p , (3.2) where C ∗ = p d ( p − p + c d (3 + 2 c ) d − dp , c d = · d d − , c = p − − p if d = 3 (cid:18) p − − p (cid:19) p − p − if d = 4 d ( p − d − (2 d +1) p if d > . (3.3) Moreover, we have k f ( λ ) − f ( µ ) k B C α | λ − µ | α k u k p k v k p , ∀ λ, µ ∈ C ± , (3.4) where α, C α are some positive constants. Below we need a simple corollary
Corollary 3.2.
Under the conditions of Theorem 3.1 the operator-valued function f : C \ [0 , d ] → B satisfies k f ( λ ) k B C ∗ k u k p k v k p max { , r ( λ ) } , ∀ λ ∈ C \ [0 , d ] , (3.5) where r ( λ ) = dist { λ, σ (∆) } , and and the constant C ∗ is defined by (3.3). Proof.
Let λ ∈ C \ [0 , d ]. If r ( λ )
1, then from (3.2) we obtain (3.5). If r ( λ ) >
1, then wehave k f ( λ ) k B k u k k v k k r o ( λ ) k k u k p k v k p / r ( λ ) , which yields (3.5).In order to prove main theorem we need a simple estimate. Lemma 3.3.
Consider a function g ( a ) = a − e − i κ a sin a in a domain S + := { a ∈ C + : | Re a | π } for some parameter κ ∈ [ − , . Then max S + | g | C g := 43 π + 5 + 3 π √ . (3.6) Proof.
We have a simple decomposition g = 1 a − e − i κ a sin a = s + f, where s = 1 a − a , f = 1 − e − i κ a sin a (3.7) and the functions s , f is analytic in the strip S = {| Re a | < π } . By the maximum principle,the function s in the strip S has maximum on the lines Re a = b := π , which yieldsmax S | s | = max t ∈ R | s ( b + it ) | max t ∈ R | sin( b + it ) | + max t ∈ R | b + it | = √ π . (3.8)Consider the function f in the half strip S + . We havemax S + | f | = max { f ± , f o } , f ± = max t > | f ( ± b + it ) | , f o = max a ∈ [ − b,b ] | f ( a ) | . (3.9)Let a = ± b + it, t >
0. We obtain | f ( a ) | = | − e − i κ a || sin a | | sin a | + | e − i κ a || sin a | √ e κ t | sin a | ,e κ t | sin a | = 2 e κ t | e ia − e − ia | = 2 e ( κ − t | e i a − | = 2 e ( κ − t | ie − t + 1 | , (3.10)since e i a = e ± i π e − t = ∓ ie − t . This yields f ± √ − b a b . Then we have | f ( a ) | = | − e − i κ a || sin a | = | sin κ a || sin a | and we obtain | f ( a ) | = 2 | sin κ a || sin a | | κ a || a sin π π | = | κ | π √ π √ , if | a | π , | f ( a ) | = 2 | sin κ a || sin a | π = 2 √ , if π | a | b. This yields f o π √ | g | √ π + π √ Proof of main theorems.
Consider the operator e h = e h o + V on e H = L ( T τ , ℓ ( Z d )),where e h o = ∂ + ∆ is the free operator. Due to the factorization V = vq , we define theoperator-valued function F ( λ ) on e H by F ( λ ) = qR o v, λ ∈ C ± , where R o ( λ ) = ( e h o − λ ) − , q = | V | , v = qe i arg V . (3.11) Theorem 3.4.
Let V satisfy Condition V and the operator F ( λ ) , λ ∈ C ± be defined by (3.11).Then F ( λ ) , V R o ( λ ) ∈ e B := B ( e H ) , ∀ λ ∈ C ± . (3.12) Moreover, if we define k V k p, = R τ k V ( t ) k ℓ p ( Z d ) dt , then the operator-valued function F : C ± → e B is analytic and H¨older continuous up to the boundary and satisfies k F ( λ ) k e B C • k V k p, , ∀ λ ∈ C ± ,C • = 1 √ C τ (cid:18) C g + √ √ πτ + C ∗ τ (cid:19) , C τ = 1 + τ dπ , C g = 43 π + 5 + 3 π √ , (3.13) and k F ( λ ) − F ( µ ) k e B C ,α | λ − µ | α k V k p, , ∀ λ, µ ∈ C ± , (3.14) where α, C ,α are some positive constants and the constant C ∗ is defined by (3.3). RACE FORMULAS FOR TIME PERIODIC COMPLEX HAMILTONIANS 15
Proof.
The results in (3.12) have been proved in Lemma 2.3.Due to (1.8) it is enough to prove (3.13) only for the case λ ∈ Λ = [0 , ω ] × i R + .Using (2.15) we present F f , where f ∈ e H , λ ∈ C + in the form given by( F ( λ ) f )( t ) = iq ( t ) e itϕ Z τ (cid:18) t − s + γ (cid:19) e − isϕ v ( s ) f ( s ) ds = ( F ( λ ) f )( t ) + ( F ( λ ) f )( t ) , ( F ( λ ) f )( t ) = iq ( t ) Z t e i ( t − s )( λ − ∆) v ( s ) f ( s ) ds, ( F ( λ ) f )( t ) = q ( t ) Z τ ϑ ( A, κ ) v ( s ) f ( s ) ds, (3.15)where ϕ = λ − ∆ , A = τ ϕ τ ( λ − ∆)2 , γ = e iτϕ − e iτϕ = − e iA i sin A , κ = 2( s − t ) τ − ∈ [ − , , ϑ ( α, κ ) = − e − i κ α α , α ∈ R , ω = 2 πτ . (3.16)The first term, the operator-valued function F : C → e B is entire and satisfies k F k e B = Z τ dt Z t k q ( t ) e i ( t − s )( λ − ∆) v ( s ) k B ds Z τ dt Z t k q ( t ) k B k q ( s ) k B e − t − s ) Im λ ds Z τ k q ( t ) k dt Z t k q ( s ) k ds = 12 (cid:18) Z τ k V ( t ) k dt (cid:19) = 12 k V k , , ∀ Im λ > , (3.17)where k q ( t ) k a = k q ( t ) k B a is the norm in ℓ a ( Z d ) , a > k q ( t ) k = k V ( t ) k .Define the interval I ( ω ) = [ λ o − ω , λ o + ω ] for any fixed λ o ∈ [0 , d ] ∩ [0 , ω ]. Define χ by χ = X j χ j , χ ( µ ) = ( , µ ∈ I ( ω )0 , µ ∈ R \ I ( ω ) , χ j = χ ( · − ωj ) , j ∈ Z . (3.18)Note that the sum χ (∆) = P j χ j (∆) is finite and the number of the function χ j (∆) = 0 isless than 1 + τdπ , since σ (∆) = [0 , d ].Consider the operator-valued function F . We rewrite one in the form( F ( λ ) f )( t ) = q ( t ) Z τ ϑ ( A, k ) χ (∆) v ( s ) f ( s ) d = X j ( F j ( λ ) f )( t ) , ( F j ( λ ) f )( t ) = q ( t ) Z τ ϑ ( A, k ) χ j (∆) v ( s ) f ( s ) ds. (3.19)We consider smoothness and estimates of F ( λ ). The proof for other F j , j = 0 is similar. Wepresent ϑ ( A, k ) in the following form ϑ ( a, κ ) = − a + g ( a )2 , g ( a ) := 1 a − e − i κ a sin a . (3.20)Thus we obtain F ( λ ) f ( t ) = q ( t ) Z τ ϑ ( A, κ ) χ (∆) v ( s ) f ( s ) ds = q ( t )2 Z τ (cid:18) − A + g ( A ) (cid:19) χ (∆) v ( s ) f ( s ) ds, (3.21) which yields F = F r + F g , where F r ( λ ) f ( t ) = q ( t ) τ Z τ r o ( λ ) χ (∆) v ( s ) f ( s ) ds,F g ( λ ) f ( t ) = q ( t )2 Z τ g ( A ) χ (∆) v ( s ) f ( s ) ds, (3.22)and r o ( λ ) = (∆ − λ ) − . In order to consider the operator-valued function F g : C → e B weneed to discuss the function g ( A ) χ (∆). We have A = τ ( λ − ∆) = ζ + A , where A o = τ ( λ o − ∆) , ζ = τ ( λ − λ o )and k A o χ (∆) k τω = π , ( | ζ | < τω = π if | λ − λ o | < ω | ζ | < τω = π if | λ − λ o | < ω . Thus we have F g ( λ ) f ( t ) = q ( t )2 Z τ g ( A o + ζ ) χ (∆) v ( s ) f ( s ) ds where g ( a ) is analytic in the domain S + := { a ∈ C + : | Re a | π } and due to (3.6) it satisfies k g ( A ) χ (∆) k C g = π + π √
2. Then we obtain k F g ( λ ) k e B = Z τ dt Z τ k q t g ( A )2 χ (∆) v s k B ds C g Z τ dt Z τ k q ( t ) k B k q ( s ) k B ds = C g Z τ k q ( t ) k dt Z τ k q ( s ) k ds = C g k V k , . (3.23)Consider the operator-valued function F r : Λ → e B . We have F r = G + G , where( G ( λ ) f )( t ) = q ( t ) τ Z τ r o ( λ ) v ( s ) f ( s ) ds, ( G ( λ ) f )( t ) = q ( t ) τ Z τ r o ( λ )( χ (∆) − v ( s ) f ( s ) ds. Due to Theorem 3.1 the operator-valued function G : C + → B is analytic and H¨oldercontinuous up to the boundary and via (3.2) satisfies k G ( λ ) k e B = 1 τ Z τ dt Z τ k q ( t ) r o ( λ ) v ( s ) k B ds C ∗ τ Z τ dt Z τ k q ( t ) k p k q ( s ) k p ds = C ∗ τ Z τ k V ( t ) k p dt Z τ k V ( s ) k p ds = C ∗ τ k V k p, . (3.24)The operator-valued function r o ( λ )( χ (∆) −
1) : I ( ω ) × R + → B is analytic and H¨oldercontinuous up to the boundary I ( ω ) ⊂ I ( ω ) and satisfies k r o ( λ )( χ (∆) − k ω . Moreover,we have k G ( λ ) k e B = 1 τ Z τ dt Z τ k q ( t ) r o ( λ )( χ (∆) − v ( s ) k B ds πτ Z τ dt Z τ k q ( t ) k k q ( s ) k ds = 2 πτ Z τ k V ( t ) k dt Z τ k V ( s ) k ds = 2 πτ k V k , . (3.25)Collecting all estimates we obtain k F ( λ ) k e B (cid:18) C g + √ √ πτ (cid:19) k V k , + C ∗ τ k V k p, . RACE FORMULAS FOR TIME PERIODIC COMPLEX HAMILTONIANS 17
Recall that the sum χ (∆) = P j χ j (∆) is finite and the number of the function χ j (∆) = 0 isless than C τ = 1 + τdπ , since σ (∆) = [0 , d ]. Thus we have k X j ( F j ( λ ) k e B C τ (cid:18) C g + √ √ πτ (cid:19) k V k , + C τ C ∗ τ k V k p, and jointly with (3.17) we obtain (3.13). Proof of Theorem 1.1
Let V satisfy (1.4) and a constant C ∗ be defined by (3.3).i) Due to Theorem 3.4 the operator-valued function F : C ± → B ( ℓ ( Z d )), defined by (1.9)is analytic and H¨older continuous up to the boundary. Moreover, it satisfies (1.11).ii) From i) we deduce that the modified determinant D ( λ ) = det h ( I + F ) e − F i ( λ ) is analytic in C ± , is H¨older up to the boundary. Identity (1.8) at V = 0 implies (1.12), i.e., D ( λ + ω ) = D ( λ )for all λ ∈ C ± . Asymptotics (2.21) yields (1.13). From (1.11) and (2.5) we obtain (1.14).Moreover, estimate (1.15) has been proved in Proposition 2.3. Proof of Theorem 1.2.
From Theorem 1.1 we obtain that the function ψ ∈ H ∞ ( D )and is H¨older up to the boundary and satisfies (1.11). The zeros { z j } Nj =1 of ψ in D satisfy P Nj =1 (1 − | z j | ) < ∞ . Moreover, the function log ψ ( z ) is analytic in D r and has the Taylorseries log ψ ( z ) = ψ z + ψ z + ψ z + ......, as | z | < r o . where coefficients ψ n are given by Theorem 2.5.We recall the standard facts about the canonical factorization of functions from Hardy space,see e.g. [10], [18] and in the needed specific form for us from [28]. Theorem 3.5.
Let a function ψ ∈ H ∞ ( D ) and be H¨older up to the boundary. Then thereexists a singular measure m > on [0 , π ] , such that ψ has a canonical factorization for all | z | < given by ψ ( z ) = B ( z ) e Ψ( z ) , Ψ( z ) = 12 π Z π − π e it + ze it − z dµ ( t ) , (3.26) where the measure dµ ( t ) = log | ψ ( e it ) | dt − d m ( t ) and log | ψ ( e it ) | ∈ L ( T ) and the measure m satisfies supp m ⊂ { t ∈ [0 , π ] : ψ ( e it ) = 0 } . Moreover, Ψ has the Taylor series Ψ( z ) = µ ( T )2 π + µ z + µ z + µ z + µ z + ..., (3.27) in some disk {| z | < r } , r > , where µ ( T ) = Z π dµ ( t ) = Z π log | f ( e it ) | dt − m ( T ) , µ n = 1 π Z π e − int dµ ( t ) , n ∈ N . We are ready to describe the function ψ . Proof of Corollary 1.3.
The proof follows from Theorem 1.2 and Theorem 3.5.We describe trace formulae.
Proof of Theorem 1.4.
Using (2.23) and the identity R = R o − R o V R o + R o V R o V R weobtain D ′ ( λ ) D ( λ ) = − Tr (cid:0) ( R o V ) R (cid:1) ( λ ) = − Tr (cid:0) R − R o + R o V R o ) (cid:1) ( λ ) . Then differentiation of ψ ( z ) = D ( λ ( z )) in (1.24) yields ψ ′ ( z ) ψ ( z ) = X (1 − | z j | )( z − z j )(1 − z j z ) + 1 π Z π e it dµ ( t )( e it − z ) = D ′ ( λ ( z )) D ( λ ( z )) λ ′ ( z ) , where λ ′ ( z ) = z ′ ( λ ) = iτz . Thus collecting the two last identities we get (1.26).Due to the canonical representation (1.24), the function ψ ( z ) B ( z ) has no zeros in the disc D andΨ( z ) = log ψ ( z ) B ( z ) (see (3.26)) satisfiesΨ( z ) = 12 π Z π − π e it + ze it − z dµ ( t ) , z ∈ D , (3.28)where the measure dµ = log | f ( e it ) | dt − d m ( t ). In order to show (1.27)–(1.29) we need theasymptotics of the Schwatz integral Ψ( z ) as z → π Z π − π e it + ze it − z dµ ( t ) = µ ( T )2 π + µ z + µ z + µ z + µ z + ... as | z | < , (3.29)where µ ( T ) = Z π dµ ( t ) = Z π log | f ( e it ) | dt − ν ( T ) , µ n = 1 π Z π e − int dµ ( t ) , n ∈ Z . We have the identity log ψ ( z ) = log B ( z ) + π R π − π e it + ze it − z dµ ( t ) for all z ∈ D r . Combin-ing asymptotics (1.19), (1.22) and (3.29) we obtain (1.27)-(1.28). In particular, we have − log B (0) = P N Im τ λ j > µ ( T )2 π > Proof of Corollary 1.5.
Let a potential V satisfy (1.4). Then substituting estimate (1.14)into (1.27) we obtain ν ( T )2 π + τ X λ ∈ Λ Im λ j = 12 π Z π − π log | ψ ( e it ) | dt π Z π − π C • k V k p, dt = C • k V k p, , which yields (1.30). Acknowledgments.
EK study was partly supported by the RSF grant No 18-11-00032.
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