Factorization of the characteristic function of a Jacobi matrix
aa r X i v : . [ m a t h . SP ] O c t Factorization of the characteristic function of aJacobi matrix
F. ˇStampach , P. ˇSˇtov´ıˇcek Department of Applied Mathematics, Faculty of Information Technology, CzechTechnical University in Prague, Kolejn´ı 2, 160 00 Praha, Czech Republic Department of Mathematics, Faculty of Nuclear Science, Czech TechnicalUniversity in Prague, Trojanova 13, 120 00 Praha, Czech Republic
Abstract
In a recent paper a class of infinite Jacobi matrices with discrete character ofspectra has been introduced. With each Jacobi matrix from this class an analyticfunction is associated, called the characteristic function, whose zero set coincideswith the point spectrum of the corresponding Jacobi operator. Here it is shownthat the characteristic function admits Hadamard’s factorization in two possibleways – either in the spectral parameter or in an auxiliary parameter which maybe called the coupling constant. As an intermediate result, an explicit expressionfor the power series expansion of the logarithm of the characteristic function isobtained.
Keywords : infinite Jacobi matrix, characteristic function, Hadamard’s factorization
MSC codes : 47B36, 33C99, 11A55
In [12] we have introduced a class of infinite Jacobi matrices characterized by a simpleconvergence condition. Each Jacobi matrix from this class unambiguously determinesa closed operator on ℓ ( N ) having a discrete spectrum. Moreover, with such a matrixone associates a complex function, called the characteristic function, which is analyticon the complex plane with the closure of the range of the diagonal sequence beingexcluded, and meromorphic on the complex plane with the set of accumulation pointsof the diagonal sequence being excluded. It turns out that the zero set of the char-acteristic function actually coincides with the point spectrum of the correspondingJacobi operator on the domain of definition (with some subtleties when handling thepoles; see Theorem 1 below). 1he aim of the current paper is to show that the characteristic function admitsHadamard’s factorization in two possible ways. First, assuming that the Jacobi ma-trix is real and the corresponding operator self-adjoint, we derive a factorization in thespectral parameter. Further, for symmetric complex Jacobi matrices we assume theoff-diagonal elements to depend linearly on an auxiliary parameter which we call, fol-lowing physical terminology, the coupling constant. The second factorization formulathen concerns this parameter.Many formulas throughout the paper are expressed in terms of a function, called F , which is defined on a suitable subset of the linear space of all complex sequences;see [11] for its original definition. This function was also heavily employed in [12]. Sowe start from recalling its definition and basic properties. Apart of the announcedHadamard factorization we derive, as an intermediate step, a formula for log F ( x ).Define F : D → C , F ( x ) = 1 + ∞ X m =1 ( − m ∞ X k =1 ∞ X k = k +2 · · · ∞ X k m = k m − +2 x k x k +1 x k x k +1 · · · x k m x k m +1 , where D = ( { x k } ∞ k =1 ⊂ C ; ∞ X k =1 | x k x k +1 | < ∞ ) . (1)For a finite number of complex variables we identify F ( x , x , . . . , x n ) with F ( x ) where x = ( x , x , . . . , x n , , , , . . . ). By convention, let F ( ∅ ) = 1 where ∅ is the emptysequence.Notice that ℓ ( N ) ⊂ D . For x ∈ D , one has the estimates | F ( x ) | ≤ exp ∞ X k =1 | x k x k +1 | ! , | F ( x ) − | ≤ exp ∞ X k =1 | x k x k +1 | ! − , (2)and it is true that F ( x ) = lim n →∞ F ( x , x , . . . , x n ) . (3)Let us also point out a simple invariance property. For x ∈ D and s ∈ C , s = 0, it istrue that y ∈ D and F ( x ) = F ( y ) , where y k − = sx k − , y k = x k /s, k ∈ N . (4)We shall deal with symmetric Jacobi matrices J = λ w w λ w w λ w . . . . . . . . . , (5)where λ = { λ n } ∞ n =1 ⊂ C and w = { w n } ∞ n =1 ⊂ C \ { } . Let us put γ k − = k − Y j =1 w j w j − , γ k = w k − Y j =1 w j +1 w j , k = 1 , , , . . . . (6)2hen γ k γ k +1 = w k .For n ∈ N , let J n be the n × n Jacobi matrix: ( J n ) j,k = J j,k for 1 ≤ j, k ≤ n , and I n be the n × n unit matrix. Then the formuladet( J n − zI n ) = n Y k =1 ( λ k − z ) ! F (cid:18) γ λ − z , γ λ − z , . . . , γ n λ n − z (cid:19) . (7)holds true for all z ∈ C (after obvious cancellations, the RHS is well defined even for z = λ k ; here and throughout RHS means “right-hand side”, and similarly for LHS).Let us denote C λ := C \ { λ n ; n ∈ N } . Moreover, der( λ ) designates the set of all accumulation points of the sequence λ . Thefollowing theorem is a compilation of several results from [12, Subsec. 3.3]. Theorem 1.
Let a Jacobi matrix J be real and suppose that ∞ X n =1 (cid:12)(cid:12)(cid:12)(cid:12) w n ( λ n − z )( λ n +1 − z ) (cid:12)(cid:12)(cid:12)(cid:12) < ∞ (8) for at least one z ∈ C λ . Then(i) J represents a unique self-adjoint operator on ℓ ( N ) ,(ii) spec( J ) ∩ ( C \ der( λ )) consists of simple real eigenvalues with no accumulationpoints in C \ der( λ ) ,(iii) the series (8) converges locally uniformly on C λ and F J ( z ) := F (cid:18)(cid:26) γ n λ n − z (cid:27) ∞ n =1 (cid:19) (9) is a well defined analytic function on C λ ,(iv) F J ( z ) is meromorphic on C \ der( λ ) , the order of a pole at z ∈ C \ der( λ ) is lessthan or equal to the number r ( z ) of occurrences of z in the sequence λ ,(v) z ∈ C \ der( λ ) belongs to spec( J ) if and only if lim u → z ( z − u ) r ( z ) F J ( u ) = 0 and, in particular, spec( J ) ∩ C λ = spec p ( J ) ∩ C λ = F − J ( { } ) . We will mostly focus on real Jacobi matrices, except Section 4. For our purposesthe following particular case, a direct consequence of a more general result derived in[12, Subsec. 3.3], will be sufficient.
Theorem 2.
Let J be a complex Jacobi matrix of the form (5) obeying λ n = 0 , ∀ n ,and { w n } ∈ ℓ ( N ) . Then J represents a Hilbert-Schmidt operator, F J ( z ) is analyticon C \ { } and spec( J ) \ { } = spec p ( J ) \ { } = F − J ( { } ) . The logarithm of F ( x ) F ( x , . . . , x n ) is a polynomial function in n complex variables, with F (0) = 1, and solog F ( x , . . . , x n ) is a well defined analytic function in some neighborhood of the origin.The goal of the current section is to derive an explicit formula for the coefficients ofthe corresponding power series.For a multiindex m ∈ N ℓ denote by | m | = P ℓj =1 m j its order and by d ( m ) = ℓ itslength. For N ∈ N define M ( N ) = ( m ∈ N [ ℓ =1 N ℓ ; | m | = N ) . (10)Obviously, ∪ ∞ ℓ =1 N ℓ = ∪ ∞ N =1 M ( N ). One has M (1) = { (1) } and M ( N ) = (cid:8)(cid:0) , m , m , . . . , m d ( m ) (cid:1) ; m ∈ M ( N − (cid:9) ∪ (cid:8)(cid:0) m + 1 , m , . . . , m d ( m ) (cid:1) ; m ∈ M ( N − (cid:9) . Hence |M ( N ) | = 2 N − ( | · | standing for the number of elements). Furthermore, foran multiindex m ∈ N ℓ put β ( m ) := ℓ − Y j =1 (cid:18) m j + m j +1 − m j +1 (cid:19) , α ( m ) := β ( m ) m . (11) Proposition 3.
In the ring of formal power series in the variables t , . . . , t n , one has log F ( t , . . . , t n ) = − n − X ℓ =1 X m ∈ N ℓ α ( m ) n − ℓ X k =1 ℓ Y j =1 ( t k + j − t k + j ) m j . (12) For a complex sequence x = { x k } ∞ k =1 such that P ∞ k =1 | x k x k +1 | < log 2 one has log F ( x ) = − ∞ X ℓ =1 X m ∈ N ℓ α ( m ) ∞ X k =1 ℓ Y j =1 ( x k + j − x k + j ) m j . The proof of Proposition 3 is based on some combinatorial notions among themthat of Dyck path is quite substantial. For n ∈ N , n ≥
2, we may regard the setΛ n = { , , . . . , n } as a finite one-dimensional lattice. We shall say that a mapping π : { , , , . . . , N } → Λ n is a loop of length 2 N in Λ n , N ∈ N , if π (0) = π (2 N ) and | π ( j + 1) − π ( j ) | = 1 for1 ≤ j ≤ N . The vertex π (0) is called the base point of a loop. The loops in Λ n with4he base point π (0) = 1 are commonly known as Dyck paths of height not exceeding n −
1. Indeed, if π is such a loop then its graph shifted by 1, { ( j, π ( j ) − j = 0 , , . . . , N } , represents a lattice path in the first quadrant leading from (0 ,
0) to (2 N,
0) whose allsteps are solely (1 ,
1) and (1 , − m ∈ N ℓ denote by Ω( m ) the set of all loops of length 2 | m | in Λ ℓ +1 whichencounter each edge ( j, j + 1) exactly 2 m j times, 1 ≤ j ≤ ℓ (counting both directions).Let Ω ( m ) designate the subset of Ω( m ) formed by those loops which are based at thevertex 1. In other words, Ω ( m ) is the set of Dyck paths with the prescribed numbers2 m j counting the steps at each level j = 1 , , . . . , ℓ . One can call m the specificationof a Dyck path. If π ∈ Ω ( m ) then the sequence ( π (0) , π (1) , . . . , π (2 N − N = | m | , contains the vertex 1 exactly m times, the vertices j , 2 ≤ j ≤ ℓ , arecontained ( m j − + m j ) times in the sequence, and the number of occurrences of thevertex ℓ + 1 equals m ℓ . Remark . It can be deduced from Theorem 3B in [4] that | Ω ( m ) | = β ( m ). Let usrecall the well known fact that there exists a bijection between the set of Dyck paths oflength 2 N and the set of rooted plane trees with N edges (one can consult, for instance, §§ I.5 and I.6 in [5]). A rooted plane tree is said to have the specification m ∈ N ℓ if ithas | m | edges and the number of its vertices of height j equals m j , j = 1 , , . . . , ℓ . Usingthe mentioned bijection one finds that β ( m ) also equals the number of rooted planetrees with the specification m [4, 9]. More recently, this result has been rediscoveredand described in [1]. For the reader’s convenience we nevertheless include this identityin the following lemma along with a short proof. The other identity in the lemmaproviding a combinatorial interpretation of the number α ( m ) seems to be, to theauthors’ best knowledge, new. Lemma 5.
For every ℓ ∈ N and m ∈ N ℓ , | Ω ( m ) | = β ( m ) and | Ω( m ) | = 2 | m | α ( m ) .Proof. To show the first equality one can proceed by induction in ℓ . For ℓ = 1 andany m ∈ N one clearly has | Ω ( m ) | = 1. Suppose now that ℓ ≥ m ∈ N ℓ .Denote m ′ = ( m , . . . , m ℓ ) ∈ N ℓ − . For any π ′ ∈ Ω ( m ′ ) put˜ π = (1 , π ′ (0) + 1 , π ′ (1) + 1 , . . . , π ′ (2 N ′ ) + 1 , N ′ = | m ′ | = | m | − m . The vertex 2 occurs in ˜ π exactly ( m + 1) times.After any such occurrence of 2 one may insert none or several copies of the two-letterchain (1 , m −
1. This way one generates all Dyck paths from Ω ( m ), and each exactly once.This implies the recurrence rule | Ω ( m , m , . . . , m ℓ ) | = (cid:18) m − m m (cid:19) | Ω ( m , . . . , m ℓ ) | , thus proving that | Ω ( m ) | = β ( m ). 5et us proceed to the second equality. Put N = | m | . Consider the cyclic group G = h g i , g N = 1. G acts on Ω( m ) according to the rule g · π = ( π (1) , π (2) , . . . , π (2 N ) , π (0)) , ∀ π ∈ Ω( m ) . Clearly, G · Ω ( m ) = Ω( m ). Let us write Ω( m ) as a disjoint union of orbits,Ω( m ) = M [ s =1 O s . For each orbit choose π s ∈ O s ∩ Ω ( m ). Let H s ⊂ G be the stabilizer of π s . Then | Ω( m ) | = M X s =1 N | H s | . Denote further by G s the subset of G formed by those elements a obeying a · π s ∈ Ω ( m )(i.e. the vertex 1 is still the base point). Then | G s | = m and O s ∩ Ω ( m ) = G s · π s .Moreover, G s · H s = G s , i.e. H s acts freely from the right on G s , with orbits ofthis action being in one-to-one correspondence with elements of O s ∩ Ω ( m ). Hence |O s ∩ Ω ( m ) | = | G s | / | H s | and | Ω ( m ) | = M X s =1 |O s ∩ Ω ( m ) | = M X s =1 m | H s | . This shows that | Ω( m ) | = (2 N/m ) | Ω ( m ) | . In view of the first equality of the propo-sition and (11), the proof is complete. Lemma 6.
For N ∈ N , X m ∈M ( N ) α ( m ) = 12 N (cid:18) NN (cid:19) . Proof.
According to Lemma 5, the sum2 N X m ∈M ( N ) α ( m ) = X m ∈M ( N ) | Ω( m ) | equals the number of equivalence classes of loops of length 2 N in the one-dimensionallattice Z assuming that loops differing by translations are identified. These classes aregenerated by making 2 N choices, in all possible ways, each time choosing either thesign plus or minus (moving to the right or to the left on the lattice) while the totalnumber of occurrences of each sign being equal to N . Remark . The sum P m ∈M ( N ) β ( m ) can readily be evaluated, too, since this is nothingbut the total number of Dyck paths of length 2 N . As is well known, this number equalsthe Catalan number C N := 1 N + 1 (cid:18) NN (cid:19) (see, for instance [3]). 6or m ∈ N ℓ let (cid:18) | m | m (cid:19) := | m | ! m ! m ! · · · m ℓ ! . Lemma 8.
For every ℓ ∈ N and m ∈ N ℓ , α ( m ) ≤ | m | (cid:18) | m | m (cid:19) , and equality holds if and only if ℓ = 1 or .Proof. Put γ ( m ) = α ( m ) / (cid:0) | m | m (cid:1) . To show that γ ( m ) ≤ / | m | one can proceed byinduction in ℓ . It is immediate to check the equality to be true for ℓ = 1 and 2. For ℓ ≥ m > γ ( m , m , m , . . . , m ℓ ) < γ ( m − , m + 1 , m , . . . , m ℓ ) . Furthermore, if ℓ ≥ m = 1 and the inequality is known to be valid for ℓ −
1, onehas γ ( m , m , m , . . . , m ℓ ) = m γ ( m , m , . . . , m ℓ )1 + m + m + · · · + m ℓ < | m | . The lemma follows.
Proof of Proposition 3.
The coefficients of the power series expansion at the origin ofthe function log F ( t , . . . , t n ) can be calculated in the ring of formal power series. Asshown in [12], one has F ( t , . . . , t n ) = det( I + T )where T = t t t . . . . . . . . .. . . . . . . . . t n − t n − t n . (13)Since det exp( A ) = exp(Tr A ) and so log det( I + T ) = Tr log( I + T ), and noticing thatTr T k +1 = 0, one getslog F ( t , . . . , t n ) = Tr log( I + T ) = − ∞ X N =1 N Tr T N . From (13) one deduces thatTr T N = X π ∈L ( N ) 2 N − Y j =0 t π ( j ) (14)7here L ( N ) stands for the set of all loops of length 2 N in Λ n . Let k = min { π ( j ); 1 ≤ j ≤ N } and put ˜ π ( j ) = π ( j ) − k +1 for 0 ≤ j ≤ N . Then ˜ π ∈ Ω( m ) for certain (unambiguous)multiindex m ∈ M ( N ) of length d ( m ) ≤ n − k . Conversely, given m ∈ M ( N ) of length d ( m ) ≤ n − k , 1 ≤ k ≤ n − d ( m ), one defines π ∈ L ( N ) by π ( j ) = k + ˜ π ( j ) − ≤ j ≤ N . Hence the RHS of (14) equals X m ∈M ( N ) d ( m ) In this section, we introduce a regularized characteristic function of a Jacobi matrixand show that it can be expressed as a Hadamard infinite product.Let λ = { λ n } ∞ n =1 , { w n } ∞ n =1 be real sequences such that lim n →∞ λ n = + ∞ and w n = 0, ∀ n . In addition, without loss of generality, { λ n } ∞ n =1 is assumed to be positive.Moreover, suppose that ∞ X n =1 w n λ n λ n +1 < ∞ and ∞ X n =1 λ n < ∞ . (15)Under these assumptions, by Theorem 1, J defined in (5) may be regarded as aself-adjoint operator on ℓ ( N ). Moreover, der( λ ) is clearly empty and the characteristic8unction F J ( z ) is meromorphic on C with possible poles lying in the range of λ . Toremove the poles let us define the functionΦ λ ( z ) := ∞ Y n =1 (cid:18) − zλ n (cid:19) e z/λ n . Since P n λ − n < ∞ , Φ λ is a well defined entire function. Moreover, Φ λ has zeros atthe points z = λ n , with multiplicity being equal to the number of repetitions of λ n inthe sequence λ , and no zeros otherwise.Finally we define (see (9)) H J ( z ) := Φ λ ( z ) F J ( z ) , and call H J ( z ) the regularized characteristic function of the Jacobi operator J . Notethat for ε ≥ F J + εI ( z ) = F J ( z − ε ) and so H J + εI ( z ) = H J ( z − ε )Φ λ ( − ε ) − exp − z ∞ X n =1 ελ n ( λ n + ε ) ! . (16)According to Theorem 1, the spectrum of J is discrete, simple and real. Moreover,spec( J ) = spec p ( J ) = H − J ( { } ) . As is well known, the determinant of an operator I + A on a Hilbert space canbe defined provided A belongs to the trace class. The definition, in a modified form,can be extended to other Schatten classes I p as well, in particular to Hilbert-Schmidtoperators; see [10] for a detailed survey of the theory. Let us denote, as usual, thetrace class and the Hilbert-Schmidt class by I and I , respectively. If A ∈ I then( I + A ) exp( − A ) − I ∈ I , and one defines det ( I + A ) := det (( I + A ) exp( − A )) . We shall need the following formulas [10, Chp. 9]. For A, B ∈ I one hasdet ( I + A + B + AB ) = det ( I + A ) det ( I + B ) exp ( − Tr( AB )) . (17)A factorization formula holds for A ∈ I and z ∈ C ,det ( I + zA ) = N ( A ) Y n =1 (1 + zµ n ( A )) exp ( − zµ n ( A )) , (18)where µ n ( A ) are all (nonzero) eigenvalues of A counted up to their algebraic multi-plicity (see Theorem 9.2 in [10] and also Theorem 1.1 ibidem introducing the algebraicmultiplicity of a nonzero eigenvalue of a compact operator). In particular, I + zA is9nvertible if and only if det ( I + zA ) = 0. Moreover, the Plemejl-Smithies formulatells us that for A ∈ I , det ( I + zA ) = ∞ X m =0 a m ( A ) z m m ! , (19)where a m ( A ) = det m − . . . A m − . . . A Tr A . . . A m − Tr A m − Tr A m − . . . A m Tr A m − Tr A m − . . . Tr A (20)for m ≥ 1, and a ( A ) = 1 [10, Thm. 5.4]. Finally, there exists a constant C such thatfor all A, B ∈ I , | det ( I + A ) − det ( I + B ) | ≤ k A − B k exp (cid:0) C ( k A k + k B k + 1) (cid:1) , (21)where k · k stands for the Hilbert-Schmidt norm.We write the Jacobi matrix in the form J = L + W + W ∗ where L is a diagonal matrix while W is lower triangular. By assumption (15), theoperators L − and K := L − / ( W + W ∗ ) L − / (22)are Hilbert-Schmidt. Hence for every z ∈ C , the operator L − / ( W + W ∗ − z ) L − / belongs to the Hilbert-Schmidt class. Lemma 9. For every z ∈ C , H J ( z ) = det (cid:0) I + L − / ( W + W ∗ − z ) L − / (cid:1) . In particular, H J (0) = F J (0) = det ( I + K ) . Proof. We first verify the formula for the truncated finite rank operator J N = P N J P N ,where P N is the orthogonal projection onto the subspace spanned by the first N vectorsof the canonical basis in ℓ ( N ). Using formula (7) one derivesdet (cid:2) ( I + P N L − / ( W + W ∗ − z ) L − / P N ) exp (cid:0) − P N L − / ( W + W ∗ − z ) L − / P N (cid:1)(cid:3) = det( P N L − P N ) det( J N − zI N ) exp (cid:0) z Tr( P N L − P N ) (cid:1) = N Y n =1 (cid:18) − zλ n (cid:19) e z/λ n ! F (cid:26) γ n λ n − z (cid:27) Nn =1 ! . Sending N to infinity it is clear, by (3) and (15), that the RHS tends to H J ( z ).Moreover, one knows that det ( I + A ) is continuous in A in the Hilbert-Schmidtnorm, as it follows from (21). Thus to complete the proof it suffices to notice that A ∈ I implies k P N AP N − A k → N → ∞ .10e intend to apply the Hadamard factorization theorem to H J ( z ); see, for example,[2, Thm. XI.3.4]. For simplicity we assume that F J (0) = 0 and so J is invertible.Otherwise one could replace J by J + εI for some ε > K defined in (22) is Hilbert-Schmidt. At thesame time, this is a Jacobi matrix operator with zero diagonal admitting applicationof Theorem 1. One readily finds that F K ( z ) = F (cid:18)(cid:26) − γ n zλ n (cid:27) ∞ n =1 (cid:19) . Hence F K ( − 1) = F J (0), and J is invertible if and only if the same is true for ( I + K ).In that case, again by Theorem 1, 0 belongs to the resolvent set of J , and J − = L − / ( I + K ) − L − / . (23) Lemma 10. If J is invertible then J − is a Hilbert-Schmidt operator and det (cid:0) I − z ( I + K ) − L − (cid:1) = det (cid:0) I − zJ − (cid:1) (24) for all z ∈ C .Proof. By assumption (15), L − / belongs to the Schatten class I . Since the Schattenclasses are norm ideals and fulfill I p I q ⊂ I r whenever r − = p − + q − [10, Thm. 2.8],one deduces from (23) that J − ∈ I .Furthermore, one knows that Tr( AB ) = Tr( BA ) provided A ∈ I p , B ∈ I q and p − + q − = 1 [10, Cor. 3.8]. HenceTr (cid:0) ( I + K ) − L − (cid:1) k = Tr (cid:0) L − / ( I + K ) − L − / (cid:1) k = Tr( J − k ) , ∀ k ∈ N , k ≥ . It follows that coefficients a m defined in (20) fulfill a m (( I + K ) − L − ) = a m ( J − ) for m = 0 , , , . . . . The Plemejl-Smithies formula (19) then implies (24). Theorem 11. Using notation introduced in (5), suppose a real Jacobi matrix J obeys(15) and is invertible. Denote by λ n ( J ) , n ∈ N , the eigenvalues of J (all of them arereal and simple). Then L − − J − ∈ I , ∞ X n =1 λ n ( J ) − < ∞ , (25) and for the regularized characteristic function of J one has H J ( z ) = F J (0) e bz ∞ Y n =1 (cid:18) − zλ n ( J ) (cid:19) e z/λ n ( J ) (26) where b = Tr (cid:0) L − − J − (cid:1) = ∞ X n =1 (cid:18) λ n − λ n ( J ) (cid:19) . roof. Recall equation (23). Since L − / ∈ I and K ∈ I one has, after somestraightforward manipulations, L − − J − = L − / K ( I + K ) − L − / ∈ I . (27)By Lemma 10, the operator J − is Hermitian and Hilbert-Schmidt. This implies (25).Furthermore, by Lemma 9, formula (17) and Lemma 10, H J ( z ) = det ( I + K − zL − )= det ( I + K ) det (cid:0) I − z ( I + K ) − L − (cid:1) exp (cid:2) z Tr (cid:0) K ( I + K ) − L − (cid:1)(cid:3) = F J (0) e bz det (cid:0) I − zJ − (cid:1) . Here we have used (27) implyingTr (cid:0) K ( I + K ) − L − (cid:1) = Tr (cid:0) L − / K ( I + K ) − L − / (cid:1) = Tr (cid:0) L − − J − (cid:1) = b. Finally, by formula (18),det (cid:0) I − zJ − (cid:1) = ∞ Y n =1 (cid:18) − zλ n ( J ) (cid:19) e z/λ n ( J ) . This completes the proof. Corollary 12. For each ǫ > there is R ǫ > such that for | z | > R ǫ , | H J ( z ) | < exp (cid:0) ǫ | z | (cid:1) . (28) Proof. Theorem 11, and particularly the product formula (26) implies that H J ( z ) isan entire function of genus one. In that case the growth property (28) is known to bevalid; see, for example, Theorem XI.2.6 in [2]. Example . Put λ n = n and w n = w = 0, ∀ n ∈ N . As shown in [11], the Besselfunction of the first kind can be expressed as J ν (2 w ) = w ν Γ( ν + 1) F (cid:18)(cid:26) wν + k (cid:27) ∞ k =1 (cid:19) , (29)as long as w, ν ∈ C , ν / ∈ − N . Using (29) and thatΓ( z ) = e − γz z ∞ Y n =1 (cid:16) zn (cid:17) − e z/n , where γ is the Euler constant, one gets H J ( z ) = e γz w z J − z (2 w ). Applying Theorem 11to the Jacobi matrix in question one reveals the infinite product formula for the Besselfunction considered as a function of its order. Assuming J (2 w ) = 0, the formula reads w z J − z (2 w ) J (2 w ) = e c ( w ) z ∞ Y n =1 (cid:18) − zλ n ( J ) (cid:19) e z/λ n ( J ) where c ( w ) = 1 J (2 w ) ∞ X k =0 ( − k ψ ( k + 1) w k ( k !) ,ψ ( z ) = Γ ′ ( z ) / Γ( z ) is the digamma function, and the expression for c ( w ) is obtainedby comparison of the coefficients at z on both sides.12 Factorization in the coupling constant Let x = { x n } ∞ n =1 be a sequence of nonzero complex numbers belonging to the domain D defined in (1). Our goal in this section is to prove a factorization formula for theentire function f ( w ) := F ( wx ) , w ∈ C . Let us remark that f ( w ) is even.To this end, let us put v k = √ x k , ∀ k , (any branch of the square root is suitable)and introduce the auxiliary Jacobi matrix A = a · · · a a · · · a a · · · a · · · ... ... ... ... . . . , with a k = v k v k +1 , k ∈ N . (30)Then A represents a Hilbert-Schmidt operator on ℓ ( N ) with the Hilbert-Schmidtnorm k A k = 2 ∞ X k =1 | a k | = 2 ∞ X k =1 | x k x k +1 | . The relevance of A to our problem comes from the equality F A ( z ) = F (cid:16)n x k z o ∞ k =1 (cid:17) = f (cid:0) z − (cid:1) , which can be verified with the aid of (4) and (9). Hence F A ( z ) is analytic on C \ { } .By Theorem 2, the set of nonzero eigenvalues of A coincides with the zero set of F A ( z ).It even turns out that the algebraic multiplicity of a nonzero eigenvalue ζ of A equalsthe multiplicity of ζ as a root of the function F A ( z ), as one infers from the followingproposition. Proposition 14. Under the same assumptions as in Theorem 2, the algebraic multi-plicity of any nonzero eigenvalue ζ of J is equal to the multiplicity of the root ζ − ofthe entire function ϕ ( z ) = F J ( z − ) = F ( { zγ n } ∞ n =1 ) .Proof. Recall that γ n γ n +1 = w n and so, by the assumptions of Theorem 2, { γ n } ∈ D .Denote again by P N , N ∈ N , the orthogonal projection onto the subspace spanned bythe first N vectors of the canonical basis in ℓ ( N ). From formula (7) we deduce that F (cid:0) { zγ n } Nn =1 (cid:1) = det( I − zJ N ) = det (cid:0) ( I − zJ N ) e zJ N (cid:1) , where J N = P N J P N . Since P N J P N tends to J in the Hilbert-Schmidt norm, as N → ∞ , and by continuity of the generalized determinant as a functional on thespace of Hilbert-Schmidt operators (see (21)) one immediately gets ϕ ( z ) = F (cid:0) { zγ n } ∞ n =1 (cid:1) = det (cid:0) ( I − zJ ) e zJ (cid:1) = det ( I − zJ ) . From (18) it follows that ϕ ( z ) = (1 − ζ z ) m ˜ ϕ ( z ) where m is the algebraic multiplicityof ζ , ˜ ϕ ( z ) is an entire function and ˜ ϕ ( ζ − ) = 0.13he zero set of f ( w ) is at most countable and symmetric with respect to theorigin. One can split C into two half-planes so that the border line passes through theorigin and contains no nonzero root of f . Fix one of the half-planes and enumerateall nonzero roots in it as { ζ k } N ( f ) k =1 , with each root being repeated in the sequenceaccording to its multiplicity. The number N ( f ) may be either a non-negative integeror infinity. Then spec p ( A ) \ { } = (cid:8) ± ζ − k ; k ∈ N , k ≤ N ( f ) (cid:9) . Since A is a trace class operator one has, by Proposition 14 and Lidskii’s theorem, N ( f ) X k =1 ζ k = 12 Tr A = ∞ X k =1 x k x k +1 . (31)Moreover, the sum on the LHS converges absolutely, as it follows from Weyl’s inequal-ity [10, Thm. 1.15]. Theorem 15. Let x = { x k } ∞ k =1 be a sequence of nonzero complex numbers such that ∞ X k =1 | x k x k +1 | < ∞ . Then zeros of the entire even function f ( w ) = F ( wx ) can be arranged into sequences { ζ k } N ( f ) k =1 ∪ {− ζ k } N ( f ) k =1 , with each zero being repeated according to its multiplicity, and f ( w ) = N ( f ) Y k =1 (cid:18) − w ζ k (cid:19) . (32) Proof. Equality (32) can be deduced from Hadamard’s factorization theorem; see, forexample, [2, Chp. XI]. In fact, the absolute convergence of the series P ζ − k in (31)means that the rank of f is at most 1. Furthermore, (2) implies that | f ( w ) | ≤ exp | w | ∞ X k =1 | x k x k +1 | ! , and so the order of f is less than or equal to 2. Hadamard’s factorization theorem tellsus that the genus of f is at most 2. Taking into account that f is even and f (0) = 1,this means nothing but f ( w ) = exp( cw ) N ( f ) Y k =1 (cid:18) − w ζ k (cid:19) c ∈ C . Equating the coefficients at w one gets − ∞ X k =1 x k x k +1 = c − N ( f ) X k =1 ζ k . According to (31), c = 0. Corollary 16. For any n ∈ N (and recalling (10), (11)), N ( f ) X k =1 ζ nk = n X m ∈M ( n ) α ( m ) ∞ X k =1 d ( m ) Y j =1 ( x k + j − x k + j ) m j . (33) Proof. Using Proposition 3, one can expand log f ( w ) into a power series at w = 0.Applying log to (32) and equating the coefficients at w n gives (33).If the sequence { x k } in Theorem 15 is positive one has some additional informationabout the zeros of f ( w ). In that case the v k s in (30) can be chosen positive, and so A is a self-adjoint Hilbert-Schmidt operator. The zero set of f is countable and all rootsare real, simple and have no finite accumulation points. Enumerating positive zerosin ascending order as ζ k , k ∈ N , factorization (32) and identities (33) hold true. Sincethe first positive root ζ is strictly smaller than all other positive roots, one has ζ = lim N →∞ X m ∈M ( N ) α ( m ) ∞ X k =1 d ( m ) Y j =1 ( x k + j − x k + j ) m j − / (2 N ) . Remark . Still assuming the sequence { x k } to be positive let g ( z ) be an entirefunction defined by g ( z ) = 1 + ∞ X n =1 g n z n = ∞ Y k =1 (cid:18) − zζ k (cid:19) , i.e. g ( w ) = f ( w ). In some particular cases the coefficients g n may be known explicitlyand then the spectral zeta function can be evaluated recursively. Put σ (2 n ) = ∞ X k =1 ζ nk , n ∈ N . Taking the logarithmic derivative of g ( z ) and equating coefficients at the same powersof z leads to the recurrence rule σ (2) = − g , σ (2 n ) = − ng n − n − X k =1 g n − k σ (2 k ) for n > . (34)15 xample . Put x k = ( ν + k ) − , with ν > − 1. Recalling (29) and letting z = w/ (cid:16) z (cid:17) − ν Γ( ν + 1) J ν ( z ) = ∞ Y k =1 − z j ν,k ! , is obtained as a particular case of Theorem 15. The zeros of J ν ( z ), called j ν,k , alsooccur in the definition of the so called Rayleigh function [8], σ ν ( s ) = ∞ X k =1 j sν,k , Re s > . Corollary 16 implies the formula σ ν (2 N ) = 2 − N N ∞ X k =1 X m ∈M ( N ) α ( m ) d ( m ) Y j =1 (cid:18) j + k + ν − j + k + ν ) (cid:19) m j , N ∈ N . Example . This examples is perhaps less commonly known and concerns the Ra-manujan function, also interpreted as the q -Airy function by some authors [7, 14], anddefined by A q ( z ) := φ ( ; 0; q, − qz ) = ∞ X n =0 q n ( q ; q ) n ( − z ) n , (35)where φ ( ; b ; q, z ) is the basic hypergeometric series ( q -hypergeometric series) and( a ; q ) k is the q -Pochhammer symbol (see, for instance, [6]). In (35), we suppose that0 < q < z ∈ C . It has been shown in [11] that A q ( w ) = q F (cid:16)(cid:8) wq (2 k − / (cid:9) ∞ k =1 (cid:17) . Denote by 0 < ζ ( q ) < ζ ( q ) < ζ ( q ) < . . . the positive zeros of w A q ( w ) and put ι k ( q ) = ζ k ( q ) , k ∈ N . Then Theorem 15 tells us that the zeros of A q ( z ) are exactly0 < ι ( q ) < ι ( q ) < ι ( q ) < . . . , all of them are simple and A q ( z ) = ∞ Y k =1 (cid:18) − zι k ( q ) (cid:19) . One has (cid:8) ι k ( q ) − / ; k ∈ N (cid:9) = spec( AAA ( q )) \{ } where AAA ( q ) is the Hilbert-Schmidtoperator in ℓ ( N ) whose matrix is of the form (30), with a k = q k/ . Corollary 16 yieldsa formula for the spectral zeta function D N ( q ) associated with A q ( z ), namely D N ( q ) := ∞ X k =1 ι k ( q ) N = N q N − q N X m ∈M ( N ) α ( m ) q ǫ ( m ) , N ∈ N , where, ∀ m ∈ N ℓ , ǫ ( m ) = P ℓj =1 ( j − m j . In accordance with (34), from the powerseries expansion of A q ( z ) one derives the recurrence rule D n ( q ) = ( − n +1 nq n ( q ; q ) n − n − X k =1 ( − k q k ( q ; q ) k D n − k ( q ) , n = 1 , , , . . . . J of the form (5) such that the diagonal sequence { λ n } is semibounded. Suppose further that the off-diagonal elements w n dependon a real parameter w as w n = wω n , n ∈ N , with { ω n } being a fixed sequenceof positive numbers. Following physical terminology one may call w the couplingconstant. Denote λ inf = inf λ n . Assume that ∞ X n =1 ω n ( λ n − z )( λ n +1 − z ) < ∞ for some and hence any z < λ inf . For z < λ inf , Theorem 15 can be applied to thesequence x n ( z ) = κ n λ n − z , n ∈ N , where { κ n } is defined recursively by κ = 1, κ n κ n +1 = ω n ; comparing to (6) one has κ k − = γ k − , κ k = γ k /w . Let F J ( z ; w ) = F (cid:18)(cid:26) γ n λ n − z (cid:27) ∞ n =1 (cid:19) = F ( { w x n ( z ) } ∞ n =1 )be the characteristic function of J = J ( w ). We conclude that for every z < λ inf fixed,the equation F J ( z ; w ) = 0 in the variable w has a countably many positive simpleroots ζ k ( z ), k ∈ N , enumerated in ascending order, and F J ( z ; w ) = ∞ Y k =1 (cid:18) − w ζ k ( z ) (cid:19) . Acknowledgments The authors wish to acknowledge gratefully partial support from the following grants:Grant No. 201/09/0811 of the Czech Science Foundation (P.ˇS.) and Grant No. LC06002of the Ministry of Education of the Czech Republic (F.ˇS.). References [1] G. M. Cicuta, M. Contedini, L. Molinari: Enumeration of simple random walksand tridiagonal matrices , J. Phys. A: Math. Gen. (2002), 1125-1146.[2] J. B. Conway: Functions of One Complex Variable , second ed., (Springer, NewYork, 1978).[3] E. Deutsch: Dyck path enumeration , Discrete Math. (1999), 167-202.[4] P. Flajolet: Combinatorial aspects of continued fractions , Discrete Math. (1980), 125-161. 175] F. Flajolet, R. 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