SPECTRAL PROPERTIES OF UNBOUNDED JACOBI MATRICES WITH ALMOST MONOTONIC WEIGHTS GRZEGORZ S´WIDERSKI 5 1 0 Abstract. We present an unified framework to identify spectra of Jacobi matrices. We give 2 applications to long-standing conjecture of Chihara ([4], [5]) concerning one-quarter class of n orthogonalpolynomials,totheconjectureposedbyRoehnerandValent[20]concerningcontin- a uous spectra of generators of birth and death processes and to spectral properties of operators J studied by Janas, Moszyn´ski [14] and Pedersen [19]. 4 1 1. Introduction ] P Given sequences a ∞ and b ∞ such that a > 0 and b R we set S { n}n=0 { n}n=0 n n ∈ . b a 0 0 0 ... h 0 0 t a0 b1 a1 0 0 ... a   0 a b a 0 ... m C = 1 2 2 .  0 0 a b a ...  [  2 3 3   ... ... ... ... ... ...  1   v   The operator C is defined on the domain Dom(C)= x ℓ2: Cx ℓ2 , where 0 { ∈ ∈ } 2 ∞ 4 ℓ2 = x CN: x 2 < n 3 { ∈ | | ∞} 0 nX=0 . and is called a Jacobi matrix. 1 0 The study of Jacobi matrices is motivated by connections with orthogonal polynomials and 5 classical moment problem (see e.g. [22]). Also every self-adjoint operator can be represented as 1 a direct sum of Jacobi matrices. In particular, generators of birth and death processes may be : v seen as Jacobi matrices acting on weighted ℓ2 spaces. i X There are several approaches to the problem of the indentification of the spectrum of un- r bounded Jacobi matrices. A method often used is based on subordination theory (see e.g. [6], a [15], [18]). Another technique uses the analysis of commutator between Jacobi matrix and a suitable chosen matrix (see e.g. [21]). The case of Jacobi matrices with monotonic weights was considered mainly by Dombrowski (see e.g. [8]), where the author developed commutator techniques which enabled qualitative spectral analysis of examined operators. The present article is motivated by commutator techniques of Dombrowski and some ideas of Clark [6]. In fact, commutators do not appear here directly but are hidden in some of our expressions. 2010 Mathematics Subject Classification. Primary: 47B36, 42C05. Secondary: 60J80. Key words and phrases. Jacobi matrix, continuous spectrum,Chihara’s conjecture. 1 2 GRZEGORZS´WIDERSKI Let C be a Jacobi matrix and assume that the matrix C is self-adjoint. The spectrum of the operator C will be denoted by σ(C), the set of all its eigenvalues by σ (C) and the set of all p − accumulation points of σ(C) by σ (C). For a real number x we define x = max( x,0). ess − Our main result is the following theorem. Theorem A. Let C be a Jacobi matrix. If there is a positive sequence α such that n { } (a) lim a = , n→∞ n ∞ ∞ − an+1αn+1 an αn−1 (b) < , n=1(cid:20) an αn − an−1 αn (cid:21) ∞ X ∞ 1 an−1 αn−1 (c) < , n=1an−1 (cid:12) an − αn (cid:12) ∞ X∞ (cid:12) (cid:12) bn+1(cid:12)(cid:12) bn αn−1 (cid:12)(cid:12) (d) < , n=0(cid:12) an − an−1 αn (cid:12) ∞ X∞ (cid:12) (cid:12) (cid:12) 1 (cid:12) (cid:12) (cid:12) (e) = , a α ∞ n n n=0 X αn−1 an (f) lim =1, n→∞ αn an−1 b n (g) limsup| | < 2 n→∞ an then the Jacobi matrix C is self-adjoint and satisfies σ (C)= , and σ(C)= R. p ∅ The importance of Theorem A lies in the fact that we have a flexibility in the choice of the sequence α . Some choices of the sequence α are given in Section 4. The simplest case is the n n following result. Corollary A. Assume (a) lim a = , n→∞ n ∞ ∞ 1 (b) = , a2 ∞ n=0 n X ∞ 2 − a n+1 (c) 1 < , a − ∞ n=0"(cid:18) n (cid:19) # X b n (d) limsup| | < 2, n→∞ an ∞ b b n+1 n (e) | − | < . a ∞ n n=0 X Then the Jacobi matrix C is self-adjoint and satisfies σ (C)= and σ(C)= R. p ∅ In [11, Lemma 2.6] it was proven that if the nonnegative sequence a2 a2 is bounded and n− n−1 b 0 then the matrix C has no eigenvalues. Corollary A gives additional information that n ≡ SPECTRAL PROPERTIES OF JACOBI MATRICES 3 in this case holds σ(C) = R. Moreover, the assumptions of Corollary A are weaker than the conditions of [11, Lemma 2.6]. In Section 6 we provide examples showing sharpness of Corollary A. In particular, condi- tion (b) is necessary in the class of monotonic sequences a and condition (c) could not be n replaced by [(a /a )2 1]− 0. Corollary B shows that{in g}eneral condition (d) is necessary. n+1 n − → Unfortunately, we do not know whether condition (d) is implied by the rest of the assumptions. Author knows only examples satisfying assumptions of Corollary A when b /a 0. n n | | → In Section 5 we apply Corollary A to resolve a conjecture (see [20]) about continuous spectra of generators of birth and death processes. We also present there applications to the following conjecture. Conjecture A (Chihara, [4], [5]). Assume that a Jacobi matrix C is self-adjoint, b , the n → ∞ smallest point ρ of σ (C) is finite and ess a2 1 lim n = . n→∞bnbn+1 4 Then σ (C) =[ρ, ). ess ∞ A direct consequence of Corollary A providing easy to check additional assumptions to Con- jecture A is the following result. Corollary B. Assume (a) lim a = , n n→∞ ∞ ∞ 1 (b) = , a ∞ n n=0 X ∞ − a n+1 (c) 1 < , a − ∞ n=0(cid:20) n (cid:21) X (d) nl→im∞[an−1−bn +an] = M. Then the Jacobi matrix C satisfies σ (C)= [ M, ). Moreover, if a /a 1 then ess n+1 n − ∞ → a2 1 lim n = . n→∞bnbn+1 4 Let us present ideas behind the proof of Theorem A. Let the difference operator J be defined by (Jx)n = iαn−1xn−1+iαnxn+1 (n 0) − ≥ ∞ for a positive sequence {αn}n=0 and α−1 = x−1 = 0. Then we define commutator K on finite sequences by the formula 2iK = CJ JC. − − The expression S = K(pn),pn , where pn = (p ,p ,...,p ,0,0,...), p is the formal eigen- n 0 1 n k h i { } vector of C and , is the scalar product on ℓ2, proved to be an useful tool to show that the h· ·i matrix C has continuous spectrum (see e.g. [8], [11], [13]). Important observation is that we can give closed form for S (see (4)). To the author’s n knowledge this closed form has been known only for α = a (see [7]). Related expression n n for α 1 was analysed in [6]. Adaptation of techniques from [6] allow us to circumvent n ≡ 4 GRZEGORZS´WIDERSKI technicaldifficultiespresentinDombrowski’sapproach. ExtendingdefinitionofS togeneralized n eigenvectors (see (1)) enable us to show that σ(C) = R. The article is organized as follows: in Section 2 we present definitions and well-known facts importantfor our argument. InSection 3we prove TheoremA,whereas inSection 4 weshow its variants. In particular, we identify spectra of operators considered in [19] and [14]. In Section 5 we present applications of Corollary A to some open problems. Finally, in the last section we discuss the necessity of the assumptions of Corollary A. We present also examples showing that in some cases Corollary A is stronger than results known in the literature. Acknowledgments. The author would like to thank Ryszard Szwarc and Bartosz Trojan for their helpful suggestions concerning the presentation of this article. 2. Tools Given a Jacobi matrix C, λ R and real numbers (a,b) = (0,0) we introduce a generalized ∈ 6 eigenvector u by asking n { } u = a, u = b, 0 1 (1) anun+1 = (λ bn)un an−1un−1 (n 1). − − ≥ Furthermore we define the sequence of polynomials p−1(λ) = 0, p0(λ) = 1, (2) anpn+1(λ) = (λ bn)pn(λ) an−1pn−1(λ) (n 0). − − ≥ The sequence p (λ) is a formal eigenvector of matrix C associated with an eigenvalue λ. n { } ∞ Observe that p () is a sequence of polynomials. Moreover, the sequence is orthonor- { n · }n=0 mal with respect to the measure µ() = E()δ ,δ , where E is the spectral resolution of the 0 0 · h · i matrix C, , is the scalar product on ℓ2 and δ = (1,0,0,...). 0 h· ·i The following propositions are well-known. We include them for the sake of completeness. Proposition 1. Let λ R. If every generalized eigenvector u does not belong to ℓ2 then the n ∈ { } matrix C is self-adjoint, λ / σ (C) and λ σ(C). p ∈ ∈ Proof. [22, Theorem 3] asserts that C is self-adjoint provided that at least one generalized eigenvector u / ℓ2. Direct computation shows that λ σ (C) if and only if p (λ) ℓ2. n p n { } ∈ ∈ { } ∈ Therefore the matrix C is self-adjoint and λ / σ (C). p ∈ Observe that the vector x such that (C λI)x = δ satisfies the following recurrence relation 0 − b x +a x = λx +1, 0 0 0 1 0 an−1xn−1+bnxn+anxn+1 = λxn (n 1). ≥ Hencexisageneralized eigenvector, thusx / ℓ2. ThereforetheoperatorC λI isnotsurjective, i.e. λ σ(C). ∈ − (cid:3) ∈ Proposition 2. Let C and C be Jacobi matrices defined by sequences a , b and a , n n n { } { } { } b respectively. Then n {− } b σ(C) = σ(C), σ (C)= σ (C), p p − − b b SPECTRAL PROPERTIES OF JACOBI MATRICES 5 Proof. Let U be the diagonal matrix with a sequence ( 1)n ∞ on the main diagonal. From { − }n=0 the identity UCU−1 = C − and equality of domains the conclusion follows. (cid:3) b Proposition 3. Let C be a self-adjoint Jacobi matrix associated with the sequence b 0. Let n C and C be restrictions of C C to the subspaces span δ : k N and span δ :≡k N e o 2k 2k+1 · { ∈ } { ∈ } respectively. Then C and C are Jacobi matrices associated with e o ae = a a , be = a2 +a2 n 2n 2n+1 n 2n−1 2n (3) ao = a a , bo = a2 +a2 . n 2n+1 2n+2 n 2n 2n+1 respectively. Moreover, C and C are self-adjoint and o e σ(C )= σ(C )= (σ(C))2, σ (C ) = σ (C )= (σ (C))2, o e p o p e p when 0 / σ (C) and 0 / σ (C), where C is a self-adjoint Jacobi matrix associated with the p p ∈ ∈ sequences a ∞ and b 0, and for a set X we define X2 = x2: x X . { n+1}n=0 n ≡ { ∈ } e e Proof. By direct computation it may be proved that C and C satisfies (3). e o e Let pe be the sequence of associated polynomials to the matrix C . Then [22, Theorem 3] { n} e asserts that C is self-adjoint provided pe(0) / ℓ2. It is known that p (x) = pe(x2) (see e.g. e { n } ∈ 2n n [12, Section 4]). Since p (0) = 0 and 0 / σ (C) we have 2k+1 p ∈ ∞ ∞ ∞ = p2(0) = p2 (0) = (pe(0))2. ∞ n 2n n n=0 n=0 n=0 X X X Therefore C is self-adjoint. e Assume that 0 / σ (C). Observe that C = C . Therefore the previous argument applied to p o e ∈ C implies also that C is self-adjoint. o The conclusion of speectra follows from e.g. [1e2, Section 4]. (cid:3) e 3. Proof of the main theorem Given a generalized eigenvector u and a positive sequence α we set n n { } { } (4) Sn = an−1αn−1u2n−1+anαnu2n−(λ−bn)αn−1un−1un (n ≥ 1). Using the identity an−1un−1 = (λ bn)un anun+1 we get an equivalent formula − − (5) S = αn−1a2u2 +a α u2 αn−1a (λ b )u u (n 1). n an−1 n n+1 n n n − an−1 n − n n+1 n ≥ The sequence S for α = a was previously used in the study of Jacobi matrices, but only n n n in the case of bounded ones (see e.g. [7], [10]). In the case of unbounded operators a sequence similar to S for α 1 was also used in [6]. n n ≡ The following proposition is an adaptation of [6, Lemma 3.1]. Proposition 4. Let u be a generalized eigenvector associated with λ R and n { } ∈ S = u2 +u2. n n+1 n e 6 GRZEGORZS´WIDERSKI Assume that a , and n → ∞ αn−1 an bn lim = 1, limsup| | < 2. n→∞ αn an−1 n→∞ an Then there exist constants c > 0,c > 0 such that for sufficiently large n 1 2 S n c a α c a α . 1 n n 2 n n ≤ S ≤ n Proof. Observethatfromtherepresentation(5)wehavethatS isaquadraticformwithrespect n e to variables u and u . Let the minimal and the maximal value of S under the condition n n+1 n S = 1 be denoted by wmin and wmax respectively. Then n n n e 2wnmin = 1+ αn−1 an 1 αn−1 an 2+ αn−1 an λ−bn 2, anαn αn an−1 −s(cid:18) − αn an−1(cid:19) (cid:18) αn an−1 an (cid:19) 2wnmax = 1+ αn−1 an + 1 αn−1 an 2+ αn−1 an λ−bn 2. anαn αn an−1 s(cid:18) − αn an−1(cid:19) (cid:18) αn an−1 an (cid:19) Letting n we see that for large n there is a positive upper and lower bound of the above expression→s. W∞hat ends the proof. (cid:3) Corollary 1. Under the assumptions of Proposition 4, together with ∞ 1 = , a α ∞ n n n=0 X if liminfS > 0 then u / ℓ2. n ∈ Proof. Since liminfS > 0 by Proposition 4 there exists a constant c > 0 such that for every n n sufficiently large we have c S n a α ≤ n n what ends the proof. (cid:3) e Now we are ready to prove Theorem A. Proof of Theorem A. By virtue of Corollary 1 it is enough to show that liminfS > 0 for every n generalized eigenvector u . n { } By Proposition 4 there exists N such that for every n N holds S > 0. Let us define n ≥ F = (S S )/S . Then S /S = 1+F , thus n n+1 n n n+1 n n − n−1 S n = (1+F ). n S N k=N Y Hence ∞ − (6) F < . n ∞ n=1 X implies liminfS > 0. Observe that by (4) and (5) we get n S S = a α αn−1a2 u2 + αn−1a (λ b ) α (λ b ) u u . n+1− n (cid:18) n+1 n+1− an−1 n(cid:19) n+1 (cid:18)an−1 n − n − n − n+1 (cid:19) n+1 n SPECTRAL PROPERTIES OF JACOBI MATRICES 7 Therefore F = Sn+1−Sn = a α αn−1a2 u2n+1 n Sn "(cid:18) n+1 n+1− an−1 n(cid:19) Sn + αne−1a (λ b ) α (λ b ) unun+1 Sn, n n n n+1 (cid:18)an−1 − − − (cid:19) Sn #Sn e where S = u2 +u2 . By Proposition 4 and u u /S 1, there exists a conestant c > 0 n n n+1 | n n+1| n ≤ such that e e − F− c a α αn−1a2 + αn−1a (λ b ) α (λ b ) . n ≤ anαn (cid:20) n+1 n+1− an−1 n(cid:21) (cid:12)an−1 n − n − n − n+1 (cid:12)! (cid:12) (cid:12) (cid:12) (cid:12) Since (cid:12) (cid:12) − − 1 a α αn−1a2 = an+1αn+1 an αn−1 anαn (cid:20) n+1 n+1− an−1 n(cid:21) (cid:20) an αn − an−1 αn (cid:21) and 1 αn−1 1 αn−1 1 bn+1 bn αn−1 a (λ b ) α (λ b ) = λ + n n n n+1 anαn (cid:12)an−1 − − − (cid:12) (cid:12) (cid:18)an−1 αn − an(cid:19) (cid:18) an − an−1 αn (cid:19)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) λ αn−1 an−1 bn+1 bn αn−(cid:12)(cid:12)1 (cid:12) (cid:12) (cid:12) | | + (cid:12) ≤ an−1 (cid:12) αn − an (cid:12) (cid:12) an − an−1 αn (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) we obtain (6). (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)(cid:3) Remark 1. If we replace the condition (b) by ∞ an+1αn+1 an αn−1 (b’) < , n=0(cid:12) an αn − an−1 αn (cid:12) ∞ X(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) thenlimsupS < andconsequently c /(a α ) S c /(a α )forc > 0,c > 0. Henceby n 1 n n n 2 n n 1 2 ∞ ≤ ≤ usingsubordinationmethodwe can show that thespectrumof thematrix C is purely absolutely continuous (see e.g. [6], [15]). e 4. Special cases In this section we are going to show a few choices of the sequence α from Theorem A. In n { } this way we show flexibility of our approach. The following theorem was proven in [14, Theorem 1.6] and is a generalization of [6, Theo- rem 1.10]. In the proof the authors analyse transfer matrices. Therefore our argument gives an alternative proof. 8 GRZEGORZS´WIDERSKI Theorem 1 (Janas, Moszyn´ski [14]). Assume that (a) lim a = , n→∞ n ∞ ∞ 1 (b) = , a ∞ n n=0 X (c) the sequences an−1 , 1 and bn are of bounded variation, a a a (cid:26) n (cid:27) (cid:26) n(cid:27) (cid:26) n(cid:27) b n (d) lim | | < 2. n→∞ an Then σ(C) = R and the matrix C has purely absolutely continuous spectrum. Proof. Let α 1. By virtue of Remark 1 we need to check the assumptions (b’), (d) and (f) n ≡ of Theorem A. Sincethesequence an−1/an isofboundedvariationitisconvergenttoanumbera. Fromthe { } condition(b)wehavea 1,whereasthecondition(a)givesa 1. Thusthesequence a /a n+1 n ≥ ≤ { } is of bounded variation as well. This proves the conditions (b’) and (f) of Theorem A. The sequence b /a is of bounded variation because bn+1 = bn+1 an+1. The proof is { n+1 n} an an+1 · an complete. (cid:3) The next theorem imposes very simple conditions on Jacobi matrices. In Section 5 we show its applications, furthermore in Section 6 we discuss sharpness of the assumptions. Theorem 2. Assume (a) lim a = , n n→∞ ∞ ∞ 1 (b) = , a2 ∞ n=0 n X ∞ 2 − a n+1 (c) 1 < , a − ∞ n=0"(cid:18) n (cid:19) # X b n (d) limsup| | < 2, n→∞ an ∞ b b n+1 n (e) | − | < . a ∞ n n=0 X Then the Jacobi matrix C is self-adjoint and satisfies σ (C)= and σ(C)= R. p ∅ Proof. Apply Theorem A with α = a . (cid:3) n n Special cases of the following theorem were examined in [19] and [14] using commutator methods. SPECTRAL PROPERTIES OF JACOBI MATRICES 9 Theorem 3. Let log(i) be defined by log(0)(x) = x,log(i+1)(x) = log(log(i)(x)). Let g (n) = j j log(i)(n). Assume that for positive numbers K,N and for a summable nonnegative se- i=1 quence c n Q (a) lim a = , n→∞ n ∞ K a 1 1 n (b) 1 c 1+ + +c for n> N, n n − ≤ an−1 ≤ n ngj(n) j=1 X ∞ b b (c) the sequence b is bounded and | n+1− n| < , n { } a ∞ n n=0 X ∞ 1 (d) < . na ∞ n n=1 X Then σ (C)= and σ(C)= R. p ∅ Proof. We can assume that log(K)(N) > 0. Set 1 for n< N, α = n (ngK(n) otherwise. an To get the conclusion we need to check the assumptions (b), (d) and (c) of Theorem A. To show Theorem A(b) let us observe that the assumption (b) of the present theorem gives 2 K a 2 2 n ′ 1+ + +c (cid:18)an−1(cid:19) ≤ n j=1 ngj(n) n X ′ for a summable sequence c . Therefore n 2 an+1αn+1 an αn−1 n+1gK(n+1) an n 1gK(n 1) = − − an αn − an−1 αn n gK(n) −(cid:18)an−1(cid:19) n gK(n) K n+1g (n+1) n 1 2 2 g (n 1) K ′ K − 1+ + +c − ≥ n g (n) − n  n ng (n) n g (n) K j K j=1 X   K n+1g (n+1) n+1 2 g (n 1) K ′ K + +c − . ≥ n g (n) − n ng (n) n g (n) K j K j=1 X   Since the functions g are increasing, we have j K n 1 g (n+1) g (n 1) g (n 1) 2 K K K ′ (7) − − − − c . ≥ n g (n) − ng (n) g (n 1) − n (cid:18) K (cid:19) K j=1 j − X Next, observe that K (log(j))′(x) ′ g (x) = g (x) . K K log(j)(x) j=1 X 10 GRZEGORZS´WIDERSKI Therefore K 1 ′ g (x) = g (x) . K K xg (x) j j=1 X Hence Taylor’s formula applied to g at the point n 1 gives K − K 2 ′′ (n 1)[g (n+1) g (n 1)] = g (n 1) +2(n 1)g (ξ) − K − K − K − g (n 1) − K j j=1 − X for ξ (n 1,n+1). Direct computation shows g′′ (x) c/x3/2 for x sufficiently large and a ∈ − | K | ≤ constant c> 0. Therefore the right-hand side of (7) is summable. Next, since bn+1 bn αn−1 bn+1 bn bn an−1 αn−1 = − + an − an−1 αn an an−1 (cid:18) an − αn (cid:19) the condition Theorem A(d) reduces to showing Theorem A(c): ∞ 1 an−1 n 1gK(n 1) an (8) − − < . n=0an−1 (cid:12) an − n gK(n) an−1(cid:12) ∞ X (cid:12) (cid:12) ′ (cid:12) (cid:12) For constants K and c> 0 we hav(cid:12)e (cid:12) ′ an−1 n 1gK(n 1) an 1 1 K c ′ − − 1 1+ +c c an − n gK(n) an−1 ≥ 1+ Kn′ +cn − (cid:18) − n(cid:19)(cid:18) n n(cid:19) ≥ −n − n ′ for a summable sequence c . On the other hand n an−1 n 1gK(n 1) an 1 1 gK(n 1) − − 1 − (1 c ) n an − n gK(n) an−1 ≤ 1−cn −(cid:18) − n(cid:19) gK(n) − g (n 1) g (n) g (n 1) K ′ K K ′ = 1 − +c = − − +c − g (n) n g (n) n K K ′ for a summable sequence c . Hence as previously Taylor’s formula applied to g at the point n K n 1 gives − an−1 n 1gK(n 1) an c ′′ − − +c an − n gK(n) an−1 ≤ n n for a constant c> 0 and summable sequence c′′. Finally, condition (d) leads to (8). (cid:3) n Remark 2. When we compare Theorem 2 with Theorem 3, we see that Theorem 3 is interesting only in the case when ∞ 1/a2 < . In this case the condition Theorem 3(d) is satified. n=0 n ∞ The sequence similaPr to α = na−1 was used in the proof of [19, Theorem 4.1] and [14, n n Theorem 2.1]. There was shown that under the stronger assumptions (which in particular imply c 0, b 0 and K = 0) the measure µ is absolutely continuous. Whether σ(C) =R was not n n ≡ ≡ investigated. Example 1. Let K > 0. Fix M such that log(K)(M) > 0. Then for the sequences a = n (n+M)g (n+M) and b 0 the assumptions of Theorem 3 are satisfied. K n ≡