Irina Pchelintseva A FIRST-ORDER SPECTRAL PHASE TRANSITION IN A CLASS OF PERIODICALLY MODULATED HERMITIAN JACOBI MATRICES

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1 Opuscula Mathematica Vol. 8 No. 008 Irina Pchelintseva A FIRST-ORDER SPECTRAL PHASE TRANSITION IN A CLASS OF PERIODICALLY MODULATED HERMITIAN JACOBI MATRICES Abstract. We consider self-adjoint unbounded Jacobi matrices with diagonal q n b nn and off-diagonal entries λ n n, where b n is a -periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum of the operator is either purely absolutely continuous or discrete. We study the situation where the spectral phase transition occurs, namely the case of b b. The main motive of the paper is the investigation of asymptotics of generalized eigenvectors of the Jacobi matrix. The pure point part of the spectrum is analyzed in detail. Keywords: Jacobi matrices, spectral phase transition, absolutely continuous spectrum, pure point spectrum, discrete spectrum, subordinacy theory, asymptotics of generalized eigenvectors. Mathematics Subject Classification: 7A0, 7B36.. INTRODUCTION In the present paper, we study a class of Jacobi matrices with unbounded entries: a periodically modulated linearly growing diagonal and linearly growing weights. First define the operator J on the linear set of finite having a finite number of non-zero elements) vectors l fin N): Ju) n λ n u n + q n u n + λ n u n+, n ) with the initial condition Ju) q u + λ u, where q n b n n, λ n n, and b n is a real -periodic sequence, generated by the parameters b and b. 37

2 38 Irina Pchelintseva Let {e n } n N be the canonical basis in l N). With respect to this basis, the operator J has the following matrix representation: q λ 0 λ q λ J 0 λ q Due to the Carleman condition cf. []) λ n, the operator J is essentially self-adjoint. So we will in what follows assume that J is a closed self-adjoint operator in l N), defined on its natural domain DJ) {u l N) : Ju l N)}. The main tool in our investigation is Gilbert and Pearson s subordinacy theory cf. [5]), which was generalized to the case of Jacobi matrices by Khan and Pearson cf. [9]). Our example is a special case of the situation investigated by S. Naboko and J. Janas in [7] and [8]. In these articles, there was shown that the space of parameters b ; b ) R can be divided into a set of regions of two types. In the regions of the first type, the spectrum of the operator J is purely absolutely continuous and covers the real line R, and in the regions of the second type, the spectrum is discrete. According to the results of [7] and [8], spectral properties of Jacobi matrices of the class considered are determined by the location of the point zero relatively to the absolutely continuous spectrum of some periodic matrix J per, constructed out of the modulation parameters b and b. In our case this leads to: J per b b b b Considering the characteristic polynomial ) )) 0 0 d Jper λ) T r λ b λ b λ b + b )λ + b b, we can find the absolutely continuous spectrum σ ac J per ) of J per from the following condition cf. []): λ σ ac J per ) d Jper λ). ) This leads to the following result cf. [8]) about the spectral structure of the operator J. If djper 0) <, then the spectrum of the operator J is purely absolutely continuous of local multiplicity a.e. on the whole real line. If djper 0) >, then the spectrum of the operator J is discrete..

3 A first-order spectral phase transition So the condition d Jper 0) b b, equivalent to {b b 0 or b b }, determines the location of the regions on the plane b ; b ) where the spectrum of the operator J is either purely absolutely continuous or discrete. Namely: If 0 < b b < then the spectrum of the operator J is purely absolutely continuous of local multiplicity a.e. on the whole real line. If b b < 0 or b b > then the spectrum of the operator J is discrete. b b b b Fig. The boundary situations, i.e., b b 0 or b b Fig. ), are of most interest to us, because they show the spectral behavior of the operator J when the parameters cross the line where the spectral phase transition occurs. The first case b 0) was studied in [3] and the following results were obtained: The interval ; 0] is covered by the purely absolutely continuous spectrum of local multiplicity a.e.) of the operator J. In the interval 0; + ), the spectrum is discrete and there is no accumulation of eigenvalues at the point zero. The n-th eigenvalue grows linearly with n: λ n n, which means that the operator J is not semi-bounded. In the present paper, we study the spectral properties of the operator J in the second boundary case, i.e., b b, where the corresponding Jacobi matrix is semi-bounded, as will be shown below. A similar example of the Jacobi matrix with modulated weights was considered in paper [0]. Remark. We note that a Jacobi matrix J b,b with modulation parameters equal to b and b, is unitarily equivalent to the matrix J b, b. Therefore, a study of the situation considered can be reduced to the case of b, b > 0. Now let us formulate the main result of this paper.

4 0 Irina Pchelintseva Theorem.. Let b b, b, b > 0, b b. Then: The interval b +b ; + ) is covered by the purely absolutely continuous spectrum of local multiplicity a.e.) of the operator J. The operator is semi-bounded from below by the constant min{ b, b }. On the interval [min{ b, b }; b +b ), the spectrum is discrete and the following estimate on the number of eigenvalues in the interval [min{ b, b }; b +b ε) for positive ε holds: { } # λ n : min{ b, b } < λ n < ε < b b ) b + b ) b + b 6εb + b ) +.. Let b b. Then the spectrum of the operator J is σj) [; + ) and in the interval ; + ), the spectrum is purely absolutely continuous. The paper is organized as follows. In Section, we establish the asymptotics of the generalized eigenvectors and determine the spectral structure of the operator J via the Gilbert-Khan-Pearson subordinacy theory, dividing the real axis into two intervals with pure point and purely absolutely continuous spectrum. In Section 3, we study the pure point part of the spectrum, proving its non-emptiness and then discreteness and obtaining the estimate for the accumulation of eigenvalues.. GENERALIZED EIGENVECTORS AND THE SPECTRUM IN THE BOUNDARY CASE b b In this section, we find the asymptotics of the generalized eigenvectors using the Birkhoff-Adams Theorem and draw first conclusions about the spectrum. Consider the spectral recurrence relation n )u n + b n nu n + nu n+ λu n, n. 3) Since the coefficients of this recurrence relation are linear with resect to n, we attempt to use the Birkhoff-Adams Theorem. It deals with the recurrence relation of the form x n+ + F n)x n+ + F n)x n 0, n, ) where the coefficients F n) and F n) have asymptotical expansions as n of the form: a k F n) n k, F b k n) n k k0 with b 0 0. Consider the characteristic equation α + a 0 α + b 0 0 and denote its roots α and α. The following result takes place cf. []): k0

5 A first-order spectral phase transition... Theorem Birkhoff-Adams). If the roots α and α coincide, α : α α, and an additional condition a α + b 0 holds, where β + b a b 0, δ 0a b b 0 δ, then there exist two linearly independent solutions x ) n and x ) n relation ) with the following asymptotics, as n : x i) n α n e δi n n β + O n )), i,. of recurrence Obviously, we cannot apply this Theorem directly, because we do not have the required asymptotic expansions. In order to negotiate this problem, we derive equations for even and odd components of the sequence {u n }. Taking recurrence relations 5) for three consecutive values of n for n, n + and n + ), λ n u n + q n u n + λ n u n+ λu n, λ n u n + q n+ u n+ + λ n+ u n+ λu n+, λ n+ u n+ + q n+ u n+ + λ n+ u n+3 λu n+, and having eliminated the values u n and u n+ from them, we obtain one equation containing the values u n, u n+ and u n+3 only: u n+3 + p n)u n+ + p n)u n 0, with the following coefficients p n) and p n): p n) λ nq n+ λ) + λ n+q n λ) q n λ)q n+ λ)q n+ λ), λ n+ λ n+ q n λ) Denote: p n) λ nλ n q n+ λ) λ n+ λ n+ q n λ). v n : u n, w n : u n, F i n) : p i n), i,, G i n) : p i n + ), i,. We get recurrence relations for odd and even components of the sequence {u n }: { vn+ + F n)v n+ + F n)v n 0, w n+ + G n)w n+ + G n)w n 0, 5) and the coefficients F n), F n) and G n), G n) having the asymptotical expansions by the powers of n with the following first terms F n), G n) b b ) + λb ) + b ) ) + O n n, F n), G n) ) n + O n.

6 Irina Pchelintseva This makes the Birkhoff-Adams Theorem applicable. Since a 0 b b, a + λb+b), b 0 and b, we get the equation α α + 0, its root α and values β 3, δ λb + b ) δ. The direct application of the Birkhoff-Adams Theorem yields the following result. Lemma. Let b b, b, b > 0 and b b. Equations 5) have bases of solutions v n +, vn and w n +, wn with the following asymptotics as n : v n ±, w n ± n 3 e ± λb +b ) n ) ) + O n, i,. In order to return from odd and even components of the sequence {u n } back to the sequence itself, we should combine the obtained asymptotics properly. The following Lemma gives the answer. Lemma. Let b b, b, b > 0 and b b. Equation 3) has a basis of solutions u + n, u n with the following asymptotics as n : u ± n b n 3 e ± λb +b ) n ) ) + O n, u ± n b n 3 e ± λb +b ) n ) ) + O n. Proof. From Lemma, we know that any solution of recurrence relation 3) can be written in the following form: u k c v k + c +v + k, u k d w k + d +w + k with some constants c, c +, d and d +. Let us substitute the solution u of this form into the recurrence relation written for the number k in order to obtain relations between these constants. k )u k + b k) λ)u k + k)u k+ k ) [e λb k +b ) c + + b d + + c + + o))+ + e λb k ] +b ) c + b d + c + o)) 0. From the condition of cancellation in the main order of this equation, we can conclude that b d ± c ± c ±, b b which means that any solution of the recurrence relation is a linear combination of the following two solutions u + n and u n : { u ± k : b v ± k, u ± k : b w ± k, which have asymptotics of the form stated.

7 A first-order spectral phase transition... 3 Now that we know the asymptotics of generalized eigenvectors, we apply the subordinacy theory and arrive at the result regarding the spectrum of the operator J. Theorem. Let b b, b, b > 0. The spectrum of the operator J is purely absolutely continuous of local multiplicity a.e.) on the interval b +b ; + ) and pure point on the interval ; b +b ). Remark. We can use the same arguments to derive asymptotics of generalized eigenvectors in the other situations, i.e. when b b, and obtain the answers concerning the spectral structure of the operator J. The answers will be the same as in the papers mentioned above, which is self-understood. 3. DISCRETENESS OF THE PART OF THE SPECTRUM IN THE BOUNDARY CASE b b In this section, we focus on the pure point part of the spectrum in the case considered. We investigate into this part of spectrum showing its discreteness and and analyzing the accumulation of eigenvalues to the edge of the essential spectrum σ ess J) [ b +b ; + ). Towards the end of the Section, we consider the particular case of b b. This case should be developed separately and we will acquire a simple result. 3.. SEMI-BOUNDEDNESS OF THE OPERATOR First we prove the semi-boundedness of the operator, which is not an idle question for us. The fact of semi-boundeness makes the investigation into the discrete spectrum much easier. We want to stress this property of the operator in our case and, although we could derive this fact from the estimates for the eigenvalues that we obtain later, it follows immediately from the simple estimate of the quadratic form, which we show below. Theorem. If b b and b, b > 0, then the operator J is semi-bounded from below by the constant min{b, b }. Proof. Write out the quadratic form of the operator, Ju, u) nb n u n + n u n u n+ + u n+ u n ) and substitute the values of the sequence {b n } into it, + [ k )b u k + kb u k ] + k [ k )uk u k + u k u k ) + k)u k u k+ + u k+ u k ) ] k

8 Irina Pchelintseva then group the items of the same type and take out the linear terms, [k )b u k + b u k + k + u k u k + u k u k ))] + b u k )+ k + b u + [k)b u k+ + b u k + k + u k u k+ + u k+ u k ))] + ) b u k+ k and finally use the fact that b b, b n u n + n b n u n + b n+ u n+. We arrive at the equality that is useful in itself, [ Ju, u) n b n u n + b n+ u n+ + bn u n ]. 6) A very rough estimate yields Ju, u) b n u n min{b, b } u, which completes the proof. 3.. NON-EMPTINESS OF THE PURE POINT PART OF THE SPECTRUM Now we show the non-emptiness of the part of the spectrum lying below the point b +b. To this end, we find an appropriate vector of norm with the real components from l fin ), which makes the quadratic form of the operator less than b +b. We need the following technical lemma first. For a finite vector {v n }, denote X {v n }) : nv n v n+ ) vn ) n this expression can take the value, when the denominator turns to zero).

9 A first-order spectral phase transition... 5 Lemma 3. inf X {v n }) P vn )n <0 inf X {v n }) 0. P vn )n >0 We will prove this lemma a little later and now we just use its result to prove the following theorem concerning the operator J. Theorem 3. Let b b, b), b > 0. If b b, then the spectrum of the operator J in the interval ; b +b is non-empty. Proof. We need to construct a vector {u n } that makes the following expression negative: ) Ju, u) u b + b [ n b n u n + b n+ u n+ + bn u n If we take u n of the form u n )n v n bn, we get ) Ju, u) u b + b using the fact that b n b n+, [ nv n v n+ ) + b ] n b n+ vn b + b taking into account that b n b n+ ) n b b ), ] 8 u n. b + b [ nv n v n+ ) + vn 8 v ] n b + b b n [ nv n v n+ ) + b ] b ) n vn b + b [ ) n vn] b b + b + b nv n v n+ ). ) n vn Lemma 3 guarantees that we can choose such finite vector {v n } that will make the latter expression negative. The choice depends on the sign and the value of the constant b b b +b. Now, when it is clear how to apply Lemma 3, we can prove it. Proof of Lemma 3. Let us construct the sequences {v n } in both cases.

10 6 Irina Pchelintseva ) In the case of vn ) n+ > 0, there is X {v n }) nv n v n+ ) vn ) n+ choose v k v k+ : w k+ for k and w : v with some sequence {w n }) n )w n+ w n ) w denote s n : w n w n+ for n ) n )s n ). s n We set s N) n { n ln n, n N, 0, n > N, w N) N+ 0 and reconstruct sequences {wn) n n ln n, n ln n <, } and {v n N) }. Due to the fact that we can claim that choosing the number N large enough we can make the value of X at the sequence {v n N) } very close to zero: ) On the other hand, in the case of X {ṽ n }) X{v N) n }) 0 as N. vn ) n > 0, there is nṽ n ṽ n+ ) ṽn ) n

11 A first-order spectral phase transition... 7 and taking now ṽ k+ : w k+, ṽ k : w k+ for k, but w : ṽ, we get w w ) + n w n+ w n+ ) w w denote s n : w n + w n+ for n ) s + n) s n+ ). s s + s n Setting again s N) n { n ln n, n N, 0, n > N, and w N) fact N+ 0, reconstructing sequences { wn) n we find that for sequences {ṽ N) n }, which completes the proof. n ln n, n ln n <, X{ṽ N) n }) 0 as N, } and {ṽ n N) } and using the same 3.3. DISCRETENESS OF THE PURE POINT PART OF THE SPECTRUM AND THE ESTIMATE FOR THE EIGENVALUES Now we prove the discreteness of the pure point part of the spectrum, which is non-empty as we have shown above. Using the Glazman Lemma we obtain an estimate characterizing the speed of possible accumulation of eigenvalues to the point b +b. Theorem. Let b b and b, b > 0. If b b, then the spectrum of the operator J in the interval ; b +b ) is discrete and for any ε > 0 the number of eigenvalues in the interval ; b +b ε) obeys the estimate { } # λ n : λ n < ε < b b ) b + b ) b + b 6εb + b ) +.

12 8 Irina Pchelintseva Proof. We use the following simple consequence of the Glazman Lemma cf. []). If for every finite vector with the first N components being zeros the estimate Ju, u) > C u holds, then the spectrum of the operator J on the interval ; C) is discrete and the operator has at most N eigenvalues on this interval. Hence in order to prove the assertion of the Theorem we need to show that the quadratic form ) Ju, u) ε u b + b is positive on every finite vector with a number of initial components being zeros. First estimate the quadratic form of the operator using formula 6) Ju, u) n b n u n + b n+ u n+ + b n u n ), the Cauchy inequality and introducing notation v n : b n u n : Ju, u) nv n v n+ ) + vn). Rewrite also the norm of the vector {u n } u v n b n vnb n+, which can be further rewritten in more symmetric fashion: u 8 v b + v 8 nb n+ + vn+b n ). Substitute this into the expression that we desire to estimate: ) Ju, u) ε u nv n v n+ ) b n + b n+ )v b + b b + b ) n b n+ v b + b ) n+ + ε 8 v b + ε v 8 nb n+ + vn+b n ). Taking into account that b n b n+ ) n b b ) and making another kind of symmetrization ) n vn v + ) n vn vn+),

13 A first-order spectral phase transition... 9 we obtain ) Ju, u) ε u nv n v n+ ) + b + b + b b ) n vn v b + b ) n+)+ + ε v 8 nb n+ + vn+b n ) + ε 8 v b b b b + b ) v. We need to show that for large n the following expression nv n v n+ ) + ε v nb n+ + v n+b n ) > b b b + b ) v n v n+ is positive then we will be able to take zeros for initial components and obtain an estimate for the quadratic form). To this end, denote y n : v n v n+, x n : B : b b b + b ) v nb n+ + v n+ b n, b +b ). Noticing that we get v n + v n+ < b + b b + b x n x n, b b nv n v n+ ) + ε v nb n+ + v n+b n ) b b b + b ) v n v n+ > ε x n B x ny n + ny n. And the latter expression is positive for any n > Nε) : B 6ε, because xn y n ) B ε xn y n ) + n ε xn B ) B y n ε ε + n ε > 0 for such n. Now it is clear that if we assume the first Nε) + components of the vector {u n } to equal zero, we get the desired estimate of the quadratic form. That completes the proof. Acknowledgements Author expresses his deep gratitude to Prof. S.N. Naboko for his constant attention to this work and to S. Simonov for his useful remarks and advices.

14 50 Irina Pchelintseva REFERENCES [] N.I. Akhiezer, I.M. Glazman, Theory of linear operators in Hilbert space, nd ed., Dover, New York, 993. [] Yu.M. Berezanskii, Expansions in eigenfunctions of selfadjoint operators, Naukova Dumka, Kiev, 965 [in Russian]. [3] D. Damanik, S.N. Naboko, A first-order phase transition in a class of unbounded Jacobi matrices: critical coupling [to appear in J. Appr. Th.]. [] S.N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York 999. [5] D. Gilbert, D. Pearson, On subordinacy and analysis of the spectrum of one dimensional Schrodinger operators, J. Math. Anal. Appl., 8 987), [6] J. Janas, S.N. Naboko, Criteria for semiboundedness in a class of unbounded Jacobi operators, translation in St. Petersburg Math. J., 003) 3, [7] J. Janas, S.N. Naboko, Multithreshold spectral phase transition examples for a class of unbounded Jacobi matrices, Oper. Theory Adv. Appl., 00), [8] J. Janas, S.N. Naboko, Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries, J. Funct. Anal., 9 00), [9] S. Khan, D. Pearson, Subordinacy and spectral theory for infinite matrices. Helv. Phys. Acta, 65 99), [0] S. Simonov, An example of spectral phase transition phenomenon in a class of Jacobi matrices with periodically modulated weights, Oper. Theory Adv. Appl., Operator Theory, Analysis and Mathematical Physics, Birkhäuser, Basel, 7 007), Irina Pchelintseva pchelintseva@yandex.ru St. Petersburg University Institute of Physics Department of Mathematical Physics Ulianovskaia, 9890, St. Petergoff, St. Petersburg, Russia Received: July 0, 007. Revised: December 6, 007. Accepted: December 6, 007.