AN EXAMPLE OF SPECTRAL PHASE TRANSITION PHENOMENON IN A CLASS OF JACOBI MATRICES WITH PERIODICALLY MODULATED WEIGHTS

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1 AN EXAMPLE OF SPECTRAL PHASE TRANSITION PHENOMENON IN A CLASS OF JACOBI MATRICES WITH PERIODICALLY MODULATED WEIGHTS SERGEY SIMONOV Abstract We consider self-adjoint unbounded Jacobi matrices with diagonal q n = n and weights λ n = c n n where c n is a -periodical sequence of real numbers The parameter space is decomposed into several separate regions where the spectrum is either purely absolutely continuous or discrete This constitutes an example of the spectral phase transition of the first order We study the lines where the spectral phase transition occurs obtaining the following main result: either the interval ; ) or the interval ; + ) is covered by the absolutely continuous spectrum the remainder of the spectrum being pure point The proof is based on finding asymptotics of generalized eigenvectors via the Birhoff-Adams Theorem We also consider the degenerate case which constitutes yet another example of the spectral phase transition Introduction In the present paper we study a class of Jacobi matrices with unbounded entries: a linearly growing diagonal and periodically modulated linearly growing weights We first define the operator J on the linear set of vectors l fin N) having finite number of non-zero elements: ) 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 = n λ n = c n n and c n is a real -periodic sequence generated by the parameters c and c Let {e n } n N be the canonical basis in l N) With respect to this basis the operator J admits the following matrix representation: q λ 0 λ q λ J = 0 λ q 3 Due to the Carleman condition [] n= λ n = the operator J is essentially self-adjoint We will therefore assume throughout the paper that J is a closed self-adjoint operator in l N) defined on its natural domain DJ) = {u l N) : Ju l N)} We base our spectrum investigation on the subordinacy theory due to Gilbert and Pearson [6] generalized to the case of Jacobi matrices by Khan and Pearson 99 Mathematics Subject Classification 47A0 47B36 Key words and phrases Jacobi matrices Spectral phase transition Absolutely continuous spectrum Pure point spectrum Discrete spectrum Subordinacy theory Asymptotics of generalized eigenvectors

2 SERGEY SIMONOV [] Using this theory we study an example of spectral phase transition of the first order This example was first obtained by Naboo and Janas in [9] and [0] In cited articles the authors managed to demonstrate that the space of parameters c ; c ) R can be naturally decomposed 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 whereas in the regions of the second type the spectrum is discrete Due to [9] and [0] spectral properties of Jacobi matrices of our class are determined by the location of the point zero relative to the absolutely continuous spectrum of a certain periodic matrix J per constructed based on the modulation parameters c and c In our case this leads to: c 0 0 c c 0 J per = 0 c c 0 0 c Considering the characteristic polynomial ) )) 0 0 d Jper λ) = T r c c λ c c c λ c = λ ) c c c c the location of the absolutely continuous spectrum σ ac J per ) of J per can then be determined from the following condition []: ) λ σ ac J per ) d Jper λ) This leads to the following result [0] concerning the spectral structure of the operator J If d Jper 0) < then the spectrum of the operator J is purely absolutely continuous covering the whole real line If on the other hand d Jper 0) > then the spectrum of the operator J is discrete Thus the condition c c c c = equivalent to { c + c = or c c = } determines the boundaries of the above mentioned regions on the plane c ; c ) where one of the cases holds and the spectrum of the operator J is either purely absolutely continuous or discrete see fig on page 8) Note also that Jacobi matrices with modulation parameters equal to ±c and ±c are unitarily equivalent Thus the situation can be reduced to studying the case c c > 0 In the present paper we attempt to study the spectral structure on the lines where the spectral phase transition occurs ie on the lines separating the aforementioned regions The paper is organized as follows Section deals with the calculation of the asymptotics of generalized eigenvectors of the operator J This calculation is mainly based on the Birhoff-Adams Theorem [4] The asymptotics are then used to characterize the spectral structure of the operator via the Khan-Pearson Theorem [] It turns out that on the lines

3 AN EXAMPLE OF SPECTRAL PHASE TRANSITION PHENOMENON 3 where the spectral phase transition occurs the spectrum is neither purely absolutely continuous nor pure point but a combination of both In Section 3 we attempt to ascertain whether the pure point part of the spectrum is actually discrete In doing so we establish a criterion that guarantees that the operator J is semibounded from below for all c ; c ) R This semiboundedness is then used in conjunction with classical methods of operator theory to prove that in at least one situation the discreteness of the pure point spectrum is guaranteed Section 4 is dedicated to the study of the degenerate case ie the case when one of the modulation parameters turns to zero In this situation one can explicitly calculate all eigenvalues of the operator On this route we obtain yet another hidden example of the spectral phase transition of the first order as the point c ; c ) moves along one of the critical lines in the space of parameters Generalized eigenvectors and the spectrum of the operator J In this Section we calculate asymptotics of generalized eigenvectors of the operator J Consider the recurrence relation [] ) λ n u n + q n λ)u n + λ n u n+ = 0 n We reduce it to a form such that the Birhoff-Adams Theorem is applicable To this end we need to have a recurrence relation of the form: ) x n+ + F n)x n+ + F n)x n = 0 n where F n) and F n) admit the following asymptotical expansions as n : a 3) F n) n F b n) n =0 with b 0 0 Consider the characteristic equation α + a 0 α + b 0 = 0 and denote its roots α and α Then [4]: Theorem Birhoff-Adams) There exist two linearly independent solutions x ) n and x ) n of the recurrence relation ) with the following asymptotics as n : x i) n = α n i n βi + O n =0 )) i = if the roots α and α are different where β i = a α i +b a 0 α i +b 0 i= )) x i) n = α n e δi n n β + O n i = if the roots α and α coincide α := α = α and an additional condition a α + b 0 holds where β = 4 + b a b 0 δ = 0a b b 0 = δ This Theorem is obviously not directly applicable in our case due to wrong asymptotics of coefficients at infinity In order to deal with this problem we study a pair of recurrence relations equivalent to ) separating odd and even components of a vector u This allows us to apply the Birhoff-Adams Theorem to each of the recurrence relations of the pair which yields the corresponding asymptotics Combining the two asymptotics together we then obtain the desired result for the solution of )

4 4 SERGEY SIMONOV Denoting v := u and w := u we rewrite the recurrence relation ) for the consecutive values of n: n = and n = + λ v + q λ)w + λ v + = 0 λ w + q + λ)v + + λ + w + = 0 Then we exclude w in order to obtain the recurrence relation for v: w = λ v + λ v + q λ 4) v + + P )v + + P )v = 0 where P ) = q + λ q λ λ q + λ)q + λ) λ + λ + + λ + λ + λ + λ + P ) = q + λ q λ λ λ λ + λ + In our case λ n = c n n and q n = n) this yields the following asymptotic expansions cf 3)) for P ) and P ) as tends to infinity: with P ) = j=0 a j j P ) = 5) a 0 = c + c a = c + c λ = a 0 c c c c + λ c c b 0 = b = The remaining coefficients {a j } + j= {b j} + j= can also be calculated explicitly On the same route one can obtain the recurrence relation for even components w of the vector u: j=0 6) w + + R )w + + R )w = 0 Note that if is substituted in 4) by + and v by w the equation 4) turns into 6) Therefore R ) = P + ) R ) = P + ) and thus as R ) = a 0 + a + O R ) = b 0 + b + O with a 0 a b 0 b defined by 5) Applying now the Birhoff-Adams Theorem we find the asymptotics of solutions of recurrence relations 4) and 6) This leads to the following result b j j ) )

5 AN EXAMPLE OF SPECTRAL PHASE TRANSITION PHENOMENON 5 Lemma Recurrence relations 4) and 6) have solutions v n + vn and w n + wn respectively with the following asymptotics as : 7) v ± w± = α ± β± + O )) if c +c c c where α+ and α are the roots of the equation α + a 0 α + b 0 = 0 and β ± = a α ± +b a 0 α ± +b 0 with a 0 a b 0 b defined by 5) Moreover if c +c c c > then α± are real and α < < α + whereas if < then α+ = α β + = β and the vectors v + v w + w c +c c c are not in l N) 8) v ± w± = α 4 e δ ± + O c +c c c = and λ where α = α + = α if Moreover if c +c c c = then δ + = whereas if c +c c c = then δ + = λ c c = δ λ c c = δ )) Proof Consider recurrence relation 4) and let the constants a 0 a b 0 b be defined by 5) Consider the characteristic equation α + a 0 α + b 0 = 0 It has different roots α < α + when the discriminant D differs from zero: D = ) c +c c c 4 0 Note that α+ α = Consider the case D < 0 A direct application of the Birhoff-Adams Theorem yields: v ± = α ± β± + O )) where β ± = aα±+b a 0 α±+b 0 Then α + = α α + = α = and β + = β Note also that v ± are not in l : Re β + = Re β = + λ ) Re = c c a 0 + α In the case D > 0 α + and α are real and α < < α + hence v lies in l Ultimately in the case D = 0 the roots of the characteristic equation coincide and are equal to α = a0 with α = and the additional condition a 0a b is equivalent to a 0 + a 0λ ) λ c c The Birhoff-Adams Theorem yields: )) v ± = α β e δ ± + O where β = 4 δ a + = 0 λ ) c c = δ If the value δ + is pure imaginary then clearly the vectors v ± do not belong to l N)

6 6 SERGEY SIMONOV In order to prove the assertion of the Lemma in relation to w ± note that in our calculations we use only the first two orders of the asymptotical expansions for P ) and P ) These coincide with the ones for R ) and R ) Thus the solutions of recurrence relations 4) and 6) coincide in their main orders which completes the proof Now we are able to solve the recurrence relation ) combining the solutions of recurrence relations 4) and 6) Lemma Recurrence relation ) has two linearly independent solutions u + n and u n with the following asymptotics as : { u ± = α ± β ± + O )) u ± = c + α ± c )α± β ± + O )) if where the values α+ α β + β are taen from the statement c +c c c of Lemma + O )) u ± = α e δ ± 4 )) u ± = c + αc )α e δ ± 4 + O if c +c c c = and λ where the values α δ + δ are taen from the statement of Lemma Proof It is clear that any solution of recurrence relation ) u gives two vectors v and w constructed of its odd and even components which solve recurrence relations 4) and 6) respectively Consequently any solution of the recurrence relation ) belongs to the linear space with the basis {V + V W + W } where V ± = v± V ± = 0 and W ± = 0 W ± = w± This 4-dimensional linear space contains -dimensional subspace of solutions of recurrence relation ) In order to obtain a solution u of ) one has to obtain two conditions on the coefficients a + a b + b such that u = a + V + + a V + b + W + + b W 9) u = a + v + + a v u = b + w + + b w Using Lemma we substitute the asymptotics of this u into ) where n is taen equal to λ u + q λ)u + λ u + = 0 As in Lemma we have two distinct cases Consider the case Then c [ c +c c c ) ] α+ a + β+ β ) + a α + c [ a + α+ α [ ) ] α+ + b + β+ β ) + b + α ) α + β + β ) + a α ]) + o)) = 0 as

7 AN EXAMPLE OF SPECTRAL PHASE TRANSITION PHENOMENON 7 Therefore for any number greater than some big enough positive K one has: ) α+ [c a + + b + + c a + α + ] β+ β ) + [c a + b + c a α ] = 0 α hence b ± = c + α ± c )a ± Thus 9) admits the following form: u = a + v + + a v u = c + α + c )a + w + c + α c )a w It is clear now that the vectors u + and u defined as follows: u + = v+ u+ = c + α + c )w + u = v u = c + α c )w are two linearly independent solutions of the recurrence relation ) The second case here c +c c c = can be treated in an absolutely analogous fashion Due to Gilbert-Pearson-Khan subordinacy theory [6] [] we are now ready to prove our main result concerning the spectral structure of the operator J Theorem 3 Depending on the modulation parameters c and c there are four distinct cases describing the spectral structure of the operator J: a) If c +c c c < the spectrum is purely absolutely continuous with local multiplicity one almost everywhere on R b) If c c = and c c 0 the spectrum is purely absolutely continuous with local multiplicity one almost everywhere on ; ) and pure point on ; + ) c) If c + c = and c c 0 the spectrum is purely absolutely continuous with local multiplicity one almost everywhere on ; + ) and pure point on ; ) d) If > the spectrum is pure point c +c c c The four cases described above are illustrated by fig Proof Without loss of generality assume that c c > 0 Changing the sign of c or c leads to an unitarily equivalent operator Consider subordinacy properties of generalized eigenvectors [] > we have α < < α + By Lemma u is a subordinate solution and lies in l N) Thus every real λ can either be an eigenvalue or belong to the resolvent set of the operator J If c +c c c < we have Re α + = Re α Re β + = Re β u + n u n as n and there is no subordinate solution for all real λ The spectrum of J in this situation is purely absolutely continuous If c +c c c If c +c c c = which is equivalent to c c = then either λ > or λ < If λ > then α = δ + = δ > 0 and u is subordinate and lies in l N) hence λ can either be an eigenvalue or belong to the resolvent set If λ < then α = both δ + and δ are pure imaginary u + n u n as n no subordinate solution exists and ultimately λ belongs to purely absolutely continuous spectrum If c +c c c = which is equivalent to c + c = the subcases λ > and λ < change places which completes the proof

8 8 SERGEY SIMONOV c b d b a a b c c b d 0 d c b c d c b a a b d b These results elaborate the domain structure described in Section : we have obtained the information on the spectral structure of the operator J when the modulation parameters are on the boundaries of regions 3 Criterion of semiboundedness and discreteness of the spectrum We start with the following Theorem which constitutes a criterion of semiboundedness of the operator J Theorem 3 Let c c 0 If c + c > then the operator J is not semibounded If c + c then the operator J is semibounded from below Proof Due to Theorem 3 there are four distinct cases of the spectral structure of the operator J depending on the values of parameters c and c see fig ) The case a) ie c +c c c < is trivial since σac J) = R We are going to prove the assertion in the case d) ie > using c +c c c the result of Janas and Naboo [8] According to them semiboundedness of the operator J depends on the location of the point zero relative to the spectrum of the periodic operator J per [8] see also Section ) It is easy to see that the absolutely continuous spectrum of the operator J per in our case consists of two intervals σ ac J per ) = [λ + ; λ ] [λ + ; λ ++ ]

9 AN EXAMPLE OF SPECTRAL PHASE TRANSITION PHENOMENON 9 where λ ±+ = ± c + c ) λ ± = ± c c and λ + < λ < < λ + < λ ++ As it was established in [8] if the point zero lies in the gap between the intervals of the absolutely continuous spectrum of the operator J per then the operator J is not semibounded If on the other hand the point zero lies to the left of the spectrum of the operator J per then the operator J is semibounded from below A direct application of this result completes the proof in the case d) We now pass over to the cases b) and c) ie c +c c c = c c 0 This situation is considerably more complicated since the point zero lies right on the edge of the absolutely continuous spectrum of the operator J per We consider the cases b) and c) separately b): We have to prove that the operator J is not semibounded By Theorem 3 σ ac J) = ; ] thus the operator J is not semibounded from below Now consider the quadratic form of the operator taen on the canonical basis element e n We have Je n e n ) = q n + n thus the operator J is not semibounded c): We will show that the operator J is semibounded from below To this end we estimate its quadratic form: for any u DJ) DJ) being the domain of the operator J) one has Ju u) = n u n + c )u u + n= = + u u ) + c )u u + + u u + ) Using the Cauchy inequality [] and taing into account that c + c = we ultimately arrive at the estimate 3) Ju u) n u n c ) u + c ) u ) n= = = c ) u + c ) u + ) = = = which completes the proof c u + c u ) min{ c c } u > 0 = The remainder of the present Section is devoted to the proof of discreteness of the operator s pure point spectrum in the case c) of Theorem 3 ie when c + c = c c 0 By Theorem 3 in this situation the absolutely continuous spectrum covers the interval [ ; + ) and the remaining part of the spectrum if it is present is of pure point type The estimate 3) obtained in the proof of the previous Theorem

10 0 SERGEY SIMONOV implies that there is no spectrum in the interval ; min{ c c }) We will prove that nonetheless if c = c the pure point spectral component of the operator J is non-empty It is clear that if c = c = the spectrum of the operator J in the interval ; ) is empty and the spectrum in the interval ; + ) is purely absolutely continuous This situation together with its generalization towards Jacobi matrices with zero row sums was considered by Dombrowsi and Pedersen in [3] and absolute continuity of the spectrum was established Theorem 3 In the case c) ie when c + c = c c 0 under an additional assumption c = c the spectrum of the operator J in the interval ; ) is non-empty Proof Without loss of generality assume that 0 < c c < Changing the sign of c c or both leads to an unitarily equivalent operator Consider the quadratic form of the operator J I for u DJ) J ) ) I u u = [q n u n + λ n u n u n+ + u n u n+ ) ] u n = = n= n= [n u n + c n n u n+ + u n u n+ u n ) u n ] Shifting the index n by in the term c n n u n+ and then using the -periodicity of the sequence {c n } we have J ) ) I u u = = [c n n u n+ + u n ] + [n u n c n n u n c n+ n ) u n ] u n = n= n= ) = [c n n u n+ + u n cn+ c n ] + u n = n= n= = [c ) u + u + c ) u + u + ] = ) c c [ u u ] We need to find a vector u DJ) which maes this expression negative The following Lemma gives a positive answer to this problem via an explicit construction and thus completes the proof Lemma 33 For 0 < c c < c + c = there exists a vector u l fin N) such that 3) [c ) u + u + c ) u + u + ] < = = ) c c < [ u u ] =

11 AN EXAMPLE OF SPECTRAL PHASE TRANSITION PHENOMENON Proof We consider the cases c > c and c < c separately Below we will see that the latter can be reduced to the former c c > 0 In this case we will choose a vector v from l fin N) with nonnegative components such that if the vector u is defined by u = v u = v + the condition 3) holds true In terms of such v the named condition admits the following form: ) 33) c [ )v v + ) c c ] < v = c c < 0 In this case we will choose a vector w l fin N) with nonnegative components and the value t such that if the vector u is defined by u = w u = w u = tw the condition 3) holds true In terms of w and t condition 3) admits the form 34) c [)w w + ) ] < t + c t + 4c ) w = Tae t such that the expression in the bracets on the right-hand side of the latter inequality is positive This choice is possible if we tae tae the maximum of the parabola yt) = t + c t + 4c located at the point t 0 = c Then the inequality 34) admits the form 35) c [)w w + ) ] < c c ) = We will now explicitly construct a vector v N) l fin N) such that it satisfies both 33) and 35) for sufficiently large numbers of N Consider the sequence v n N) = N =n for n N and put vn) n = 0 for n > N It is clear that as N + N ) v N) ) = ln N) and [ )v N) = v N) + ) ] = [)v N) = which completes the proof of Lemma 33 w v N) + ) ] ln N = o v N) ) ) We are now able to prove the discreteness of the pure point spectral component of the operator J in the case c) of Theorem 3 which is non-empty due to Theorem 3 Theorem 34 In the case c) ie when c + c = and c c 0 under an additional assumption c = c the spectrum of the operator J in the interval ; min{ c c }) is empty the spectrum in the interval [min{ c c }; ) is discrete and the following estimate holds for the number of eigenvalues λ n in the interval ; ε) ε > 0: #{λ n : λ n < ε} ε

12 SERGEY SIMONOV Proof According to the Glazman Lemma [] dimension of the spectral subspace corresponding to the interval ; ε) is less or equal to the co-dimension of any subspace H ε l N) such that ) 36) Ju u) ε u for any u DJ) H ε Consider subspaces ln of vectors with zero first N components ie ln := {u l N) : u = u = = u N = 0} For any ε 0 < ε < we will find a number Nε) such that for any vector from H ε = lnε) the inequality 36) is satisfied We consider ε such that 0 < ε < only since the spectrum is empty in the interval ; 0] see the estimate 3)) The co-dimension of the subspace lnε) is Nε) so this value estimates from above the number of eigenvalues in the interval ; ε) Consider the quadratic form of the operator J for u DJ): Ju u) = n[ u n + c n Re u n u n+ )] n= [ n u n c n I n u n + )] u n+ = I n= n ] = u c I ) + n u n [ n c n I n c n I n n n= where we have used the Cauchy inequality [] Re u n u n+ ) I n u n + I n u n+ with the sequence I n > 0 n N which we will fix as I n := φ n in order that the expression 37) c n I n c n I n ) n taes its simplest form This choice cancels out the first order with respect to n The value of φ in the interval 0 < φ < will be fixed later on We have: 38) I n = + φ n + φ n where φ n = O n ) n Moreover as can be easily seen φ n = φφ + ) nn φ) After substituting the value of φ n into 38) and then into 37) we obtain: 39) c n I n c n ) = n n φ c n + φ) c n ) + θ n I n with θ n := c n φ ) ) n φ n n = O ) n as n Choose φ in order to mae the right hand side of expression 39) symmetric with respect to the modulation parameters c and c : φ = Then c n I n c n I n n ) = n + θ n

13 AN EXAMPLE OF SPECTRAL PHASE TRANSITION PHENOMENON 3 Consequently Ju u) u c ) + n= ) + θ nn u n Since nθ n 0 as n we can choose Nε) such that for any n > Nε) the condition nθ n > ε holds Thus condition 36) will be satisfied for all vectors from DJ) lnε) since their first components are zeros The discreteness of the pure point spectrum is proved We pass on to the proof of the estimate for Nε) We start with θ n : θ n φ φφ + ) n n nn φ) n = n 3 n n 3 We have { n > > n } { n 3 > n θ n < } 4n Taing Nε) = ε 0 < ε < we see that for any n > Nε) > the condition nθ n > ε holds Thus the condition 36) is satisfied for all vectors from DJ) lnε) which completes the proof 4 The degenerate case Now we consider the case when one of the modulation parameters turns to zero we call this case degenerate) Formally speaing we cannot call such matrix a Jacobi one but this limit case is of certain interest for us supplementing the whole picture Theorem 4 If c c = 0 c 0 denoting c := max{ c c }) then the spectrum of the operator J is the closure of the set of eigenvalues λ n : The set of eigenvalues is {λ n n N} = where eigenvalues λ ± n λ ± n σj) = {λ n n N} { {λ + n λ n n N} if c 0 c = 0 { λ + n λ n n N} if c = 0 c 0 have the following asymptotics: λ + n λ + n = + c)n + O) n λ n = c)n + c ) ) 6cn + O n n λ n = c)n + c 3 ) ) 6cn + O n 3 n Proof When one of the parameters c or c is zero the infinite matrix consists of x or x) blocs Thus the operator J is an orthogonal sum of finite x or x) matrices J n J = n= J n Then the spectrum of the operator J is the closure of the sum of spectrums of these matrices σj) = n= σj n) Let us calculate σj n ) If c 0 c = 0 then ) n c n ) J n = c n ) n

14 4 SERGEY SIMONOV and σj n ) = {λ + n λ n } where λ ± n = 4n ± 4c n ) + and it is easy to see that ) λ ± n = ± c)n ± c ± ) 6cn + O n n If c = 0 c 0 then J = σj ) = {} ) n c n ) J n = n c n ) n and σj n ) = { λ + n λ n } n where λ ± n = 4n 3± 4c n ) + and it is easy to see that ) 3 λ n ± = ± c)n ± c ± ) 6cn + O n n which completes the proof Remar 4 From the last Theorem it follows that as n λ + n λ + n + As for λ n and λ n their asymptotic behavior depends on the parameter c: If c > then λ n λ n If c = then λ n λ n Finally if 0 < c < then λ n λ n + Hence if 0 < c the operator J is semibounded from below and if c > the operator J is not semibounded This clearly corresponds to results obtained in Section 3 When we move along the side of the boundary square see fig case c)) towards one of the points {D j } 4 j= = {; 0); 0; ); ; 0); 0; )} the absolutely continuous spectrum covers the interval [ ; + ) At the same time at each limit point D j j = 3 4 the spectrum of J becomes pure point which demonstrates yet another phenomenon of the spectral phase transition Moreover note that the spectrum at each limit point consists of two series of eigenvalues one going to + another accumulating to the point λ = both points prior to the spectral phase transition having been the boundaries of the absolutely continuous spectrum Remar 43 The proof of discreteness of the spectrum in the case c) of Theorem 3 essentially involves the semiboundedness property of the operator J In the case b) one does not have the advantage of semiboundedness and due to that reason the proof of discreteness supposedly becomes much more complicated Remar 44 The choice q n = n was determined by the possibility to apply the Birhoff-Adams technique It should be mentioned that much more general situation q n = n α 0 < α < may be considered on the basis of the generalized discrete Levinson Theorem Proper approach has been developed in [] see also [5] One can apply similar method in our situation Another approach which is also valid is so-called Jordan box case and is presented in [7] Acnowledgements The author expresses his deep gratitude to Prof S N Naboo for his constant attention to this wor and for many fruitful discussions of the subject and also to Dr A V Kiselev for his help in preparation of this paper

15 AN EXAMPLE OF SPECTRAL PHASE TRANSITION PHENOMENON 5 References [] N I Ahiezer I M Glazman Theory of linear operators in Hilbert space nd edition) Dover New Yor 993 [] Yu M Berezansii Expansions in eigenfunctions of selfadjoint operators Russian) Nauova Duma Kiev 965 [3] J Dombrowsi S Pedersen Absolute continuity for unbounded Jacobi matrices with constant row sums J Math Anal Appl vol 6700) no pp [4] S N Elaydi An Introduction to Difference Equations Springer-Verlag New Yor 999 [5] D Damani S N Naboo A first order phase transition in a class of unbounded Jacobi matrices: critical coupling to appear in Journal Approximation Theory) [6] D Gilbert D Pearson On subordinacy and analysis of the spectrum of one dimensional Schrödinger operators J Math Anal Appl vol 8 987) pp [7] J Janas The asymptotic analysis of generalized eigenvectors of some Jacobi operators Jordan box case to appear in J Difference Eq Appl) [8] J Janas S N Naboo Criteria for semiboundedness in a class of unbounded Jacobi operators translation in St Petersburg Math J vol 4 003) no 3 pp [9] J Janas S N Naboo Multithreshold spectral phase transition examples for a class of unbounded Jacobi matrices Oper Theory Adv Appl vol 4 00) pp [0] J Janas S N Naboo Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries J Funct Anal vol 9 00) no pp [] J Janas S N Naboo E Sheronova Asymptotic behaviour of generalizes eigenvectors of Jacobi matrices in Jordan box case submitted to Rocy Mount Math J) [] S Khan D Pearson Subordinacy and spectral theory for infinite matrices Helv Phys Acta vol 65 99) pp Department of Mathematical Physics Institute of Physics St Petersburg University Ulianovsaia St Petergoff St Petersburg Russia address: sergey simonov@mailru