PERIODIC SOLUTIONS OF DISCRETE GENERALIZED SCHRÖDINGER EQUATIONS ON CAYLEY TREES

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1 Communications on Stochastic Analysis Vol. 9, No. (5) 8-96 Serials Publications PERIODIC SOLUTIONS OF DISCRETE GENERALIZED SCHRÖDINGER EQUATIONS ON CAYLEY TREES FUMIO HIROSHIMA, JÓZSEF LÖRINCZI, AND UTKIR ROZIKOV Abstract. In this paper we define a discrete generalized Laplacian with arbitrary real power on a Cayley tree. This Laplacian is used to define a discrete generalized Schrödinger operator on the tree. The case discrete fractional Schrödinger operators with index < α < is considered in detail, and periodic solutions of the corresponding fractional Schrödinger equations are described. This periodicity depends on a subgroup of a group representation of the Cayley tree. For any subgroup of finite index we give a criterion for eigenvalues of the Schrödinger operator under which periodic solutions exist. For a normal subgroup of infinite index we describe a wide class of periodic solutions.. Introduction The spectral properties of Schrödinger operators on graphs have many applications in physics and they have been intensively studied since the late 99s. In [4] Klein proved the existence of a purely absolutely continuous spectrum on the Cayley tree, under weak disorder. The paper [8] reviews the main aspects and problems in the Anderson model on the Cayley tree. It shows whether wave functions are extended or localized is related to the existence of complex solutions of a certain non-linear equation. In [5] sufficient conditions for the presence of the absolutely continuous spectrum of a Schrödinger operator on a rooted Cayley tree are given; see [] for more results on the model. In [8] the spectrum of a Schrödinger operator on the d-dimensional lattice Z d is studied. In recent years, fractional calculus has received a great deal of attention. Equations involving fractional derivatives and fractional Laplacians have been studied by various authors, see e.g. [, 6, ] and the references therein. In probability theory fractional Laplacians appear as generators of stable processes []. Nonlocal (such as fractional) Schrödinger operators are currently increasingly studied [7, 9,,, 6, 7]. Their counterparts on lattices are currently much less understood than local discrete Schrödinger operators. In [] a new phenomenon on the spectral edge behaviour of non-local lattice Schrödinger operators with a δ-interaction has been reported. In this paper we study solutions of generalized Schrödinger equations on Cayley trees. Received 4--; Communicated by Hui-Hsiung Kuo. Mathematics Subject Classification. Primary 8B6; Secondary 9A. Key words and phrases. Cayley tree, fractional Laplacian, Schrödinger equation, periodic solution. 8

2 84 FUMIO HIROSHIMA, JÓZSEF LÖRINCZI, AND UTKIR ROZIKOV The paper is organized as follows. In Section we define a discrete generalized Laplacian (with power α R) on a Cayley tree. This Laplacian is then used to define a discrete generalized Schrödinger operator on the Cayley tree. In Section we consider fractional Schrödinger operators with exponents α, and describe periodic solutions of the corresponding fractional Schrödinger equations. This periodicity depends on a subgroup of a group representation G k of the Cayley tree, i.e., for a given subgroup Ĝ of G k we define a notion of a Ĝ-periodic solution. Related ideas of Ĝ-periodic Gibbs measures and periodic p harmonic functions have been considered in [,, 4] before. The notion of Ĝ-periodic solutions can, however, be considered for arbitrary graphs and groups. For any subgroup of finite index we give a criterion for eigenvalues of the fractional Schrödinger operator under which there are periodic solutions. Also, we describe a wide class of periodic solutions for a normal subgroup of infinite index.. A Discrete Fractional Laplacian on Cayley Trees Recall that a Cayley tree Γ k is an infinite k-regular tree (k ), i.e., a connected graph on a countably infinite set of vertices, with no cycles and in which every vertex has degree k +. We denote Γ k = (V, L), where V is the vertex set and L the edge set of Γ k. Two vertices x and y are called adjacent if there exists an edge l L connecting them. In such a case we will use the notation l = x, y. Also, we denote the set of adjacent vertices of x by S(x) = {y V : x, y L} and call it the neighbourhood of x. An alternating sequence (x, x, x, x,..., x n, x n, x n, x n ) of vertices and edges is a path. The length of a path is the number of edges connecting the two end-vertices of the path. On Γ k one can define a natural distance d(x, y) as the length of the shortest path connecting x and y. For any x V denote W m (x) = {y V : d(x, y) = m}, m. Let u : V C be a function. The Laplacian on a Cayley tree is defined by ( u)(x) = (u(y) u(x)). (.) For n N we denote by n the n-fold iteration of, i.e., n = ( n ). We have the following expression. Lemma.. For any n N and x V the following holds n ( ) n ( n u)(x) = ( (k + )) j j j= y S(x) y S(y ) y n j S(y n j ) u(y n j ) with y = x. +( (k + )) n u(x), (.)

3 SOLUTIONS OF A DISCRETE SCHRÖDINGER EQUATION 85 Proof. We proceed by induction. For n = this is true by (.). For n = using (.) we obtain ( u)(x) = ( u)(x) = (( u)(y) ( u)(x)) = y S(x) y S(y ) u(y ) (k + ) y S(x) u(y ) + (k + ) u(x). Now we assume that the formula (.) is true for n and shall prove it for n + : ( n+ u)(x) = ( n u)(x) = (( n u)(y) ( n u)(x)) = = n ( ) n ( (k + )) j j j= y S(y) y S(y ) +( (k + )) n u(y) n ( ) n ( (k + )) j j j= y S(y) y S(y ) n ( ) n + ( (k + )) j j j= y S(x) y S(y ) +( (k + )) n u(x)) u(y n ) y n S(y n ) y S(y) y S(y ) +( (k + )) n y n j S(y n j ) y n j S(y n j ) u(y) y n j S(y n j ) u(y n j ) u(y n j ) u(y n j ) n ( ) n (k + ) ( (k + )) j u(y n j ) j j= y S(x) y S(y ) y n j S(y n j ) ( ) n (k + )( (k + )) n u(x). (.) n Now denote variable y by y and y i by y i+ for i =,..., n, then using ( ) n+ j = ( n ( j ) + n j) we get RHS of (.) = u(y n+ ) + n ( ) n + ( (k + )) j j j= n ( ) n ( (k + )) j j j = +( (k + )) n+ u(x) y S(x) y S(y ) y S(x) y S(y ) y S(x) y S(y ) y n+ S(y n) y n j+ S(y n j) y n j + S(y n j ) u(y n j+ ) u(y n j +)

4 86 FUMIO HIROSHIMA, JÓZSEF LÖRINCZI, AND UTKIR ROZIKOV + = y S(x) y S(y ) n (( n ( (k + )) j j j= y n+ S(y n) u(y n+ ) ) ( )) n + j y S(x) +( (k + )) n+ u(x) n ( ) n + = ( (k + )) j j j= +( (k + )) n+ u(x). It is easily seen that y S(x) y S(y ) y S(x) y S(y ) u(y ) = y W (x) y n j+ S(y n j) y n j+ S(y n j) u(y n j+ ) u(y n j+ ) u(y) + (k + )u(x). (.4) Next we define real powers α of the Laplacian in such a way that when α takes a positive integer value n, then n is recovered. To define a fractional Laplacian on the Cayley tree we make the change ( α j ) Γ(α + ) Γ(j + )Γ(α + j). Note that this is the multi-dimensional version of the discrete fractional difference operator introduced by Andersen for a sequence (u n ) n N of real numbers (see [6]) ( ) α α u n = ( ) j u n+j. j j= Definition. (Generalized Laplacian on a Cayley tree). We define the generalized Laplacian of order α on a Cayley tree by [α] ( α u)(x) = A(α, j) u(y [α] j ) j= y S(x) y S(y ) y [α] j S(y [α] j ) +( (k + )) [α] u(x), (.5) where A(α, j) = ( (k+))j Γ(α+) Γ(j+)Γ(α+ j) and [α] denotes the integer part of α. When < α <, we call α fractional Laplacian of order α. A Schrödinger equation of a fractional power α can be written as ( α u)(x) = Eu(x) v(x)u(x). (.6) Here u(x) C is the value of the wave function at vertex x, E R is the energy of the particle, and v = {v(x)} x V is a given real-valued function.. Periodic Wave Functions In what follows we consider the case α.

5 SOLUTIONS OF A DISCRETE SCHRÖDINGER EQUATION 87.. Case: α <. We have by (.5) ( α u)(x) = ( (k + )) [α] u(x) = u(x), (.) i.e., the operator α is an identity operator. Consequently, from (.6) we obtain u(x) = Eu(x) v(x)u(x). (.) From this equation we get (E v(x))u(x) =. A straightforward analysis of this equation gives the following. Proposition.. For given E and function v the equation (.) has the solution {, if x {y : v(y) + E} u(x) = u (x), where u is an arbitrary function. if x {y : v(y) + = E}, Remark.. If one wants to have a fractional Laplacian α with the property that it is not identical for any α >, then this operator can be defined using operator (.5) as { ( α+ u)(x), if α [, + ) \ {,,,... } ( α u)(x) = (.) ( α u)(x), if α {,,,... }. The Schrödinger equation (.6) corresponding to the operator α is ( α u)(x) = Eu(x) v(x)u(x). (.4) We note that if we know the solutions of the equation (.6), then using (.) we can find solutions of (.4). Thus it is sufficient to consider the equation (.6) for the operator α... Case: α =. In this case the Schrödinger equation reads u(y) = Eu(x) v(x)u(x). (.5)... Properties of group representations of the Cayley tree. To study solutions of equation (.5) and equations considered below first we give some properties of a group representation of the Cayley tree. Let G k be a free product of k + cyclic groups of the second order with generators a, a,..., a k+, respectively. Any element x G k has the following form: x = a i a i... a in, where i m k +, m =,..., n. It is known [] that there exists a one-to-one correspondence between the set of vertices V of the Cayley tree Γ k and the group G k. In Fig. some elements of G k are shown (see [, Chapter ] for detailed properties of this group representation). Let K G k be an arbitrary normal subgroup of index r of the group G k (see [4,, ] for examples of subgroups of G k ). We introduce the following equivalence relation on the set G k : x y if xy K. Let G k /K = {K, K,..., K r } be the factor-group with respect to K. Denote q i (x) = W (x) K i, s i (x) = W (x) K i, i =,..., r, x G k, where is the counting measure of a set. The following lemma is known [9, ].

6 88 FUMIO HIROSHIMA, JÓZSEF LÖRINCZI, AND UTKIR ROZIKOV a a a a a a a a a a a a a e a a a a a Figure. Cayley tree of order two, i.e., k =. Lemma.. If x y, then q i (x) = q i (y) and s i (x) = s i (y) for any i =,..., r. By Lemma. it follows that q j (x) and s j (x) corresponding to K have the following form q j (x) = q ij, s j (x) = s ij for all x K i. Since the set V of vertices has the group representation G k, without loss of generality we identify V with G k, i.e., we may replace V with G k. Definition.4. Let K be a subgroup of G k, k. A function u : x G k u(x) C is called K-periodic if u(yx) = u(x) for all x G k and y K. A G k -periodic function u is called translation-invariant.... The case of finite index. Let G k /K = {K,..., K r } be a factor group, where K is a normal subgroup of index r. Note that any K-periodic function u has the form u(x) = u j, x K j, j =,..., r. (.6) Assume that v = {v(x)} x V is K-periodic, i.e. v(x) = v j for any x K j then from (.5) we get r q ij u j = (E v i )u i, i =,,..., r. (.7) j= This system can be written as r q ij u j (E q ii v i )u i =, i =,..., r. (.8) j=:j i

7 SOLUTIONS OF A DISCRETE SCHRÖDINGER EQUATION 89 For a given normal subgroup K of index r we denote by D K (E, v) the determinant of the system (.8). Thus the following holds. Theorem.5. For given K and a K-periodic v there exists a K-periodic wave function u if and only if E is a solution to D K (E, v) =. Example.6. () In case K = G k, we have translation invariant wave functions, i.e., u(x) = u for all x V. In this case D K (E, v) = E v k =, thus a translation invariant wave function exists if and only if E = v + k +. () Let K = G () k be the subgroup in G k consisting of all words of even length. Clearly, it is a subgroup of index. In this case we have D K (E, v) = (E v )(E v ) (k + ) =. For any E satisfying this equation there are G () k -periodic wave functions having the form u, if x G () k u(x) = u, if x G k \ G () k. () Let k =, K = H, with H = {x G : ω x (a ) even, ω x (a ) even}, where the number of letters a i of the word x is denoted by ω x (a i ). Note that H is a normal subgroup of index 4. The group G /H = {H, H, H, H } has the following elements: H = {x G : ω x (a ) even, ω x (a ) odd} H = {x G : ω x (a ) odd, ω x (a ) even} H = {x G : ω x (a ) odd, ω x (a ) odd}. In Fig. the partitions of Γ with respect to H are shown. The elements of the class H i, i =,,, are denoted by i. In this case an H -periodic wave function u has the form u(x) = u i if x H i. Such a non-zero function exists if and only if E satisfies the equation v + E v + E D H (E, v) = =. v + E v + E... The case of infinite index. We consider a normal subgroup H of infinite index constructed for M = {, } as follows. Let the mapping π M : {a,..., a k+ } {a, a } {e} be defined by { ai, if i M π M (a i ) = e, if i / M. Denote by G M the free product of cyclic groups {e, a i }, i M. Consider f M (x) = f M (a i a i...a im ) = π M (a i )π M (a i )...π M (a im ).

8 9 FUMIO HIROSHIMA, JÓZSEF LÖRINCZI, AND UTKIR ROZIKOV Figure. Partition of Γ by subgroup H of index 4. Then it is seen that f M is a homomorphism and hence H = {x G k : f M (x) = e} is a normal subgroup of infinite index. The factor group has the form G k /H = {H, H (a ), H (a ), H (a a ),... }, where H (y) = {x G k : f M (x) = y}. With the notations H m = H (a a...), H }{{} m = H (a a...) }{{} m m the factor group can be represented as G k /H = {..., H, H, H, H, H,... }. The partition of the Cayley tree Γ with respect to H is shown in Fig. (the elements of the class H i, i Z, are denoted by i). Any H -periodic wave function has the form We note that (see [4]) if x H m, then u(x) = u m if x H m, m Z. q m (x) = W (x) H m =, q m (x) = k, q m+ (x) =. Using these equalities we get (see Fig. ) s m (x) = W (x) H m =, s m (x) = (k ) s m (x) = (k )(k ), s m+ (x) = (k ), s m+ (x) =.

9 SOLUTIONS OF A DISCRETE SCHRÖDINGER EQUATION Figure. The partition of the Cayley tree Γ w.r.t. elements of the class H i, i Z, are denoted by i. H, the Assuming that v is H -periodic, i.e., v(x) = v m for any x H m, equation (.5) gives (see Fig. ) (E v n )u n = u n + (k )u n + u n+, (.9) where n Z. For simplicity, assume that v n v, i.e., v is translation invariant. Then equation (.9) becomes a linear homogeneous recurrence equation of order, i.e. u n+ + (k + v E )u n + u n =. (.) Recall that the solution procedure of a general linear homogeneous recurrence equation with constant coefficients of order d is given by u n + c u n + c u n + + c d u n d =. (.) Let p(t) be the characteristic polynomial, i.e., p(t) = t d + c t d + c t d + + c d. Suppose λ i is a root of p(t) having multiplicity r i. Write u n as a linear combination of all the roots counting multiplicity with arbitrary coefficients b ij : u n = i ( bi + b i n + b i n + + b iri n ri ) λ n i. (.) This gives the general solution to equation (.). Applying this procedure to equation (.), we have p(λ) = λ + (k + v E)λ + =. (.) An analysis of this equation by using formula (.) yields the following result.

10 9 FUMIO HIROSHIMA, JÓZSEF LÖRINCZI, AND UTKIR ROZIKOV Theorem.7. For H and a translation invariant v the following hold: () If E (k + v, k + v + ), then an H -periodic wave function is u(x) = C λ n + C λ n for any x H n, n Z, where λ, λ are (complex) solutions to equation (.). () If E = k+v, then there is an H -periodic function u(x) = ( ) n (C + nc ), x H n, n Z. () If E = k + v +, then there is an H -periodic function u(x) = C + nc, x H n. (4) If E (, k + v ) (k + v +, + ), then an H -periodic wave function is u(x) = C λ n + C λ n for any x H n, n Z, where λ, λ are (real) solutions to equation (.). These solutions are in each case unique up to the choice of the constant prefactors... Case: < α <. In this case we have from (.5) ( α u)(x) = A(α, ) u(y)+a(α, )u(x) = u(y) (k + )Γ(α + ) u(x). Γ(α) Consequently, from (.6) we get an equation similar to equation (.5), with E replaced by E = E + (k+)γ(α+) Γ(α). Thus the analysis of this equation is very similar to the analysis of equation (.5)..4. Case: α =. Using formula (.4) for α = n =, we get ( u)(x) = u(z) (k + ) u(y) + (k + )(k + )u(x). (.4) z W (x) y W (x) The Schrödinger equation corresponding to the bi-laplacian is Now we discuss the periodic solutions of (.5). ( u)(x) = Eu(x) v(x)u(x). (.5).4.. The case of finite index. Let G k /K = {K,..., K r } be a factor group, where K is a normal subgroup of index r. Assume that v = {v(x)} x Gk is K-periodic, i.e., v(x) = v j for any x K j. Then by (.5) and (.4) for any i =,,..., r we get r r s ij u j (k + ) q ij u j + (k + )(k + )u i = (E v i )u i. (.6) j= j= This system can be written as r (s ij (k+)q ij )u j (E (k+)(k+)+(k+)q ii s ii v i )u i =, (.7) j=:j i where i =,,..., r. For a given normal subgroup K of index r we denote by D K (E, v) the determinant of the system (.7). Thus we have the following result. Theorem.8. For a given K and a K-periodic v there exists a K-periodic solution u to the equation (.5) if and only if E is a solution of D K (E, v) =.

11 SOLUTIONS OF A DISCRETE SCHRÖDINGER EQUATION 9 Example.9. We consider some examples of subgroups introduced in Subsection... () In case K = G k we have translation invariant solutions, i.e., u(x) = u for all x V. In this case D K (E, v) = E v =. Thus a non-zero, translation invariant wave function exists if and only if E = v. () For K = G () k we have D K (E, v) = (E v )(E v ) (k + ) (E (v + v )) =. For any E satisfying this equation there exist G () k -periodic solutions of equation (.5). () Let K = H. By Fig. it is clear that q = q = q = q = q = q = q = q = q = q = q = q = q = q = q = q = {, if i = j s ij =, if i j. Hence an H -periodic function exists if and only if E satisfies the equation v + 6 E v + 6 E 4 D H (E, v) = =. 4 v + 6 E v + 6 E.4.. The case of infinite index. Consider H constructed in Subsection... Then from equation (.5) using (.4) and the above expressions of q m (x) and s m (x) we obtain (E v m )u m = u m+ 4u m+ + 6u m 4u m + u m, (.8) where m Z. For simplicity, we assume that v n v, i.e., v is translation invariant. Then equation (.8) yields the characteristic equation λ 4 4λ + (6 + v E)λ 4λ + =. (.9) We make use again of the general argument (.)-(.). Denoting ξ = λ + /λ, from (.9) we get (ξ ) = E v. (.) By a simple analysis of (.) and then of λ + /λ = ξ, we obtain the following results. () If E < v, then equation (.9) has the four distinct complex solutions ˆλ, = ( z ± ) z 4, ˆλ,4 = ( z ± z ) 4, where z = + i v E, z = i v E. () If E = v, then equation (.9) has the unique solution λ = with multiplicity 4.

12 94 FUMIO HIROSHIMA, JÓZSEF LÖRINCZI, AND UTKIR ROZIKOV () If v < E < 6 + v, then equation (.9) has the two real solutions λ, = + E v ± ( + E v ) 4 and the two complex non-real solutions (.) λ,4 = E v ± i 4 ( E v ). (.) (4) If E = 6 + v, then equation (.9) has three real solutions where has multiplicity. λ, = ±, λ =, (.) (5) If E > 6 + v, then equation (.9) has the four real solutions λ, λ given by (.) and λ,4 = E v ± ( E v ) 4. (.4) Thus we have proved the following result. Theorem.. For H and a translation invariant v (i.e., v v ) the following hold: () If E < v, then there exists an H -periodic function of the form u(x) = u m = C ˆλm + C ˆλm + C ˆλm + C 4ˆλm 4, x H m, m Z, where ˆλ i, i =,,, 4 are the complex numbers above. () If E = v, then equation (.5) has an H -periodic solution u(x) = u m = C + C m + C m + C m, x H m, m Z. () If v < E < 6 + v, then equation (.5) has an H -periodic solution u(x) = u m = C λ m + C λ m + C λm + C 4 λ4, x H m, m Z, where λ, λ are given by (.), and λ, λ 4 are defined in (.) above. (4) If E = 6 + v, then equation (.5) has an H -periodic solution u(x) = u m = C ( ) m + C ( + ) m + (C + mc 4 )( ) m, for all x H m and m Z. (5) If E > 6 + v, then equation (.5) has an H -periodic solution u(x) = u m = C λ m + C λ m + C λ m + C 4 λ m 4, x H m, m Z, where λ, λ, λ, λ 4 are given by (.) and (.4). These solutions are in each case unique up to the choice of the constant prefactors. Acknowledgment. U.Rozikov thanks Aix-Marseille University Institute for Advanced Study IMéRA (Marseille, France) for support by a residency scheme. We thank the referee for careful reading of the manuscript and helpful suggestions which have led to improvement of the paper.

13 SOLUTIONS OF A DISCRETE SCHRÖDINGER EQUATION 95 References. Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods, World Scientific,. Derrida, B. and Rodgers, G. J.: Anderson model on a Cayley tree: the density of states, J. Phys. A 6 (99), no. 9, L457 L46.. Ganikhodjaev, N. N.: Group presentations and automorphisms of the Cayley tree, Doklady AN Uzbekistan 4 (994), Ganikhodjaev, N. N. and Rozikov, U.A.: Classes of normal subgroups of finite index of the group representation of the Cayley tree, Uzbek. Mat. Jour. 4 (997), 9.(Russian) 5. Helms, L. L.: Biharmonic functions and Brownian motion, J. Appl. Prob. 4 (967), Hilfer, R.: Applications of Fractional Calculus in Physics, World Scientific,. 7. Hiroshima, F., Ichinose, T., and Lőrinczi, J.: Path integral representation for Schrödinger operators with Bernstein functions of the Laplacian, Rev. Math. Phys. 4 (), Hiroshima, F., Sasaki, I., Shirai, T., and Suzuki, A.: Note on the spectrum of discrete Schrödinger operators, J. Math-for-Ind. 4B (), Hiroshima, F. and Lőrinczi, J.: Lieb Thirring bound for Schrödinger operators with Bernstein functions of the Laplacian, Commun. Stoch. Anal. 6 (), Hiroshima, F. and Lőrinczi J.: The spectrum of non-local discrete Schrödinger operators with a δ-potential, Pacific J. Math. Ind. 6 (4), 6.. Kaleta, K. and Lőrinczi, J.: Fractional P (φ) -processes and Gibbs measures, Stoch. Proc. Appl. (), Kaleta, K. and Lőrinczi, J.: Pointwise estimates of the eigenfunctions and intrinsic ultracontractivity - type properties of Feynman Kac semigroups for a class of Lévy processes, Ann. Probab. 4 (5), Kaleta, K. and Lőrinczi, J.: Fall-off of eigenfunctions of non-local Schrödinger operators with decaying potentials, arxiv:5.58 (5), submitted for publication 4. Klein, A.: Extended states in the Anderson model on the Bethe lattice, Advances in Math. (998), Kupin, S.: Absolutely continuous spectrum of a Schrödinger operator on a tree, J. Math. Phys. 49 (8), Lőrinczi, J., Kaleta, K., and Durugo, S.O., Spectral and analytic properties of some non-local Schrödinger operators and related jump processes, to appear in Commun. Appl. Ind. Math. (5). 7. Lőrinczi, J. and Malecki, J.: Spectral properties of the massless relativistic harmonic oscillator, J. Diff. Equations 5 (), Miller, J. and Derrida, B.: Weak-disorder expansion for the Anderson model on a tree. J. Statist. Phys. 75 (994), no. -4, Normatov, E. P.: On partitions of sets of m-neighboring vertices of a Cayley tree. Uzbek. Mat. Jour. (), (Russian). Podlubny, I.: Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, Rozikov, U. A.: Structures of partitions of the group representation of the Cayley tree into cosets by finite index normal subgroups, and their applications to the description of periodic Gibbs distributions, Theor. Math. Phys. (997), no., Rozikov, U. A.: Countably periodic Gibbs measures of the Ising model on the Cayley tree, Theor. Math. Phys. (), no., 9.. Rozikov, U. A.: Gibbs Measures on Cayley Trees, World Scientific, Singapore,. 4. Rozikov, U. A. and Ishankulov, F. T.: Description of periodic p-harmonic functions on Cayley trees, NoDEA. Nonlinear Differential Equations and Appl. 7 (), no., 5 6.

14 96 FUMIO HIROSHIMA, JÓZSEF LÖRINCZI, AND UTKIR ROZIKOV Fumio Hiroshima: Faculty of Mathematics, Kyushu University, Fukuoka, 89-95, Japan. address: József Lőrinczi: Department of Mathematics, Loughborough University, LE TU, UK address: Utkir Rozikov: Institute of mathematics, 9, Do rmon Yo li str., 5, Tashkent, Uzbekistan. address:

On disordered phase in the ferromagnetic Potts model on the Bethe lattice. Osaka Journal of Mathematics. 37(2) P.373-P.383

Title Author(s) On disordered phase in the ferromagnetic Potts model on the Bethe lattice Ganikhodjaev, Nasir; Rozikov, Utkir Citation Osaka Journal of Mathematics. 37(2) P.373-P.383 Issue Date 2000 Text