Int. J. Contemp. Math. Sci., Vol., 26, no. 6, 245-254 On the Regularized Trace of a Fourth Order Regular Differential Equation Azad BAYRAMOV, Zerrin OER, Serpil ÖZTÜRK USLU and Seda KIZILBUDAK C. ALIṢKAN Department of Mathematics Faculty of Arts and Science, Yıldız Technical University (342), Davutpaṣa, İstanbul,Turkey Abstract We shall obtain a formula for the regularized trace of a fourth order regular differential equation. Mathematics Subject Classification: 34L5, 47A Keywords: Asymptotics of eigenvalues and eigenfunctions, unitary matrix, regularized trace. STATEMENT OF THE PROBLEM In the space L 2 [,π] we consider the self-adjoint operators L are generated by the following expressions: and L which l (y) =y (4), l(y) =y (4) + p(x)y with the same boundary conditions y (ν) () = y (ν) (π) (ν =,, 2, 3) respectively. Here p(x) is a real valued, continuous function in [, π].
246 A. Bayramov et. al. { } The spectrum of operatorl coincides with the set 6n 4 Every point of. the spectrum is an eigenvalue with multiplicity two except point zero. { Zero } is the simple eigenvalue. We denote the eigenvalues of operator L by μ k k= (k +) 4 if k is odd and k μ = and μ k = k 4 if k is even and k 2 Orthonormal eigenfunctions corresponding to this eigenvalues are denoted by ψ = π, ψ = 2π sin 2x, ψ 2 = 2π cos 2x, We denote the eigenvalues of operator L by λ <λ λ 2 <λ 3 and corresponding orthonormal eigenfunctions by ϕ,ϕ,ϕ 2,ϕ 3,. In this paper, by Dikiy s method, we will obtain a formula for the sum of series (λ n μ n ) which is called regularized trace of operator L. Firstly the regularized trace formula for the Sturm-Liouville operator have been found by Gelfand-Levitan []. The some regularized trace formula for the same problem was obtained with different method by Dikiy [2]. Later study of regularized trace was generalized for different differential operators ( see, for example [3]-[8]) 2 SOME ESTIMATES. In this section we prove the formula [ ] lim (ϕ n,lϕ n ) (ψ n,lψ n ) = (2.) which will be used later. For this purpose we investigate the transfer matrix (u ik ) i, k= from the orthonormal basis {ϕ k} to orthonormal basis {ψ k } as in [2]: ψ k = u ik ϕ i, where u ik =(ϕ i,ψ k )
Fourth Order Differential Equation 247 (u ik ) i, k= is the a unitary matrix, that is u 2 ik = u 2 ik =. k= First of all, let us give some estimate for numbers u ik. It is clear that Lψ k = μ k ψ k + pψ k (2.2) Scalar producting both side of equality (2.2) by ϕ i we obtain (Lψ k,ϕ i )=(μ k ψ k,ϕ i )+(pψ k,ϕ i ) or and λ i (ψ k,ϕ i )=μ k (ψ k,ϕ i )+(pψ k,ϕ i ) (λ i μ k )(ψ k,ϕ i )=(pψ k,ϕ i ) According to [2] taking the square of both sides of the last equality and summing according to i we obtain π (λ i μ k ) 2 (ψ k,ϕ i ) 2 = (pψ k,ϕ i ) = pψ k 2 = [p(x)ψ k (x)] 2 dx π p 2 ψ 2 k (x)dx = p2 where p = max p(x) x π Hence (λ i μ k ) 2 u 2 ki <C (C = const.) (2.3) Suppose that p(x) is a continous function such that the following conditions hold:. For eigenvalues and eigenfunctions of operator L holds the asymptotic formulas ( ) ( ) λ k = μ k + O ϕ k = ψ k + O k + (k +) 2 In this paper constants C may be different
248 A. Bayramov et. al. 2. p(x)dx =. We shall use condition. below estimating. From inequality (2.3) it follows that (λ i μ k ) 2 u 2 ik <C i=n+ for all integer N and. (λ i μ k )u 2 ik <C i=n+ Then, it is obvious that i=n+ (λ i μ k )(λ i λ k )u 2 ik <C. From here we obtain i=n+ (λ i λ k ) 2 u 2 ik <C (λ N+ μ k )(λ i λ k )u 2 ik (λ i μ k )(λ i λ k )u 2 ik <C i=n+ i=n+ i=n+ (λ i λ k )u 2 ik < C (k N) (2.4) λ N+ μ k Now let us prove formula (2.). We have ( (ψ k,lψ k )= u ik ϕ i, ) λ i u ik ϕ i = λ i u 2 ik Summation on k from to N we have (ψ k,lψ k )= λ i u 2 ik k=
Fourth Order Differential Equation 249 Taking into account u 2 ik = we have (ψ k,lψ k )= λ k = λ i u 2 ki k= k= Hence we must prove that we have [ N ] lim λ i u 2 ik λ i u 2 ki = (2.5) k= k= λ i u 2 ik λ i u 2 ki = (λ i λ k )u 2 ik k= k= k= i=n+ + λ k (u 2 ik u2 ki ) (2.6) k= i=n+ Let us calculate first sum on the right side of equality (2.6). For convenience while let N + be even number then we have k= i=n+ N (λ i λ k )u 2 ik = k= i=n+ (λ i λ k )u 2 ik +(λ N+ λ N )u 2 N+ N + i=n+2 (λ i λ N )u 2 in (2.7) By inequality (2.4) we shall calculate first and third sum on the right side of equality (2.7), when N N k= i=n+ (λ i λ k )u 2 ik < = N k= k= C (N +) 4 (k +) 4 = C (N +) 4 k 4 N (N +) 4 N + dx 4 (N +) 4 x < 4 < N 3 + N + (N +) 4 N d( x N+ ) ( x N+ )4 =
25 A. Bayramov et. al. = N 3 + (N +) 3 = N 3 + (N +) 3 N N+ N+ N N+ N+ du u 4 = N + ( N 3 2 (N +) 3 N + ) + N + 2 ( +u + ) du 2 u 2 and + 2(N +) 3 N N+ N+ du u 2 ln N N 3 asn (2.8) (λ i λ N )u 2 C in < i=n+2 (N +3) 4 (N +) < C 4 (N +2) asn 3 (2.9) Now let us calculate the second term on the right side of equality (2.7) when N. Assume that N + is even. Using the condition. above, we have (λ N+ λ N )u 2 N+N λ N+ λ N = ( ) =(N +) 4 (N +) 4 + O asn (2.) N + Thus for even number N + from the expressions (2.7), (2.8), (2.9) and (2.) we have lim k= i=n+ (λ i λ k )u 2 ik = (2.) Formula (2.) can be proved for odd number N + similarly. Now we shall calculate second sum on the right side of equality (2.6) we have u ik + u ki =(ϕ i,ψ k )+(ϕ k,ψ i )= (ϕ i ψ i,ϕ k ψ k ) (2.2) By equality (2.2) and condition., we have
Fourth Order Differential Equation 25 C u ik + u ki ϕ i ψ i ϕ k ψ k (i +) 2 (k +) < C 2 (i + )(k +) (2.3) By using Cauchy-Schwarz inequality we have Hence i=n+ < (λ i μ k ) u 2 ik u 2 ki = i=n+ u ik + u ki 2 i=n+ i=n+ (λ i μ k ) u ik u ki u ik + u ki (λ i μ k ) 2 u ik u ki 2 C N +2(k +) (2.4) u 2 ik + u2 ki < C i=n+ (k +) (2.5) N +2(λ N μ k ) Now we shall estimate the second sum on the right side of equality (2.6), means that λ k u 2 ik u 2 N ki = λ N u 2 in u 2 Ni + λ k u 2 ik u 2 ki k= i=n+ i=n+ k= i=n+ = λ N u 2 N+N u2 NN+ + λ N i=n+2 N u 2 in u2 Ni + λ k k= i=n+ By inequality (2.3) we have λ N u 2 N+N u 2 NN+ = λ N u N+N u NN+ u N+N + u NN+ u 2 ik u2 ki (2.6) C(N +) 4 (N +2) 2 (N +) u 2 N+N u NN+ as N (2.7) By the expression (2.5) we estimate the second and third sum on the right side of equality (2.6) λ N u 2 in u2 Ni < i=n+2 (N +) 2 <C asn (2.8) (N + )(N + 2)[(N +3) 4 (N +) 4 ]
252 A. Bayramov et. al. and N λ k k= i=n+ = C k= N u 2 ik u2 ki <C k 4 k= k N +2[(N +) 4 k 4 ] <CN3 (k +) 2 (k +) N + 2[(N +) 4 (k +) 4 ] 2 k= (N +) 4 k 4 CN 5 ln N 2 as N (2.9) (N +) 3 From the expressions (2.6), (2.7),(2.8) and (2.9) we have lim k= i=n+ λ k (u 2 ik u2 ki )= (2.2) Thus from the expressions (2.6), (2.), and (2.2) we have formula (2.5). Hence formula (2.) have proved. 3 CALCULATION OF THE REGULARIZED TRACE It is easy to see that (ϕ n,lϕ n )=λ n and (ψ n,lψ n )=μ n +(ψ n,pψ n ) Putting these into formula (2.) we have [(ψ n,lψ n ) (ϕ n,lϕ n )] = (μ n λ n )+ (ψ n,pψ n ) asn (3.) Now we shall calculate lim (ψ n,pψ n ) By condition 2. we have for even number N (ψ n,pψ n )= π N 2 2 p(x)dx + n=( π p(x) sin 2 2nxdx+ 2 π ) p(x) cos 2 2nxdx
Fourth Order Differential Equation 253 = p(x)dx + N π π Analogically we have for odd number N p(x)dx = (3.2) (ψ n,pψ n )= π = π π + 2 π N 2 2 p(x)dx+ n=( π π p(x) sin 2 2( N 2 p(x) sin 2 2nxdx+ 2 π +)xdx π ) p(x) cos 2 2nxdx + p(x) cos 4( N + )x asn (3.3) 2 From the expressions (3.2) and (3.3) we have lim Hence from the expressions (3.) and (3.4) we have (ψ n,pψ n )= (3.4) lim (λ n μ n )= (3.5) Thus we have proved the following theorem. Theorem 3. If p(x) is continous function such that conditions. and 2. above are fulfilled, then the formula (3.5) is true. References [] Gelfand, I. M. and Levitan, B. M., On a formula for eigenvalues of a differential operator of second order, Dokl. Akad. Nauk SSSR 88(4), 593-596, (953) [2] Dikiy, L. A., About a formula of Gelfand-Levitan, Usp. Mat. Nauk 8(2), 9-23, (953) [3] Gelfand, I. M., About an identity for eigenvalues of a differential operator of second order, Usp. Mat. Nauk (67), 9-98, (956)
254 A. Bayramov et. al. [4] Halberg, C. J. and Kramer, V. A., A generalization of the trace concept, Duke Math. J., 27(4), 67-68, (96) [5] Maksudov, F. G., Bairamoglu, M. and Adiguzelov, E. E., On regularized trace of Sturm-Liouville operator on a finite interval with the unbounded operator coefficient, Dokl. Akad. Nauk SSSR 3(), 69-73, (984) [6] Lax, P. D., Trace formulas for the Schrödinger operator, Comm. Pure Appl. Math. V. 47, 53-52, (994) [7] Podol skii, V. E., On the summability of regularized sums of eigenvalues of the Laplace-Beltrami operator with potential on symmetric spaces of rank one, Russian J. Math. Phys., 4(), 23-3, (996) [8] Dostaniċ, M., Spectral properties of the operator of Riesz potential type, Proc. Amer. Math. Soc., 26(8), 229-2297 (998) Received: November 3, 25