The Almost Everywhere Convergence of Eigenfunction Expansions of Schrödinger Operator in L p Classes.

Transcription

1 Malaysian Journal of Mathematical Sciences (S) April : 9 36 (7) Special Issue: The nd International Conference and Workshop on Mathematical Analysis (ICWOMA 6) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: The Almost Everywhere Convergence of Eigenfunction Expansions of Schrödinger Operator in L p Classes. Jamaludin, N. A. and Ahmedov, A. Department of Mathematics, Centre Foundation Studies, National Defence University of Malaysia, Malaysia Faculty of Industrial Sciences & Technology, Universiti Malaysia Pahang, Malaysia amalinajamaludin@upnm.edu.my Corresponding author ABSTRACT In this paper the eigenfunction expansions of the Schrödinger operator with the potential having singularity at one point are considered. The uniform estimations for the spectral function of the Schrödinger operator in closed domain are obtained. The almost everywhere convergence of the eigenfunction expansions by Riesz means in the classes L p classes is proven by estimating the maximal operator in L and L and applying the interpolation theorem for the family of linear operators. Keywords: Schrödinger operator, almost everywhere convergence and eigenfunction expansions.

2 Jamaludin, N. A. and Ahmedov, A.. Introduction Let be an arbitrary bounded domain in R N (N 3) with smooth boundary. Let fix the point x. We assume that the potential function q L () has a following form: q(x) = a( x x ), x, x x where a C (, ) is non-negative function satisfying the condition: for some τ >. t k d k a(t) dt k Ctτ, k =,,,..., [N/] We consider Schrödinger operator H = + q with domain C (). We denote an arbitrary nonnegative self-adjoint extension of H with discrete spectrum by Ĥ. Let 3... be the eigenvalues of Ĥ, and let {u n } n= be the corresponding complete orthonormal system of eigenfunctions. For each Re(s), we define the s th order Riesz mean of the eigenfunction expansion of a function f L () as follows E s f(x) = n< ( ) s n (f, u n )u n (x), where (f, u n ) denotes the Fourier coefficients of the function f: (f, u n ) = f(y)u n(y)dy, n =,, 3,... The current work is devoted to the investigations connected with the problems of the almost everywhere convergence of the eigenfunction expansions of the Schrödinger operator by Riesz means. The main result of the paper is the following theorem: Theorem (. ) If f(x) L p (), p. Then Riesz means E s f(x) of order s > N p almost everywhere in converges to f(x). The similar statement for the eigenfunction expansions established by Alimov (97b). When we consider multiple Fourier series and integrals the con- Malaysian Journal of Mathematical Sciences

3 An Almost Everywhere Convergence of Eigenfunction Expansions of Schrödinger Operator in L p Classes on Closed Domain ( ) dition for the almost everywhere convergent guranteed if s > (N ) p for the values of p : p. As we can see the gap between these cases. In paper Bastis (983) has been constructed the example of the eigenfunction expansion corresponding to the self adjoint extension of the Laplace ( operator ) that the Riesz means of eigenfunction expansions of order s : s < N p 3 almost everywhere diverges to infinity for certain function from L p. In this work we are extending the result of the Alimov (97b) for the case of Schrodinger operator with nonsmooth potential. To prove the statement in the Theorem we estimate the maximal operator E f s = sup > E sf in the classes L () and L (), and apply interpolation to the family of linear operators. In this work we investigate the convergence almost everywhere of the eigenfunction expansions of the Schrödinger operator on closed domain where E s f(x) converge to f(x) L p ().. Preliminaries In this section we obtain a suitable representation for the Riesz means of the eigenfunction expansions of the Schrodinger operator. The eigenfunction expansions of the Schrödinger operator with special type potential function are considered in Alimov and Joo (983a) and Alimov and Joo (983b). Let denote by r = x x, x. Mean value formula for the eigenfunctions of the Schrödinger operator has following form θ [ ( ) u n (x +rθ)dθ = u n (x ) N N Γ (r n ) N where the function ν(r, n ) satisfies the estimation (r n ) + ν(r, ] n ) () ν(r, n ) const( n ) τ ω(r n ) here ω(t) = min{, ( ) N t }, t >. For arbitrary R > we define the following set: R = {y : dist(y, ) > R} and for any x, y we introduce the following function: Malaysian Journal of Mathematical Sciences

4 Jamaludin, N. A. and Ahmedov, A. R s (x, y, ) = s Γ(s + ) (π) N ( ) N +s(r ) (r, x y R ) N +s, x y > R, and D s (x, y, ) = s Γ(s + ) N 4 s u n (x)u n (y) ( n ) N I s(, n ), () where I s (, n ) denotes the integral I s (, n ) = R +s(r ) (r n )r s dr. The following representation is very important for the estimation of the maximal operator in the classes of L p (). Lemma. The Riesz means has the following representation Ef(x) s = f(x)r s (x, y, )dx + f(x)d s (x, y, )dx f(x)p s (x, y, )dx, r R where P s (x, y, ) = s Γ(s + ) (π) N R u n (x)u n (y) +s(r ) (r ) ( ) N ν(r, N n )r N dr. +s Proof : Applying the Parseval formula for f L () and g x (x) = x x N s +s( x x ), x x R, x x > R. Malaysian Journal of Mathematical Sciences

5 An Almost Everywhere Convergence of Eigenfunction Expansions of Schrödinger Operator in L p Classes on Closed Domain where the number R is chosen with the condition that the ball of radius R: B R (x ) = {x : x x < R}. (3) We obtain f(x)g x (x)dx = x x R f(x) x x N s +s( x x )dy = f n d n The Fourier coefficients d n = g x (x)u n (x)dx,n =,, 3,... of the function g x (x) can be calculated using the property that it is radial function and applying mean value formula () as follows: d n = C N u n (x )( n ) N + u n (x ) R R +s(r ) (r n )r s dr r N s +s(r )ν(r, n )dr To the first term of d n account the formula after substitution R = R and taking into +s(r ) N 4 n (r n )r s dr = s N 4 s Γ(s + ) ( ) s n δ n, (4) where δ n is a Kronecker we have d n = C u n (x ) s N 4 + u n (x ) R ( ) s n δ n C N u n (x )( n ) N Is (, n ) r N s +s(r )ν(r, n )dr Malaysian Journal of Mathematical Sciences 3

6 Jamaludin, N. A. and Ahmedov, A. where C = C N s Γ(s+), C N = N Γ( N ), then, C = N s Γ( N ) Γ(s+). Substitute d n into the Parseval formula : fgdx = r R = C s N 4 + f(x)r N s +s(r )dx n R c n u n (x ) ( c n u n (x ) ) s n C N c n u n (x )( n ) N Is (, n ) r N s +s(r )ν(r, n )dr. (5) We obtain the following representation for the Riesz means: Ef(x s ) = f(x)r N C s N s +s(r )dx 4 r R + C N c C s N n u n (x )( n ) N Is (, n ) 4 R c C s N n u n (x ) r N s +s(r )ν(r, n )dr 4 which completes proof of Lemma. 3. Estimation for P s (x, y, ) First two terms of Riesz means can be estimated by the methods in Alimov (97a). Let estimate third term. Lemma 3. We have P s (x, y, ) dx C N τ, τ > N. (6) 4 Malaysian Journal of Mathematical Sciences

7 An Almost Everywhere Convergence of Eigenfunction Expansions of Schrödinger Operator in L p Classes on Closed Domain uniformly for all x and >. Proof: Let expand P s (x, y, ) into eigenfunction expansions by eigenfunction {u n (x)}: P s (x, y, ) = c n (y)u n (x) where R c n (y) = C N u n (y) +s(r ) (r ) ( ) N ν(r, N n )r N dr +s Application of Parseval s formula gives: R P s (x, y, ) dx = c n (y) = C u n(y) N +s(r ) (r ) ν(r, N n )r N dr +s C R J u n(y) N N +s(r ) n< (r ) ν(r, N n )r N dr +s + C R J u n(y) N N +s(r ) (r ) ν(r, N n )r N dr +s n> (7) Let denote I = R +s(r ) (r ) ν(r, N n )r N +s dr where ν(r, n ) C ν ( n ) τ ω(r n ), C ν is a constant: Malaysian Journal of Mathematical Sciences 5

8 Jamaludin, N. A. and Ahmedov, A. ω(r n ) = (, < r n < r n ) N, r n > We have the following lemma to estimate I in (7): Lemma 4. Let s > N. For any r > we have I = R +s(r ) (r ) ν(r, N n )r N dr +s = O() τ+n n, n < τ+n, n. Proof: Firstly, let estimate I for the case n < ; r < : +s(r ) (r ) ν(r, N n )r N +s dr = C( n ) τ r N dr = C N (r ) N +s ( n ) τ ( ) N C N (r ) N +s ( n ) τ r N dr τ+n n. Secondly, let estimate I for the case n < ; r < n : n +s(r ) (r ) ν(r, N n )r N +s dr C = C( ( ) N+ n ) τ +s n n ( ) N C ( N s ) s+τ ( ) N+ +s n C ( N s ) τ+n n. (r ) (r ) ( N n ) τ r N dr +s ( ) τ ( ) N+ r N N s +s n dr = C n ( ) τ ( ) N C ( N s ) n r N s 3 dr 6 Malaysian Journal of Mathematical Sciences

9 An Almost Everywhere Convergence of Eigenfunction Expansions of Schrödinger Operator in L p Classes on Closed Domain Thirdly, let estimate I for the case n < ; r > n : n +s(r ) (r ) ν(r, N n )r N +s dr C ( ) N +τ ( ) N+ +s = C n r n r N +s (r n ( ) (r ) ( N n ) τ +s r n ( ) N r N dr r ) N r N dr ( ) N +τ ( ) N+ +s ( ) N = C r s +τ ( ) N+ +s [ r s dr = C n n s n ] n = C s ( ) N +τ s ( ) N+ +s C n s τ+n n. Fourthly, let estimate I for the case n > ; r < n : n +s(r ) (r ) ν(r, N n )r N +s dr C = C( n ) τ n r N dr = C N n (r ) N +s ( ) N+τ C τ+n. n N (r ) N +s ( n ) τ r N dr Fifthly, let estimate I for the case n > ; n r < : C n +s(r ) (r ) ν(r, N n )r N +s dr C (r n ( ) N +τ ( ) N = C r N dr = C n n n = N ) +s (r ) ( N n ) τ +s +τ [ r N+ ( ) [ N C +τ ( ) N+ ( ) N+ ] (N + ) n n ( ) N C +τ ( ) N+ (N + ) n N+ ] n ( ) N+τ C (N + ) n ( r n C τ+n N +. ) N r N dr Malaysian Journal of Mathematical Sciences 7

10 Jamaludin, N. A. and Ahmedov, A. Lastly, let estimate I for the case n > ; r > : +s(r ) (r ) ν(r, N n )r N +s dr C ( ) N +τ ( ) N+ +s = C n C r r N +s ( ) (r ) ( N n ) τ +s r n ( ) N r N dr r (r ) N ( ) N +τ ( ) N+ +s ( ) N = C r s +τ ( ) N+ +s [ r s dr = C n n s r N dr ] = C s ( ) N +τ ( ) N+ C τ+n. n s Finally, we obtain P s (x, y, ) dx = = C p n (y) C R u n(y) N n< u n(y) τ n + C +s(r ) (r ) ν(r, N n )r N dr +s n> u n(y) τ. (8) where C = [ C N + ] C ( N s) + C s and C = [ C N + ] C N+ + C s. We refer the following formula from Alimov and Joo (983b) n u n(x) = O() N, >, x. firstly, we have to estimate the first term of (8) as follows 8 Malaysian Journal of Mathematical Sciences

11 An Almost Everywhere Convergence of Eigenfunction Expansions of Schrödinger Operator in L p Classes on Closed Domain C n< u n(y) τ n C m= [ ] m= m τ m< n<m+ [ ] [ ] C m τ m N = C t τ+n dt = C [ t τ+n τ + N ] [ ] u n(y) = C N τ [ τ+ N ] C3 N τ. where C 3 = C N τ. Next we estimate the second term of equation (8), C n> u n(y) τ C m=[ ] m τ C t τ+n dt = C [ t N τ N τ m< n<m+ ] = C 4 N τ u n(y) C m=[ ] m τ m N where C 4 = Then, C N τ. P s (x, y, ) dx C N τ, τ > N, s > N, where C = C 3 + C 4. This is completing proof. Lemma 5. Let 4. Estimation in L p, p > P s (x, y) = sup P s (x, y, ). Malaysian Journal of Mathematical Sciences 9

12 Jamaludin, N. A. and Ahmedov, A. If α N = ɛ >, s = α + iδ, and y we have the relations as follows [P s (x, y)] dx C. Proof. We know P s (x, y, ) = µ P s (x, y, t) t (P µ s(x, y, t))dt P s (x, y, t) + t (P s(x, y, t)) dt Taking supremum, we have P s (x, y, ) µ P s (x, y, t) + t (P s(x, y, t)) dt We integrate both sides with respect to x, we obtain [ µ ] Ps (x, y, ) dx P s (x, y, t) + t (P s(x, y, t)) dt dx [ µ ] = P s (x, y, t) + t (P s(x, y, t)) dx dt We have the following estimate if we choose τ > N +, then we obtain Ps (x, y, ) dx [ ] P s (x, y, t) + t (P s(x, y, t)) dx dt = t N τ dt = C. Lemma 5 is proved. 3 Malaysian Journal of Mathematical Sciences

13 An Almost Everywhere Convergence of Eigenfunction Expansions of Schrödinger Operator in L p Classes on Closed Domain 5. Estimation in L As we established in the beginning for the Riesz means we have: E s f(x) = σ s f(x) + ξs f(x) + γs f(x). From Alimov (97a) s result we conclude that: ξ s+iδ f L() C s N f L (), f L (), σ s+iδ f Lp() C p f L p(), p. Collecting all estimations we obtain ( ) E s+iδ f Lp() C p + + f Lp(). p >, (p ). s N τ N Lemma 6. The functions (9) W s (x, f) = [ ] [ Ef(x) s E s f(x) d, Q s (x, f) = sup () E s f(x, t) dt ] Re(s) = α >. W s (x, f) dx C f(x) dx, Q s (x, f) dx C f(x) dx. () Proof. To prove () we note that Malaysian Journal of Mathematical Sciences 3

14 Jamaludin, N. A. and Ahmedov, A. [ µ [ µ + E s f(x, t) dt ] E s+m f(x, t) dt k= ] [ m µ E s+k f(x, t) E s+k f(x, t) dt m W s+k (x; f) + Q s+m+ (x, f). k= () ] From () and () the estimate m Q s (x, f) W s+k (x; f) + Es+m(x, f). k= Choose m such that m + α > N, it follows from (9) and () we obtain (). From Corollary, for any s = α + iδ,re(s) >, E s, (x, f) dx C f(x) dx. (3) This follows from the obvious relation E s ( ) = s Γ(s + ) [ Γ( s+ )] µ E s (t)t s (µ t ) s dt, E s (x, s, [ ) C µ ] E s (t) dt CQ s+ (x, f). (4) From (4) and () implies (3). 3 Malaysian Journal of Mathematical Sciences

15 An Almost Everywhere Convergence of Eigenfunction Expansions of Schrödinger Operator in L p Classes on Closed Domain 6. The analytic family of linear operators and L p -spaces We say that a function ϕ(τ), τ R, has admissible growth if there exist constants a < π and b > such that ϕ(z) exp(b exp a τ ). (5) Let A z be a family of operators defined for simple functions (i.e. functions which are finite linear combinations of characteristic functions of measurable subsets of ). We term the family A z admissible if for any two simple functions f and g the function ϕ(z) = f(x)a z g(x)dx is analytic in the strip Rez and has admissible growth in Im z, uniformly in Rez (this means that we have an estimate in Imz which is analogous to (5), with constants a and b independent of Rez). Theorem 7. Let A z be an admissible family of linear operators such that A iτ f Lp () M (τ) f Lp (), p, A +iτ f Lp () M (τ) f Lp (), p, for all simple functions f and with M j (τ) independent of τ and of admissible growth. Then there exist for each t, t, a constant M t such that for every simple function f holds A t f Lpt () M t f Lpt (), = t + t. p t p p We apply the interpolation theorem on an analytic family of linear operators in L p space. Let µ(x) be a measurable function on such as µ(x) µ < Malaysian Journal of Mathematical Sciences 33

16 Jamaludin, N. A. and Ahmedov, A. and α(z) = ( N + ɛ) z, Rez. We define an analytic family of linear operators: A z f(x) = E α(z) µ(x) f(x), Rez. We have A iy f(x) L() E α(iy) f(x) L() Be π y f L(). Secondly on the line z = + iy, we have A +iy f(x) Lp () E α(+iy) y f(x) Lp Deπ f Lp () (). Therefore by the interpolation we get E αt f(x) L M t f Lp(). (6) And if note that p = t + t, p p ( = t ), p then, t = p p > p ( = p ), then, 34 Malaysian Journal of Mathematical Sciences

17 An Almost Everywhere Convergence of Eigenfunction Expansions of Schrödinger Operator in L p Classes on Closed Domain α(t) = ( ) ( ) ( N N + ɛ t > + ɛ p ) ( > N p ), p >. Let us denote by Λf(x) the fluctuation of E s nf(x): Λf(x) = lim sup Es f(x) lim inf Es f(x). It is obvious, that Λf(x) E s f(x). From density of C L p, p, we have that for any ɛ > the function f L p, p can be represented as the sum of two functions: f(x) = f (x)+f (x), where f x C, and f L ɛ. Then we have Λf(x) = lim sup Es f (x) lim inf Es f(x) f Lp ɛ. Therefore almost everywhere ( ) Λf(x) =. So almost everywhere on for Riesz means of order s > N p, p, we have lim E s f(x) =. 7. Conclusion The almost everywhere convergence of the eigenfunction expansions of the Schrödinger operator by Riesz means of order s > N(/p /) in the classes of L p, p is established by estimating the maximal operators in the classes L and L and application of the interpolation Theorem for the family of linear operators. This result is extending the similar result for the eigenfunction expansions of the Laplace operator obtained in the work Alimov (97b). References Alimov, S. A. (97a). On summability of eigenfunction expansions from lp classes. Journal of Differential Equation, 4:3. Malaysian Journal of Mathematical Sciences 35

18 Jamaludin, N. A. and Ahmedov, A. Alimov, S. A. (97b). The summability of the fourier series of functions in l p in terms of eigenfunctions. Differentsialnye Uravnenija, 6: Alimov, S. A. and Joo, I. (983a). On the eigenfunction expansions associated with the schrodinger operator having spherically symmetrical potential. Journal of Acta. Math. Hung., 4(-): 9. Alimov, S. A. and Joo, I. (983b). On the riesz summability of eigenfunction expansions. Acta. Sci. Math., 45:5 8. Bastis, A. I. (983). Convergence almost everywhere of spectral expansions of functions l α.mat.zametki, 34(4) : Malaysian Journal of Mathematical Sciences