Reconstruction of the Diffusion Operator from Nodal Data

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1 Recostructio of the Diffusio Operator from Nodal Data Chua-Fu Yag Departmet of Applied Mathematics, Naig Uiversity of Sciece ad Techology, Naig 94, Jiagsu, People s Republic of Chia Reprit requests to C.-F. Y.; chuafuyag@yahoo.com.c Z. Naturforsch. 65a, 6 (); received November 7, 8 / revised April 9, 9 I this paper, we deal with the iverse problem of recostructig the diffusio equatio o a fiite iterval. We prove that a dese subset of odal poits uiquely determie the boudary coditios ad the coefficiets of the diffusio equatio. We also provide costructive procedure for them. Key words: Diffusio Operator; Iverse Nodal Problem; Recostructio Formula. 99 Mathematics Subect Classificatio: 34A55, 34L4, 35K57. Itroductio Iverse spectral problems cosist i recoverig operators from their spectral characteristics. Such problems play a importat role i mathematics ad have may applicatios i atural scieces (see, for example, 6]). I 988, the iverse odal problem was posed ad solved for Sturm-Liouville problems by J. R. McLaughli 7], who showed that the kowledge of a dese subset of odal poits of the eigefuctios aloe ca determie the potetial fuctio of the Sturm-Liouville problem up to a costat. This is the so-called iverse odal problem 8]. Iverse odal problems cosist i costructig operators from the give odes (zeros) of the eigefuctios. Recetly, some authors have recostructed the potetial fuctio for geeralizatios of the Sturm-Liouville problem from the odal poits (for example, 7 ]). I this paper we cocer ourselves with the recostructio of the diffusio equatio from odal data. The ovelty of this paper lies i the use of a dese subset of odal poits for the eigefuctios as the give spectra data for the recostructio of the diffusio equatio.. Mai Results The problem of describig the iteractios betwee collidig particles is of fudametal iterest i physics. Oe is iterested i collisios of two spiless particles, ad it is supposed that the s-wave scatterig matrix ad the s-wave bidig eergies are exactly kow from collisio experimets. With a radial static pote- tial V(x) the s-wave Schrödiger equatio is writte as y +E V(E,x)]y =, where V(E,x) has the followig form of the eergy depedece: V(E,x)= EP(x)+Q(x). Before givig the mai results of this work, we metio some properties of the diffusio equatio. The diffusio operator is writte as Ly] y +q(x)+λ p(x)]y, x,], where the fuctio q(x) L,] ad p(x) W,]. Let λ be the spectrum of the problem Ly]=λ y, y (,λ ) hy(,λ )=, y (,λ )+Hy(,λ)=, () h y() + H y() + y (x) + q(x) y(x) ]dx >, h,h R + ad y(x,λ ) the eigefuctio correspodig to the eigevalue λ. It is well kow that the sequece {λ : Z} satisfies the classical asymptotic form, ] λ = + c + c + c,, () / / $ 6. c Verlag der Zeitschrift für Naturforschug, Tübige Uautheticated Dowload Date /3/8 :6 PM

2 C. F. Yag Recostructio of the Diffusio Operator from Nodal Data where c, < ad G c = x ( ) p(t)dt p(x)dx, c = { h + H + } (3) + ( )c ] (7) x p(t)cos( + c )t]dt, q(x)+p (x)]dx. H The solutio of the equatio Ly]=λ l ], (8) y with the iitial coditios y()=ady ()=h is K (h + H) c ] + p (x)dx y(x,λ )=cos λ x α(x) ( ) + 3 l + p(t)dt + A(x,t) cos λt x dt ( ) + c + B(x,t) si λt dt, (9) + p(t)cos( + c )t]dt x ] where the kerels A(x,t),B(x,t) L + c + (,],]) p(t)dt. x ad The mai theorems are the followig. x α(x)= p(t)dt. (4) Theorem.. For x,]. Let{x } X be chose such that x Let < x <... < x <... < x = x. The the followig fiite < be the its exist odal poits of the -th eigefuctio y(x,λ ).Iother words, y(x,λ )=, =,,,. Let be I g (x) F =, g (x) G () (x,x + ) ad the odal legth l be ad l = x + x. (5) g (x)= p(x)dx c x, Defie x = adx =. We also defie the fuctio (x) to be the largest idex such that x x. g (x)= h + q(x)dx c p(x)dx () Thus, = (x) if ad oly if x x,x + ). Defie X {x } +(c c )x., =,. X is called the set of odal poits of the diffusio operator (). Let us ow formulate a uiqueess theorem ad provide a costructive procedure for the solutio of the Uder the coditio that h, H, adp(x) i () are kow the paper 3] gave the recostructio of the potetial fuctio q(x) of the diffusio operator by odal Theorem.. Let X X be a subset of odal poits iverse odal problem. data. which is dese o (,). The, the specificatio of Whe we solve the iverse problem from the spectra data, the obvious questio occurs: What if h, H, X uiquely determies p(x) p(x)dx ad q(x) ad p(x), q(x) i () are all ukow? Our motivatio q(x)dx o (,), ad the coefficiets h ad H of the boudary coditios i (). p(x) i cosiderig odal poits of eigefuctios as data is p(x)dx ad our desire to obtai more iformatio o the diffusio operator. I this paper, we prove that a dese sub- costructed via the formulae q(x) q(x)dx, ad the umbers h ad H ca be set X of odal data uiquely determie the coefficiets p(x) p(x)dx = d q(x) ad p(x), adh ad H i (). dx g (x), Defie F x ( ), (6) h = g (), H = 3h p (x)dx g (), () Uautheticated Dowload Date /3/8 :6 PM

3 C. F. Yag Recostructio of the Diffusio Operator from Nodal Data ad q(x) + h + H q(x)dx = d dx g (x) + c p(x)+ p (x)dx c, (3) where g (x) ad g (x) are calculated by (). Theorem.3. Give x,]. Let{x } X be chose such that x = x.theh coverges to p(x) p(x)dx a. e. x,] ad i L (,)-orm, ad K coverges to q(x) q(x)dx a. e. x,] ad i L (,)-orm. Usig oly the odal data ad the costats, amely p(x)dx ad q(x)dx, we ca recostruct these ukow coefficiets. Our recostructio formulae are direct ad automatically imply the uiqueess of this iverse problem. 3. Proofs Before provig the theorems we shall derive some results that will be used later o to establish our pricipal results. For odal poits x (the zero poits of the -th eigefuctio), the asymptotic formula for odal poits ( ) follows from 3] x = ( ) λ h λ + λ + cos(λ t)]q(t)+λ p(t)]dt ( ) + O. (4) λ 4 Takig () ito accout ad usig Taylor s expasios for ( + x) α ad cosx, we shall obtai the refiemet of odal poits. Simple calculatios show that ( λ = + c + c + c ), = c + c c 3 c, 3 + O ( λ = + c + c = c 3 + O + c, ( 4 ) ), ( ) 4, (5) (6) ad ( cos(λ t)=cos + c + c + c ) ], t ( c = cos( + c )t]cos + c ) ], t ( c si( + c )t]si + c ) ], t ( )] = cos( + c )t] c t + o si( + c )t] ( c + c, ) t + o ( = cos( + c )t] c t cos( + c )t] (c ( ) + c, )t si( + c )t] + o. (7) )] Pluggig these expressios for (5), (6), ad (7) ito (4) ad usig the Riema-Lebesgue Lemma ad Lemma 3., we obtai asymptotic formulae for odal poits as uiformly i, =, : x = ( ) p(t)dt ( )c p(t)cos( + c )t]dt h c p(t)dt q(t)dt c p(t)cos( + c )t]dt (8) q(t)cos( + c )t]dt (c + c, ) tp(t)si( + c )t]dt + (c c c, )( ) 3 + O ( ) 3. We ote that the set X is dese o (,). By the asymptotic formula (8) for odal poits above, usig the Riema-Lebesgue Lemma ad Lemma 3., we ca obtai the asymptotic expasio of odal legths as follows: l = x + x p(t)dt c p(t)cos( + c )t]dt + O ( ) 3. (9) I the above results, the order estimate is idepedet Uautheticated Dowload Date /3/8 :6 PM

4 C. F. Yag Recostructio of the Diffusio Operator from Nodal Data 3 of. As a result, l = + o ( ). () Lemma 3.. Suppose that f L (,). The for x (,), with = (x), + =, () x ad + x + x f (t)cos( + c )t]dt =, () f (t)si( + c )t]dt =. (3) Proof. We first show that the result () holds if f C,].LetM = max x,] f (x). The + x ( ) M(x + x )=Ml = O. Therefore, if f C,],the + x ca be arbitrarily small for large which implies () is true. Sice C,] is dese i L (,), for ay f L (,) there exists a sequece f k C(,) that coverges to f i L (,).Now + x + (4) ( f (t) f k (t))dt + x + f k (t)dt. x For ay ε >, fix k large eough ad large eough such that the first term of the right-had side i (4) is small tha ε, together with (). For all large eough, the last term is small tha ε by above. Hece, for f L (,), () is true. Usig the same method above, we ca verify that () ad (3) are true. The proof is fiished. Lemma 3.. Suppose that f L (,). The, with = (x), + x x = f (x) a. e. x,] (5) ad + x f (x) as. (6) Proof. Note that x + x = λ + + λ = O() x + x + λ + x + x. Applyig the result i, ]: Suppose that f L (,), with = (x), there hold ad λ x + = f (x) a. e. x,] x λ x + f (x) as, x we coclude that (5) ad (6) hold. The proof is complete. Now we ca give the proofs of the theorems i this paper. Proof of Theorem. Usig the asymptotic expasio for odal poits i (8) we get F = p(t)dt ( )c + h c c + (c + c, ) p(t)cos( + c )t]dt p(t)dt + q(t)dt p(t)cos( + c )t]dt q(t)cos( + c )t]dt tp(t)si( + c )t]dt + (c c c, )( ) + O ( ) Uautheticated Dowload Date /3/8 :6 PM

5 4 C. F. Yag Recostructio of the Diffusio Operator from Nodal Data ad G = h c p(t)dt + + (c c c, )( ) c p(t)cos( + c )t]dt + q(t)cos( + c )t]dt q(t)dt ( ) (c + c, ) tp(t)si( + c )t]dt + O. Usig the Riema-Lebesgue Lemma we obtai as F = p(t)dt ( )c + o() (7) ad G = h c p(t)dt + q(t)dt + (c c )( ) + o(). (8) Also, the fact that x = x implies that ( ) = x from (8), ad x f (x)dx = f (x)dx. From (7) ad (8) it follows that ad F g (x)= p(x)dx c x (9) G g (x)= h + (3) q(x)dx c p(x)dx +(c c )x. This proves the theorem. Proof of Theorem. Now give a odal subset X, by Theorem. we ca build up the recostructio formulae. Formulae () ad (3) ca be derived directly from (9) ad (3) stepwise. We preset the followig procedure. Step : Takig derivatives i (9) we obtai p(x) p(x)dx = d dx g (x). (3) Step : Takig x = i(3),itfollowsg ()= h. Hece, h = g (). (3) Step 3: After h ad p are recostructed, takig x = i (3), it follows g ()= h + q(x)dx c p(x)dx +(c c ) = h + q(x)dx c + c h H q(x)+p (x)]dx = 3h H p (x)dx, which yields H = 3h p (x)dx g (). (33) Step 4: After h,h,adp are recostructed, we get thus, d dx g (x)= q(x) c p(x)+c c = q(x) c p(x)+c h + H q(x)dx p (x)dx, q(x) q(x)dx = d dx g h + H (x)+ + c p(x) + p (x)dx c. (34) Sice each odal data oly determie a set of recostructio formulae which oly deped o odal data, the uiqueess holds obviously. Uautheticated Dowload Date /3/8 :6 PM

6 C. F. Yag Recostructio of the Diffusio Operator from Nodal Data 5 Proof of Theorem.3 Usig the asymptotic expasio for the odal legth i (9) ad usig the Riema-Lebesgue Lemma ad Lemma 3. we obtai as ad H = K = + x + x p(t)dt c + o() (35) q(t)dt From this we get H p(x) x + p(t)dt p(x) + o(). x q(x)dx + o(). (36) Usig Lemmas 3. ad 3. yields H p(x) = a.e.x,]. Moreover, H p(x) dx x + p(t)dt p(x) dx + o(). x Lemma 3. tells us H p(x) dx =. Thus, H coverges to p(x) p(x)dx a. e. x,] ad i L (,) orm. Also, we get K q(x) x + q(t)dt q(x) + o(). x Usig Lemma 3. yields K (x) q(x) = a. e. x,]. Moreover, K q(x) dx x + q(t)dt q(x) dx + o(). x Lemma 3. tells us K q(x) dx =. Therefore K coverges to q(x) q(x)dx a. e. x,] ad i L (,) orm. The proof of theorem is complete. Ackowledgemets The author ackowledges helpful commets ad suggestios of the referees. ] G. Freilig ad V. A. Yurko, Iverse Sturm-Liouville Problems ad Their Applicatios, NOVA Sciece Publishers, New York. ] B. M. Levita, Iverse Sturm-Liouville Problems, VNU Sciece Press, Utrecht ] V. A. Marcheko, Sturm-Liouville Operators ad Their Applicatios, Naukova Dumka, Kiev 977; Eglish trasl.: Birkhäuser, ] J. Pöschel ad E. Trubowitz, Iverse Spectral Theory, Academic Press, Orlado ] V. A. Yurko, Iverse Spectral Problems for Differetial Operators ad Their Applicatios, Gordo ad Breach, Amsterdam. 6] V. A. Yurko, Itegral Trasforms Spec. Fuct., 4 (). 7] J. R. McLaughli, J. Diff. Equa. 73, 354 (988). 8] O. H. Hald ad J. R. McLaughli, Iverse Problems 5, 37 (989). 9] P. J. Browe ad B. D. Sleema, Iverse Problems, 377 (996). Uautheticated Dowload Date /3/8 :6 PM

7 6 C. F. Yag Recostructio of the Diffusio Operator from Nodal Data ] Y. T. Che, Y. H. Cheg, C. K. Law, ad J. Tsay, Proc. Amer. Math. Soc. 3, 39 (). ] Y. H. Cheg, C. K. Law, ad J. Tsay, J. Math. Aal. Appl. 48, 45 (). ] S. Currie ad B. A. Watso, Iverse Problems 3, 9 (7). 3] H. Koyubaka ad E. Yilmaz, Z. Naturforsch. 63a, 7 (8). 4] C. K. Law, C. L. She, ad C. F. Yag, Iverse Problems 5, 53 (999); Errata, 7, 36 (). 5] C. K. Law ad J. Tsay, Iverse Problems 7, 493 (). 6] C. K. Law ad C. F. Yag, Iverse Problems 4, 99 (998). 7] C. L. She, SIAM J. Math. Aal. 9, 49 (988). 8] C. L. She ad C. T. Shieh, Iverse Problems (). 9] C. T. Shieh ad V. A. Yurko, J. Math. Aal. Appl. 347, 66 (8). ] X. F. Yag, Iverse Problems 3, 3 (997). ] M. G. Gasymov ad G. Sh. Guseiov, SSSR Dokl. 37, 9 (98). ] G. Sh. Guseiov, Soviet Math. Dokl. 3, 859 (985). 3] W. H. Steeb, W. Strampp, Physica D3, 637 (98). Uautheticated Dowload Date /3/8 :6 PM