The Uniqueness Theorem for Inverse Nodal Problems with a Chemical Potential

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1 Iraia J. Mah. Chem. 8 4 December Iraia Joural of Mahemaical Chemisr Joural homepage: The Uiueess Theorem for Iverse Nodal Problems wih a Chemical Poeial SEYFOLLAH MOSAZADEH Deparme of Pure Mahemaics Facul of Mahemaical Scieces Uiversi of Kasha Kasha I R Ira ARTICLE INFO Aricle Hisor: Received 3 Februar 6 Acceped 6 April 6 Published olie 5 November 6 Academic Edior: Ali Reza Ashrafi Kewords: Boudar Value Problem Iverse Nodal Problem Eigevalues Nodal Pois ABSTRACT I his paper a iverse odal problem for a secod-order differeial euaio havig a chemical poeial o a fiie ierval is ivesigaed. Firs we esimae he odal pois ad odal leghs of differeial operaor. The we show ha he poeial ca be uiuel deermied b a dese se of odes of he eigefucios. 7 Uiversi of Kasha Press. All righs reserved INTRODUCTION There are ma problems i mahemaics chemisr phsics ad some egieerig scieces which are coeced o he secod-order differeial euaios. For eample i he process of he formaio of mehliodide CH 3 I b he biological ad phoochemical producio mechaisms i a biogeochemical module he followig euaio appears: dc dc P S Fair Sea A d z dz which describes he evoluio of mehl iodide coceraio c [mmolm 3 ] over ime uder producio P degradaio S air sea echage F as well as urbule verical diffusio Aν diffusio coefficie see [6]. Usig he separaio of variables echiue Correspodig Auhor address: s.mosazadeh@ashau.ac.ir DOI:.5/imc.6.398

2 44 MOSAZADEH oe euaio we ca rasform he euaio o he followig secod-order differeial euaio: A where λ is he specral parameer A is a real umber he poeial is real-valued. Euaio has a sigulari a he edpoi =. For oher eamples i uaum chemisr or uaum mechaics we refer o he uaum modelig of he hdroge aom or he Hellma euaio o fidig a approimaio for he simplified descripio of comple ssems which ca be rasformed o see also [ ]. Iverse problems associaed wih he euaio wih A= have various versios. The firs versio was sudied b Borg ad Leviso ad i is show ha he poeial ca be uiuel deermied from he give boudar codiio ad oe possible reduced specrum [5 8]. For he secod versio usig wo specra λ ad λˊ Marcheo uiuel deermied he poeial ad he correspodig boudar codiios []. Fiall Gelfad ad Levia proved ha uiuel deermied b he specral fucio []. Some iverse problems havig sigulariies or urig pois ad/or discoiui codiios were sudied b he above mehods i ma wors see [ ]. Noe ha i [] we cosidered a secod-order differeial euaio of Surm- Liouville pe havig wo urig pois ad sigulariies i a fiie ierval. The is asmpoic form of he soluios was sudied ad obaied he ifiie represeaio of he soluios of differeial euaio which plas a impora role i ivesigaig he correspodig iverse problem. I laer ears i some ieresig wors bu wihou sigulari iverse problems were ivesigaed usig a ew specral daa which are so-called odal pois ad heir correspodig iverse problems are so-called iverse odal problems. Mclaughli seems o have bee he firs o cosider his mehod for he oe-dimesioal Schrödiger euaios []. For oher wors see also [7 4 5]. I his wor we cosider he iverse odal problem associaed wih he sigular differeial euaio ad he Dirichle boudar codiio 3 o he ierval. We also assume ha L 4 where is a member of {34 }. The problem -3 has ifiiel ma orivial soluios. The values of λ for which here eis orivial soluios are so-called eigevalues ad heir correspodig orivial soluios λ are so-called eigefucios. All he eigevalues are real ad he se of he eigevalues is couabl ifiie ad also he eigevalues ca be arraged i icreasig order as follows 3...

3 The Uiueess Theorem for Iverse Nodal Problems wih a Chemical Poeial 45 such ha λ as. I he prese paper firs we obai he asmpoic formula for he eigevalues he odes of he eigefucios ad he odal leghs he secio. The we prove ha he se of he odal pois of he boudar value problem -3 is dese i he ierval ad he poeial ca be uiuel deermied from his ew id of specral daa see he Secio 3. ASYMPTOTIC FORMULA FOR NODAL POINTS We cosider he boudar value problem L=L defied b -3. Assume ha i A From [] we ow ha he euaio has wo soluios λ ad λ which are liearl idepede wih respec o ad also have he followig asmpoic form as λ : / / i i e [] e [] 6 i i i e [] e [] 7 4 where [] O. Therefore he soluio λ of he euaio uder he codiio = ca be wrie as a liear combiaio of ad. Also sice he boudar value problem L is self-adoi ad are eire i λ hus all of he eigevalues of L are real ad simple. I he case whe is odd i follows from 3 7 ha λ= λ ad he asmpoic form of he eigevalues as follows O. 8 Similarl i he case whe is eve we derive from 3 6 ha λ= λ ad also he eigevalues of L ma be calculaed as 8. For he boudar value problem L a aalog of Surm's oscillaio heorem is rue. More precisel he eigefucios = λ has eacl - simple zeros iside he ierval amel: The se.... L X : 9 is called he se of odal pois of he problem L. Also le I : [ be he h odal domai of he h eigefucio ad le : I ]

4 46 MOSAZADEH be he associaed odal legh. Iverse odal problems cosis i recoverig he poeial from he give se X L of odal pois or from a cerai is par. Now i he followig heorem we develop asmpoic epressios for odal pois ad he odal leghs =3 = - a which he eigefucio correspodig o he eigevalue λ of he problem L vaishes. Theorem. We cosider he euaio uder Dirichle boudar codiio 3. Le saisfies 4 he he odal pois of he problem L defied b -3 are O ad he odal leghs are O. Proof. Suppose ν= -/ ad is odd. The b 7-8 ad solvig λ= we approimae he odal pois of he form. Similarl i he whe is eve usig 6 8 ad from λ= we arrive a. Moreover O O O. Theorem speciall he relaio provide he sufficie codiios for he uiueess heorem i he e secio. 3 THE UNIQUENESS THEOREM I his secio we show ha he se of he odal pois of he form is dese i. The we prove a uiueess heorem for he soluio of he iverse odal problem associaed wih he boudar value problem L. Firs we cosider he euaio wih he boudar codiios w w w w.

5 The Uiueess Theorem for Iverse Nodal Problems wih a Chemical Poeial 47 I is easil show ha he soluio of he problem - is w si. Furhermore he eac eigevalues of he problem L defied b - are 3 ad heir correspodig eigefucios are w w si. 4 Sice for each {34 } here eis { } ad m { } such ha = + -m+ so accordig o 3-4 he se m... m... cosiss of all eigevalues of - ecep =. Moreover he eigefucio correspodig o he eigevalue = + -m+ is w si m so ha m/ + -m+ is a zero of he eigefucio w. Therefore he se of he odal pois of L is X : L m... m... m. 5 Lemma. The se X L defied b 5 is dese i []. Proof. For each fied Moreover { } we have m m ad for m= m m m m : m 6. m m m Hece here eiss a sufficiel large umber such ha for each we have

6 48 MOSAZADEH m. 7 Now le } such ha m : m/ + m+. The for each [] here eiss m { [ ] [ m m ] [ ]. 8 O he oher had he righ sides of euaios 6 ad 7 ed o zero as. This ogeher wih euaio 8 complees he proof. Theorem. The se of he odal pois of he boudar value problem L X L is dese i he ierval. Proof. I follows from 5 ha he odal pois of L have he form Thus usig we obai O. 9 B 9 ad Lemma we coclude ha X L is dese i. Now we prove he mai resul of his secio. Theorem 3. Cosider he boudar value problems defied b ad Dirichle codiio A i i. Le saisf he codiio 4 ad. The a.e.. Proof. Firs we cosider he case whe is odd i 5. Le be a arbirar fied umber i he ierval []. Sice he se of he odal pois X L defied i 9 is dese i he ierval b Theorem i follows ha here eiss a subseuece { } =3 such ha lim.

7 The Uiueess Theorem for Iverse Nodal Problems wih a Chemical Poeial 49 Le i i be he soluio of - wih he poeial i. The usig we derive ' ' d d. 3 Iegraig 3 from o : we ge ' ' d. 4 Sice he lef side of 4 is eual o zero for each {3 }. Thus d for We are doe if we ca show d. For his goal b 8 we have as. Hece ogeher wih ad 4 hese resuls impl lim d. 5 Moreover i follows from 7 ha here eiss a cosa C such ha for sufficiel large we have 3 si C. So si 6 Therefore b 56 we ge d. 7 Fiall sice was chose arbirar i he ierval [] ogeher wih 7 his ields = a.e.. I he case whe is eve Theorem 3 ca be proved similarl b 6 ad he same wa as above.

8 4 MOSAZADEH Theorem 3 show ha he soluio of he iverse odal problem associaed wih 3 he poeial fucio ca be uiuel deermied b a dese se of odes of he eigefucios. ACKNOWLEDGMENT This research is pariall suppored b he Uiversi of Kasha uder gra umber 4645/. REFERENCES. R. Kh. Amirov O ssem of Dirac differeial euaios wih discoiui codiios iside a ierval Uraiia Mah. J R. Amirov ad N. Topsaal O iverse problem for sigular Surm-Liouville operaor wih discoiui codiios Bull. Iraia Mah. Soc W. O. Amrei A. M. Hiz ad D. B. Pearso SurmLiouville Theor: Pas ad Prese Birhäuser Verlag Basel R. P. Bell The Tuel Effec i Chemisr Spriger-Sciece Uiversi Press Cambridge G. Borg Eie umehrug der Surm Liouvillesehe eigeweraufgabe Aca Mah J. P. Bod Surm-Liouville eigevalue problems wih a ierior pole J. Mah. Phsics Y. H. Cheg C. K. Law ad J. Tsa Remars o a ew iverse odal problem J. Mah. Aal. Appl W. Eberhard G. Freilig ad K. Wilce-Soeber Idefiie eigevalue problems wih several sigular pois ad urig pois Mah. Nachr G. Freilig ad V. Yuro O cosrucig differeial euaio wih sigulariies from icomplee specral iformaio Iv. Prob G. Freilig ad V. Yuro O he deermiaio of differeial euaios wih sigulariies ad urig pois Resuls Mah G. Freilig ad V. Yuro Iverse problems for differeial operaors wih sigular boudar codiios Mah. Nachr I. M. Gelfad ad B. M. Levia O he deermiaio of a differeial euaio from is specral fucio Amer. Mah. Soc. Tras D. M. Haalad ad R. T. Meer Reacio of eied iodie aoms wih mehl iodide rae cosa deermiaios I. J. Chem. Kieics O. Hald ad J. R. McLaughli Soluios of iverse odal problems Iv. Prob

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