0109, Tbilisi, Georgia. St. Andrea the First-called Georgian University of Patriarchy of Georgia, Chavchavadze Ave. 53a, 0162, Tbilisi, Georgia

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1 Singul Behvio of the Lplce Opeto in ol Spheicl Coodintes nd Some of Its Consequences fo the Rdil Wve Function t the Oigin of Coodintes Anzo A. Khelshvili, nd Teimuz. Ndeishvili,3 Institute of High Enegy hysics, Iv. Jvkhishvili Tbilisi Stte Univesity, Univesity St. 9, 9, Tbilisi, Geogi St. Ande the Fist-clled Geogin Univesity of tichy of Geogi, Chvchvdze Ave. 53, 6, Tbilisi, Geogi E-mil: nzokhelshvili@homil.com 3 Iv. Jvkhishvili Tbilisi Stte Univesity, Fculty of Exct nd Ntul Sciences, Chvchvdze Ave. 3, 79, Tbilisi, Geogi E-mil: teimuz.ndeishvili@tsu.ge Abstct: Singul behvio of the Lplce opeto in spheicl coodintes is investigted. It is shown tht in couse of tnsition to the educed dil wve function in the Schodinge eqution thee ppes dditionl tem consisting the Dic delt function, which ws unnoted duing the full histoy of physics nd mthemtics. The possibility of voiding this contibution fom the educed dil eqution is discussed. It is demonstted tht fo this im the necessy nd sufficient condition is equiement the fst enough flling of the wve function t the oigin. The esult does not depend on chcte of potentil is it egul o singul. The vious mnifesttions nd consequences of this obsevtion e consideed s well. The conestone in ou ppoch is the ntul equiement tht the solution of the dil eqution t the sme time must obey to the full eqution. Keywods: Lplcin, Delt Function, dil eqution, boundy condition. ACS numbes: w, 3.65.C, 3.65.Ge, 3.65.T

2 . INTRODUCTION The im of this ppe is to suvey the singul behvio of the Lplcin in spheicl coodintes. Lplcin is encounteed lmost in ll disciplines of Theoeticl physics s well s in mthemticl physics. In this ticle ou ttention is pid mostly to the Schodinge eqution, which in the Ctesin coodintes hs fom (in units = c = ) Δ+ V ( ) ψ ( ) = Eψ ( ); m () whee Δ = + + () x y z is Lplcin. In spheicl coodintes the vibles e septed nd the totl wve function is epesented s m u( ) m ψ ( ) = R( ) Yl ( θϕ, ) = Yl ( θϕ, ) (3) The Lplcin is lso ewitten in tems of these coodintes nd fte the substitution of Eq. (3) into the Eq. () we deive the dil equtions d d l( l+ ) + R ( ) + R ( ) + V ( ) R ( ) = E R( ) (4) m d d m o d l( l+ ) + + V ( ) u( ) = Eu( ) (5) md m All of this is well known fom the clssicl textbooks on quntum mechnics, electodynmics nd etc. We disply them hee fo futhe pcticl puposes. It will be shown below tht the sttus of the Eq. (5) is poblemtic. Fom both mthemticl nd physicl points of view it is vey impotnt tht the solutions of dil equtions wee comptible with the full Schodinge eqution ().This is veblly mentioned in books, not only elie [,] but lso in the moden ones [3]. Fo exmple, Dic [] wote: Ou equtions... stictly speking, e not coect, but the eo is esticted by only one point =.It is necessy pefom specil investigtion of solutions of wve equtions, tht e deived by using the pol coodintes, to be convince e they vlid in the point = (p.6) We e sue tht mthemticins new bout this poblem (singulity of the Lplcin) fo long time, but chcte of singulity neve been specified. It ws lwys undelined in mthemtics tht > stictly, but = is not somehow pominent point fo the 3-dimensionl eqution. Theefoe efinement of the behvio of the dil wve function t tht point hs bsic mening by ou opinion. The fist ppes [4-7] on this poblem ppeed ecently lmost in pllel. Becuse of eltive novelty of this subject below we will tke some ttention to its substntition. To complete the pictue we fist discuss biefly the essence of this poblem nd then some of its ppliction will be consideed. In the teching books nd scientific ticles two methods wee pplied in the tnsition fom Eq. (4) to Eq. (5):

3 . Substitution u( ) R( ) = (6) into the Eq. (4) o. Replcement of the diffeentil expession in the penthesis of Eq. (4) s [8-] d d d + (. ) (7) d d d We demonstte below tht in both cses the mistkes wee mde. Becuse ll the pincipl infomtion is concentted in the Lplce opeto, we begin by considetion the clssicl Lplce eqution in the vcuum (electosttic eqution). THE LALACE EQUATION Let us conside the Lplce eqution in vcuum ϕ ( ) = (8) which in Ctesin coodintes hs the fom ϕ( ) = + + ϕ ( xyz,, ) = (9) x y z This eqution my be solved simply by seption of vibles. The solution hs the fom [] ± iαx ± iβy ± α + β z ϕ( x, y, z) = e e e () Clely the solution is egul eveywhee nd t the oigin is constnt ϕ ( ) = const () Thee e nothe foms of solution of Eq.(9) depending on ltente wys of seption, but ll of them give the constnt vlues t the oigin. Now, let us find the spheiclly symmetic solution. The coesponding eqution is witten s [8] d d + () = ϕ () d d Cetinly, it ws possible pssing to spheicl coodintes in Eq. (9), substituting (3) nd tking zeo ngul momentum. We ll ive gin to the Eq. (). The opeto in penthesis of Eq. () often is ewitten ([8], Ch., [9] etc.) ccoding to (7) nd subsequently, eqution () tkes the fom d ( ϕ ) = (3) d the solution of which is u () ϕ = + b (4) But, detemining fom hee the function In the fundmentl book of J.D.Jckson [] this eltion is even exhibited on the cove-pge in the list of the most fundmentl foms! 3

4 b ϕ = + (5) does not obey to Eq. (), becuse d d ( 3 4 ) + = πδ ( ) (6) d d i.e. the function (5) is the solution eveywhee except the oigin of coodintes. It does not stisfy the boundy vlue () s well. Wht hppens? It seems tht we mde n illegl ction somewhee (see, Feynmn [8]). It is possible to conside this poblem by nothe wy lso, nmely, following to the substitution (6), tke u( ) ϕ ( ) = (7) in ode to emove the fist deivtive tem fom the Eq. (). Then we obtin d d d d du d () () = u u + (8) d d d d d d The lst tem cncels the fist deivtive tem in the fist penthesis nd thee emins d u d d + u = + (9) d d d but, ccoding to Eq. (6), insted of Eq. (3), it follows du ( 3 4πδ ) ( ) u( ) = () d The ppence of the delt function hee is unexpected. Comping this one with Eq. (3) we conclude tht the epesenttion of the Lplce opeto in the fom (7) is not vlid eveywhee. The coect fom is [5, 7] d d d ( 3 + = ( ) 4πδ ) ( ) () d d d This expession defines the fom of the Lplsin pecisely eveywhee including the oigin of coodintes. It is evident, tht fte substitutions d d d + ϕ ( ϕ ) nd u= ϕ, () d d d the solution ϕ = u/, obtined fom the eqution (3), neve stisfies to the initil eqution () eveywhee. By unknown fo us esons this simple fct styed unnoted till now nd in ll ppes s well s in ll books the expession (7) ws used. As we mde cle up bove, in this cse the obtined solution (5) looks like if thee is point souce t the oigin. Howeve it is not so mthemtic eson is tht in spheicl coodintes the point = is bsent. The Jcobin of tnsfomtion to spheicl coodintes hs fom J = sinθ nd is singul t points = ndθ = nπ ( n=,,,... ). Singulity in ngles is eliminted by equiements of continuity nd uniqueness, which led to spheicl hmonics Υ ( θ, ϕ). As egds of the dil vible, thee is no such estiction fo. m l 4

5 it. Theefoe if we wnt to deive the solution vlid eveywhee, we e foced to include the delt function into the considetion. It must be noted tht the ppence of the delt function in the Lplce eqution ws discussed n n lso in ticle [6], whee the diffeence between spces R nd R /{ } is studied fom the positions of distibution theoy. The question is: how to fomulte the poblem in such wy tht to emin ll esults deived elie fo the centl potentils with the id of tditionl educed dil eqution (5) contining the second deivtive only? One of the esonble wys is the following: Becuse in ( 3 spheicl coodintes ) δ ( ) () δ = [], the Eq. () cn be educed to 4π d u δ ()() u = (3) d o d u u() δ () = (4) d Let us equie tht the dditionl tem does not pesent i.e. u ( ) = (5) Moeove the delt function be ovecome if t lest limu( ) (6) Then, owing to the eltion δ () =, the ext tem flls out nd the stndd eqution (3) follows. Let us look fist wht the condition (5) gives in bove consideed solution (4). Requiing (5), it follows b =, i.e. u = nd ϕ ( ) = = const. Hence we obtin the coect, consisting with the full eqution (8) vlue (). It is consisting lso with the el physicl pictue. Theefoe in the educed dil eqution (5) we must conside only such clss of solutions, which vnish t the oigin. The othe entie boundy conditions loss the physicl mening nd hve only mthemticl inteest. It is pecisely the min esult of this section the eqution (5) gives the consistent with the pimy eqution in Ctesin coodinte s solution only if the estiction (5) is stisfied. Appence of this condition is puely geometicl (not dynmicl) tefct. In shot wods, the Eq. (5) nd the condition (5) ppe simultneously. 3. THE RADIAL SCHRODINGER EQUATION AND u () As n exmple let us conside the dil Schodinge eqution (4) Afte the substitution (6), ccoding to bove mentioned bout the Lplce opeto, we obtin the following fom of this eqution d u l l + δ u d () ( ) () ( ) () [ ()] ( ) u u + m E V To single out the tue singulity let us multiply this eqution on nd integte by sphee of smll dius. We deive () = d u u( ) d l( l + ) d u( ) + ( me V ( ) ) u( ) d = d Fom hee we detemine (7) d in (8) 5

6 () d u u( ) u( ) = d l( l + ) d + ( me V () ) u( ) d (9) d substitute hee the symptotic fom of wve function t the oigin u Becuse of smllness of () s nd the potentil s g V( ) ; n> n (3) Then the integtion in Eq. (9) my be esily pefomed nd we obtin s ( ) ( s ) l( l + ) s me s+ mg s+ n u = + s s + s + n (3) We must emove the ext delt tem fom Eq. (7), becuse othewise we do not get the usul fom of dil eqution (5). If we etin () in the Eq. (8) then thee e 3 possible vlues fo it:, is finite nd u =. Note tht ll the enumeted vlues do not contdict to nomliztion ( ) (3) u u ( ) = u ( ) condition ne the oigin ud<, but not ll of them e useful. The fist vlue is pefeble mong them, becuse in opposite cses - finite u ( ) will give const R t the oigin nd in Eq. (7) the delt function eppes gin. Theefoe this solution will not obey to full Schodinge eqution. The lst vlue, u ( ) = of couse is uncceptble, becuse to hve n infinite numbe in eqution is senseless. Thee emins only one esonble vlue, Eq. (5). Moeove this estiction tkes plce in spite of the potentil is egul o singul. Singulity of the potentil effects only on the lw of tuning of u() to zeo. This follows fom the eltion (3) s ll the exponents hee must be positive. We ll hve theefoe s >, s + >, s + n > It follows fom the lst inequlity tht when the index of singulity of potentil n inceses, the index of wve function behvio s must lso incese. Moeove we must hve s in ode the wve function t the oigin ovecomes the delt function in the temu() δ (). Theefoe thee emins the finl llowed inequlities s, s+ n> (33) If in ddition we equie tht this poduction be distibution, it becomes necessy tht u( ) be n infinitely smooth function [, 3], i.e. in Eq. (3) we must hve s nd the index s is n intege numbe. Thus the wve function must be sufficiently egul one t the oigin. This fct my hve f eching consequences. 4. SOME ALICATIONS 6

7 The fist question, tht ppes hee, is the following: unde wht conditions cn we mintin the stndd fom of educed wve eqution? Bsing on the pevious considetions we suppose tht the eqution in the stndd fom (5) tkes plce nd clify fo which potentils it hppens, i.e. when we cn stisfy the estiction (5)? 4A) Regul potentils Let us conside fist the egul potentils lim V ( ) = Then in the Schodinge eqution (5) the leding symptotic t the oigin is detemined by s s = l l+. So. centifugl tem nd the chcteistic eqution tkes the fom ( ) ( ) l + l u ~ c + c ; l =,,... (35) We must etin only the fist solution becuse now s = l + nd the deived epesenttion is stisfied (s is n intege numbe!). At the sme time the second solution with s = l must be ignoed even fo l = [4]. The second solution does not stisfy to the 3-dimensionl Schodinge eqution (), s fte its substitution the Lplcin poduces l -fold deivtives of delt function [4]. Resuming bove sying we conclude tht in cse of egul potentils (34) the dil eqution (5) emins, becuse in this cse the ll equiements e elized nd consequently, the esults obtined elie by this eqution emin vlid without ny chnges! 4B) Wekly singul tnsitive potentils Let us now conside potentils tht e intemedite between singul nd egul ones, so-clled wekly-singul potentils of the fom lim V = V = (36) ( ) const Hee V > coesponds to the ttction, while V < - to epulsion. Now the behvio of u() t the oigin is + + = + u ~ d d ; l mv > (37) In ode tht the usul eqution (5) will still emin, ccoding to Eq.(33) we must hve ( s ), i.e. fo ll l, including l =, nd t the sme time, ccoding to equiement of the distibution theoy / + = N ; N =,,3... So it esults stnge quntiztion of V, which is lso senseless. It follows tht in this cse thee e no solutions except fo quntized V. We see tht the second solution in Eq. (37) must be discded. Note tht in scientific litey thee is no definite viewpoint concening to this. (see, e.g. book by R. Newton [5] nd vious moden ticles [6,7]). Theefoe the bove mentioned deivtion is fist coect one. 4C) The poblem of Self-djoint extention (SAE) Lst decdes the poblem of self-djoint extension (SAE) of dil Hmiltonin (34) 7

8 H ( + ) d l l + + mv d (38) ws often consideed in cses of singul potentils, like bove one. In this poblem the essentil ole plys the behvio of dil wve functions u() t the oigin of coodintes. Fo exmple, the condition of self-djointicity of Hmiltonin (38) hs fom [8] ˆ ˆ u H u d u H ud = lim [ u ( ) u ( ) u ( ) u ( )] = (39) whee ( ) ( ) u R,, ( ) e two linely independent solutions of the educed dil eqution (5) coesponding to diffeent eigenvlues of the Hmiltonin (38).Thee wee consideed vious boundy conditions such s ones of Diichlet, Neumn, nd the most genel condition of Robin [9]. While s we mde cle bove only the Diichlet condition (5) is ight In most ticles in couse of discussion of SAE pocedue with the Hmiltonin (38) uthos tke ttention only on sque integbility of the wve function []. But it is not sufficient in ll lim R, is not cses. Still W.uli [] noted tht the eigenfunction, fo which ( ) pemissible (even if R R d exists fo such functions). The sme is confimed in the moe moden books (fo exmple, in [3, pf 5] the utho sys: It cn be shown tht the condition u()= follows fom the equiement tht the solution of the Schodinge eqution in spheicl coodintes must be lso solution when the eqution is witten in Ctesin coodintes ). But unfotuntely, this thesis is not shown egully in this book, especilly fo singul potentils. If we impose the boundy condition with the indices s we must estict ouselves only by the fist (egul) solution, i.e. d = (See, Eq. (37)). Then the dil Hmiltonin (38) becomes self-djoint one utomticlly nd the SAE is not needed. As fo the fist solution the condition / is chieved only if l ( l + ) > mv (4) i.e. fo l = only V < is pemissible nd s egds of othe dmissible vlues, fom the condition / + = N it follows stnge quntiztion of V ( l + / ) ( N / ) V = ; N =,,3... m (4) Hence even fo such simple singul potentil (36) the eqution (5) meets the seious physicl difficulties. We do not conside hee the othe, moe singul potentils, becuse the genel tendency is obvious. The Hmiltonin (38) by itself is lwys self-djoint on the egul solutions, stisfying to (5), s it follows fom the condition (39) nd estictions (33) fo ny singul potentils. Fo ll othe boundy conditions the Hmiltonin (38) will not hve be eltion to physics, becuse this fom of Hmiltonin emeges only togethe with condition (5). We conclude tht the educed dil eqution (5) my be pplied fo ll egul potentils, nevetheless fo singul potentils one must wok only with the totl dil eqution (4) nd, consequently, use the full dil Hmiltonin d d l ( l+ H ) ( ) R = + + mv (4) d d 8

9 but sech fo egul solutions only. In the ticle [4] we hve shown tht fom the finiteness of the diffeentil pobbility ( ) dw = R d nd the time independence of the nom it follows tht R() is less singul t the oigin, thn / o lim R = u =. which, evidently is consistent to ( ) Moeove in cse of fulfillment this condition the dil eqution (4) fo full dil function R() is equivlent to the Schodinge eqution (). This equivlence tkes plce only on nonsingul solutions. In othe wods, the Eq. (4) is equivlent to the 3-dimensionl Schodinge eqution only fo egul solutions. This ws poved lso in ppe [6] in the fmewok of the distibution theoy. Fo demonsttion of pincipl diffeence between the full nd educed dil Hmiltonins let us conside now the sme poblem in view of the full dil function R(). The condition (43) is the only boundy estiction fo it, which is not so sevee. Theefoe thee ppes possibility to etin the second solution s well in the cse of singul potentils behving like (36) The following sttement cn be poved: Theoem.The dil Schodinge eqution (4) except the stndd (non-singul) solutions hs lso dditionl solutions fo ttctive potentils, like (36), when the following condition is stisfied l ( l + ) < mv (44). The poof of this theoem is stightfowd. Indeed, fo the ttctive potentil (36) t smll distnces this eqution educes to / 4 R + R R = (45) whee is defined by (37). Theefoe, Eq. (45) hs following solution /+ / lim R = + R + R (46) st dd st dd So we hve two egions fo this pmete. In the intevl < < / (47) / the second tem dd = Rdd must be lso etined, becuse the boundy condition (43) is fulfilled fo it. The potentil like (36) ws fist consideed by K.Cse [], but he ignoed the / + second tem in solution. As egds of egion, only the fist tem st = Rst must be etined. Fom eqs. (37) nd (47) it follows the condition (44) fo existence of dditionl sttes. If we demnd the elity of (othewise flling to cente tkes plce [,3,4]) the pmete V would be esticted by condition mv < l( l + ) + / 4 (48) The lst two inequlities estict mv in the following intevl l ( l + ) < mv < l( l + ) + / 4 (49) Intevls fom the left nd fom the ight sides hve no cossing nd theefoe, if dditionl solution exists fo fixed V nd fo some l, then it is bsent fo nothe l. (43) 9

10 Thus we see fom (44) tht in the l = stte except the stndd solutions thee e dditionl solutions too fo bity smll V, while fo l the stong field is equied in ode to fulfil (44). Becuse the dditionl solutions obey ll physicl equiements in the intevl (47), one hs to etin this solution s well nd study its consequences. Fo definiteness conside the potentil V V =, V (5) > When E= the solution of the full dil eqution (4) hs the fom in whole spce /+ / R = A + B (5) Thee is only one wothy cse, nmely < < /. We see tht the wve function hs simple zeo, detemined by / B = = (5) A (It is evident fom this eltion tht constnts A nd B must hve opposite signs in ode fo to be el numbe). Hence, the wve function hs only one node nd ccoding to well-known theoem (the numbe of bound sttes coincides with the numbe of nodes of dil wve function R() in E = stte []), we hve exctly one bound stte. This esult diffes fom tht consideed in ny textbooks on quntum mechnics. We cn give vey simple physicl pictue of how the dditionl solutions ise. Fo this pupose, let us ewite the Schodinge eqution ne the oigin fo ttctive potentil (36) in the fom R + R + m[ E Vc ( ) ] R = (53) whee / 4 V c = (54) m Conside the following possible cses: i). If > /, then V c > nd it is epulsive centifugl potentil nd s we sw, one hs no dditionl solutions. ii). If < </, then V c <. Theefoe, it becomes ttctive nd is clled s quntum nticentifugl potentil [5]. This potentil hs R dd sttes, becuse the condition (43) is fulfilled in this cse. iii). If <, then Vc becomes stongly ttctive nd one hs flling to the cente. Theefoe, the sign of the potentil detemines whethe we need dditionl solutions o not. V c 4D) SAE pocedue fo full Rdil Hmiltonin in pgmtic ppoch Consideing some consequences fom the point of bove mentioned esults, let us fist of ll emembe some issues of SAE pocedue. If fo ny functions u ndυ, given opeto  stisfies to the condition υ Au ˆ = Aˆυ u (55)

11 then this opeto is clled hemitin (o symmetic). Fo self-djointness it is equied in + ddition tht the domins of functions of opetos  nd  would be equl. As ule, the + domin of the  is wide nd it becomes necessy to mke self-djoint extension of the opeto Â. Thee exists well-known poweful mthemticl pptus fo this pupose [6, 7]. It my hppen tht the opeto is hemitin, but its self-djoint extension is impossible, i.e. hemiticity is the necessy, but not sufficient condition fo self-djointness. Good exmple is the opeto of the dil momentum p which is hemitin on functions tht stisfy the condition (43), but its extension to self-djoint one is impossible (see, L.D.Fddeev s emk in the A.Messih s book Russin tnsltion, footnote in p.336 [8]). Ou subject of inteest is the dil Hmiltonin (4) nd, consequently, the eqution (4) It is esy to see tht fo ny two eigenfunctions R nd R coesponding to the levels E nd E of the dil Hmiltonin H, the condition (55) tkes the following fom ˆ R ˆ ˆ R R = ( ) R H R d R H R d m E E R R d It follows tht self djoint condition is popotionl to the othogonlity integl, theefoe these two conditions e mutully dependent. Becuse the self-djoint opeto hs othogonl eigenfunctions, equiement of othogonlity utomticlly povides self-djointness of, which mens tht this wy povides eliztion of SAE pocedue. It is n essence of so-clled pgmtic ppoch [9], which is much simple nd gets the sme esults s the stong mthemticl full SAE pocedue, povided the fundmentl condition (43) is not violted. Moeove this method is physiclly moe tnspent. Just this method hd been used by Cse in his well-known ppe [], but he did conside only the egul solution. Notice tht ll bove considetions e tue only fo the dil Hmiltonin opeto H, becuse fo othe opetos popotionlity like (56) does not ise. 4E) Explicit solution of the Schodinge eqution fo the invese squed potentil It ws thought tht potentil (5) hs no levels out of egion of flling to the cente (See e.g. [,3]), but in [6,,3] single level ws found by complete SAE pocedue, while the boundy condition nd the nge of pmete, like e questionble thee. Hee we ll show explicitly tht this potentil hs exctly single level, which depends on the SAE pmete τ. Let us now study in which cses the ight-hnd sides of (56) is vnishing.we must distinguish egul nd tnsitive potentils. As we e inteested of bound sttes we suppose tht the full dil function deceses suficiently fst t infinity. So, the behvio t the oigin is elevnt fo ou ims. In cse of egul potentils (34), s ws mentioned bove, we etin only fist, egul (o stndd) solution t the oigin l+ R ~ (57) st st Clculting the.-h.-side of (56) by this function, we get zeo. Theefoe fo egul potentils the dil Hmiltonin H is self-djoint on egul solutions nd it does not need SAE. ˆ R ˆ R (56) H R

12 Conty to this cse, fo tnsitive ttctive (36) potentil one hs to etin the dditionl / solution R dd ~ s well, becuse thee e no esons to neglect it. Now fo both solutions, the.-h.-sides of (56) e not zeo in genel. Indeed they equl to st dd st dd ( ) m ( E E ) R R d = Remk. The cse = must be consideed septely, when the genel solution of (4) behves s limr = + ln = R + R st So, insted of (58) one obtins dd st dd st dd m ( E E ) R R d = ( st dd (58) (59) ) (6) Thus etining dditionl solution cuses the bekdown of othogonlity condition nd consequently, H is no moe self-djoint opeto. ˆ R It is ntul to sk how to fulfil the othogonlity condition? It is cle, tht in both nd = cses one must equie o equivlently st dd st dd = In this cse the dil Hmiltonin st (6) dd dd = (6) st Hˆ R becomes self-djoint opeto. This genelizes the Cse esult [], who consideed only stndd solution. So it is necessy to intoduce so clled SAE pmete, which in ou cse my be defined s dd τ (63) st τ pmete is the sme fo ll levels (fo fixed obitl l momentum) nd is el fo bound sttes. Now let us etun to the solution of the Schodinge eqution fo potentil (5) d R dr / = k R (64) d d whee is given by (37) nd k = me > ; ( E < ) (65) One cn educe Eq. (64) to the eqution fo modified Bessel functions by substitutions f ( ) R ( ) = ; x = k (66) leding to the following eqution d f ( x) df ( x) x + x ( x + ) f ( x) = (67) dx dx This eqution hs 3 pis of independent solutions: I (k) nd I (k), I (k) nd iπ iπ e K (k), I (k) nd e K (k),whee I (k) nd K (k) e Bessel nd McDonld modified functions, espectively [3].

13 Ceful nlysis gives tht the elevnt pi is the fist one only, i.e. the pi I (k) nd I (k) : So, the genel solution of (64) is R = [ AI( k) + BI ( k) ] (68) Conside the behviou of this solution t smll nd lge distnces: ) Smll distnces In this cse (see, [3]) z I ( z) (69) z Γ( + ) Then it follows fom (68) nd (69) tht k k lim R( ) A + B (7) Γ( + ) Γ( ) Fom (46), (6), (7) nd the definition (63) we obtin B k τ = A (7) At lge distnces, we hve [3] z e I ( z) z πz (7) nd R ( ) { A + B} e k π (73) Theefoe, equiing vnishing of R( ) t infinity, we hve to tke B = A (74) nd fom (7), (74) nd (65) we obtin one el level (fo fixed obitl l momentum, stisfying (44)), E = m ; < </ (75) τ Eq. (75) is new expession deived s consequence of othogonlity condition in the fmewok of pgmtic ppoch. Relity of enegy in (75) esticts τ pmete to be negtive τ <. In genel τ is fee pmete, but some physicl equiements my estict its mgnitude. Note tht this level is bsent in stndd quntum mechnics ( τ = ) - it ppes when one pefoms SAE pocedue. To obtin coesponding wve function, tke into ccount well-known eltion [3] π K ( z) = [ I ( z) I ( z) ] (76) sin π Then the wve function coesponding to the level (75) be z R = A sin π K π Becuse of exponentil dmping π z K ( z) e z ( k) (77) (78) 3

14 the function (77) coesponds to the bound stte. It is lso known tht K (z) function hs no zeoes fo el ( < < / ) nd theefoe (75) coesponds to single bound stte. Moeove, wve function (77) stisfies the fundmentl condition (43) fo < < /. Let us mke some comments ) In [] it ws noticed tht single bound stte my be obseved expeimentlly in pol molecules. Fo exmple, H S nd HCl toms exhibit nomlous electon sctteing [3,33], which cn be explined only by electon cptue. Indeed, fo those molecules electon is moving in point dipole field, nd, in this cse the poblem is educed to the Schodinge eqution with potentil (5). Thus, level (75) obtined theoeticlly my be obseved in those expeiments. b) It ws commonly believed, tht the potentil V V = (79) sh α hs no levels in egion (47) (see, fo exmple, poblem 4.39 in [34]). In [34] by the guments of well-known compison theoem [6], which in this cse looks like V V (8) sh α α it is concluded tht the potentil (79) cnnot hve level in the e (47), becuse the potentil (5) hs no levels in this e. But, s we know, thee is τ depended one level (75), theefoe the levels fo (79) e expected. Indeed, in [35] by using the Nikifoov-Uvov method [36], it ws shown tht the potentil (79) hs infinite numbe of levels in the egion (47). 5. OTHER ALICATIONS Thee e physiclly moe elistic potentils, which diffe fom (5), but behve s t the oigin. Fmous exmples e molecul potentil (vlence electon model), Coulomb potentil in Klein- Godon eqution nd etc. 5A) Vlence electon model Let us conside molecul potentil, hving the following fom V α V = ; ( V, ) α > (8) Becuse of singul like behvio t the oigin one must conside eqution fo the R() function, which in dimensionless vibles tkes fom whee nd is gin given by Eq.(37). If we substitute d dρ d / 4 λ + + R = ρ dρ ρ ρ 4 (8) mα ρ = 8mE = ; λ = >, E < (83) 8mE ρ + R = ρ e F( ρ), (84) 4

15 the eqution fo confluent hypegeometic functions follows ρ F + ( + ρ) F (/ + λ) F = (85) This eqution hs fou independent solutions, two of which constitute fundmentl system of solutions [37]. They e (in nottions of [37]): y = F(, b; ρ) y y 5 = ρ b = Ψ(, b; ρ) ρ F( + b, b; ρ) y7 = e Ψ( b, b; ρ) whee = / + λ, b = + (87) Only y is consideed in the scientific ticles, s well s in ll textbooks (see, e.g. [3,38]). Requiing = n ( n =,,,...) the stndd levels follow. Othe solutions ( y, y5, y7 ) hve singul behvio t the oigin nd usully they e not tken into ccount. But the singulity in cse of ttctive potentils like (36) hs the fom nd in the egion < </ solutions must be consideed s well. Theefoe, the poblem becomes moe ich. Let us conside pi nd y.the genel solution of (85) is y ρ ρ / + ρ e F ρ / ( / + λ, + ; ρ ) + C ρ e F (/ λ, ; ) (86) othe R = C (88) Consideing Eq. (88) t the oigin nd ccounting Eq. (63), we obtin the following expession fo SAE τ pmete C τ = (89) C ( 8mE ) On the othe hnd tht, R must decese t infinity. Fom well-known symptotic popeties of confluent hypegeometic function F, we find the following estiction Γ( + ) Γ( ) C + C = (9) Γ(/ + λ) Γ(/ λ) It gives n eqution fo eigenvlues in tems of τ pmete Γ ( / λ ) Γ ( ) = τ ( 8mE ) (9) Γ ( / λ + ) Γ ( + ) This is vey complicted tnscendentl eqution fo E, depending on τ pmete. Thee e two vlues of τ, when this eqution cn be solved nlyticlly: i) τ =. In this cse we hve only stndd levels, which cn be found fom the poles of Γ( / λ + ) / λ + = n ; n =,,... (9) ii) τ = ±. In this cse we hve only dditionl levels, obtined fom the poles of Γ( / λ ) / λ = n ; n =,,... (93) Thus, in these cses one cn obtin explicit expessions fo stndd nd dditionl levels m α m α E, = = (94) st dd [ / + n ± ] / + n ± ( l + / ) mv whee signs (+) o ( ) coespond to stndd nd dditionl levels, espectively. We note tht only the Eq. (9) ws known till now. So the eqution (9) nd its consequences e new esults. 5

16 Notice lso tht, in cse V < we obtin well-known Ktze potentil [38], but now the condition (44) is not stisfied. Theefoe thee e no dditionl levels fo Ktze potentil. It is emkble tht the function (88) my be ewitten in unified fom by using the following eltion fo the Whittke functions [39] x + b π F( / + b, + b; x) b F( / b, b; x) ( ) ( ) ( ) ( ) W b ( x) = e x, x sin π ( + b) Γ / b Γ + b Γ / + b Γ b (95) Then fom (83), (88), (9) nd (95) we deive sinπ ( ) ( ) ( + ) R( ) = CΓ + Γ / λ Wλ, ( π 8mE) (96) Becuse the Whittke function ( ) hs n exponentil dmping [39] W, b x x W b x e, ( ) x, (97) x (97) coesponds to bound stte wve function which stisfies to the fundmentl condition (43) fo < < / intevl. Theefoe, fo τ =, ± the stndd nd dditionl levels e obtined fom (94) with coesponding wve functions ρ Ψ 5 / + R ρ st = C e F( / + λ, + ; ρ ) (98) ρ / Rdd = Cρ e F ( / λ, ; ρ ) (99) Fo bity τ, ± the enegy cn be obtined fom the tnscendentl eqution (9), while the wve function is given by (96). The unified fom (96) is lso new esult nd it is consequence of the SAE pocedue. Accoding to [39] ou function (96) tkes the following fom ρ sinπ ( ) ( ) ( + ) R( ) = CΓ + Γ / λ e ρ Ψ λ, ; ρ () πρ whee (, b, x) is one of the bove mentioned solutions, (86), nmely y. Its zeos e well- studied [39]: Fo el, b (note tht in ou cse = λ ; b = e el numbes) this function hs finite numbes of positive oots. Howeve, fo the gound stte thee e thee cses whee this function hs no zeos: ) > ; ) b + > ; 3) < < nd < b <. Only the lst cse is inteesting fo us, becuse = λ ; b = nd is in the intevl (47). It mens mα < / < () 8mE In othe wods, the gound stte enegy, which is given by tnscendentl eqution (9), must obey this inequlity. The wve function in fom of () is lso new. In monogph [38] enegy levels fo lkline metl toms e witten in Bllme s fom E n = R () n whee R is Rydbeg constnt nd n is the effective pincipl quntum numbe n = n + l + ( n =,,...) (3) while 6

17 l = / ± = / ± ( l + / ) mv (4) Only (+) sign ws consideed in font of the sque oot until now. In [38] V ws consideed to be smll nd fte expnsion of this oot, ppoximte expession fo the stndd levels ws deived E st = R ; n = n + l + (5) ( n + Δ l ) whee st mv Δ l Δ l = (6) l + is so - clled Rydbeg coection (quntum defect) [3,38]. As egds of dditionl levels, this pocedue is invlid, becuse V is bounded fom below ccoding to (44). Appoximte expnsion fo dditionl levels is possible only fo l =. We hve in this cse = mv ( 4mV ) (7) 4 V my be bitily smll, but diffeent fom zeo, becuse in this cse = / nd we hve no dditionl levels. One cn esily obtin the existence condition of dditionl levels fom (5) nd (44) in divese fom l < Δl < l + (8) If we use dt of monogph [38], we obtin tht fo l = sttes only Li, fo l = only K nd fo l = only Cs stisfy (8) (i.e. they hve dditionl solutions nd it is necessy to cy out SAE pocedue), nd N nd Rb hve no dditionl levels. The condition (8) helps us to detemine which lkline metls need SAE extension of Hmiltonin. 5B) The Klein-Godon eqution Let us conside the Klein-Godon eqution in centl potentil ( Δ + m ) ψ ( ) = E V( ) ψ ( ) (9) Afte the seption of ngles, we deive the dil fom of this eqution d d l( l+ ) + + m ( E V) R( ) = d d () nd fo the function u= R, tking into ccount condition (5), we hve l( l+ ) u + ( E V) m u = () It seems tht even the Coulomb potentil is singul by this eqution. Now the following clssifiction must be ccounted fo this eqution lim V ( ) = - Regul () lim V () = V = const - Singul (3) i.e. the e of ppliction of Eq. () becomes nowe. It is pplicble only fo potentils, stisfying to (). Theefoe the eqution () my be used fo potentils, which hve less singulity thn the Coulomb one, whees in using of Eq. () no toubles ppe. 7

18 5C. Hydino sttes in the Klein-Godon eqution with Coulomb potentil We note tht the poblems of dditionl levels wee discussed by othe uthos s well [39-4]. In α pticul, in [4] the Klein-Godon eqution s consideed with V = Coulomb potentil l( l + ) Eα α R + R + E m + + = R (4) The utho undelines, tht thee must be levels below the stndd levels (clled, hydino eigensttes), but he/she did not pefom the SAE pocedue. Let conside this poblem in moe detil. Fist of ll note tht the eqution (4) coincides with Eq. (8), but now Eα ρ = m E ; λ = ; = ( l + / ) α > (5) m E We must equie m > E fo bound sttes. Theefoe one cn use ll the pevious eltions fom vlence electon model tking into ccount the definitions (5).In pticul the SAE pmete now is C τ = (6) C m E nd fo eigensttes we hve the following eqution Γ ( / λ ) Γ ( ) = τ m E (7) Γ ( / λ + ) Γ ( + ) This is new fom, tht follows by SAE pocedue in the Klein-Godon eqution. Fo the edge points we deive the stndd nd dditionl levels in nlogy with (7) m Est = ; n =,,... (8) α + ( / + n + ) m E = ; n =,,... (9) dd α + ( / + n ) Exctly these (9) levels e clled s hydino levels in [39-4]. It is evident tht the hydino levels e nlogicl to E dd sttes Eq.( 94), but these two cses diffe fom ech othes. ticully, it is possible to pss the limit V in the eqution (8) nd obtin Hydogen poblem. Usully this limiting pocedue is used in tditionl textbooks to choose between two signs in (94), while in (4) coupling constnts fo both tems in potentil tems e mutully popotionl ( α ndα ), nd vnishing of one of them cuses vnishing of nothe, so we tun to the fee pticle poblem insted of Coulomb one. Moeove, s we mensioned bove, in those ppes [39-4] the SAE pocedue ws not used. They consideed only two signs in font of sque oot in eqution nlogous to (94) nd only (8) nd (9) levels e consideed, which coespond only to cses τ = nd τ = ±. Conty to tht cse we pefomed SAE pocedue, deived the Eq.(7) nd tke ttention to the hydino (when τ = ± ) poblem. 8

19 The diffeence between stndd nd hydino sttes mnifests clely in the noneltivistic limit when α, which must be pefomed by definite cution. The hydino existence condition fo such sttes folows fom elie constints nd the estiction < </ It hs fom l ( l + ) < α () nd it is evident tht fo sttes with l in tnsition to the noneltivistic α limit the dditionl (hydino) sttes disppe. Theefoe we must conside only l = sttes. Fo the gound sttes ( = l = ) we hve n () E = m st + 4α () ( ) m EHyd Edd = 4α () Expnsion in powes of α gives 4 ( ) α α E st = m (3) 8 () E HYD 3 = m( α +α / ) It follows tht the hydino is vey tightly bound system nd sensitive to the sign of α. If we expnd l ; n sttes till to ode of α, we deive = (4) () α Est = m ( n + ) (5) () α EHYD = m ( n ) (6) Compison of these two expessions shows tht thee ppes some kind of degenecy between the levels with n + nodes of hydino nd enegies fo n nodes of stndd sttes. This degenecy disppes in the next ode. The fct tht the dditionl (hydino[39-4] o peculi [43,44] ) sttes of the ( n + )th stte is nely degenete with the usul n th S stte my fcilitte tunneling tnsition. Ou desciption by the unified function nlogous of (96), s esult of SAE pocedue, gives possibility of intepoltion between them. 5D) The Yukw potentil As lst ppliction of Eq. () let us conside the Yukw potentil. Accoding to common viewpoint (see, e.g. [8], Ch.8) the Yukw potentil is spheiclly symmetic solution of the Helmholz wve eqution ϕ μ ϕ = (7) If we do not tke ttention to the ppence of the delt function, we would hve dil eqution like [8] d ( ϕ) μ ϕ = (8) d S 9

20 ± μ the solution of which is ϕ = Ce nd in cse of decying symptotic t infinity the Yukw potentil follows μ e ϕ = C ( 9) Howeve the ppliction of the coect eltion () gives μ μ e e ( 3) μ = μ 4πδ ( ) e (3) Inteesting enough we found this eqution in the elie book [45]. It follows tht the Yukw potentil is not solution eveywhee, but only outside the oigin of coodintes. The Yukw potentil is the solution of the Helmholz wve eqution with souce tem on the RHS: ϕ μ ϕ = 4 πc δ ( 3) ( ) (3) It ws mentioned incoectly in [7] tht thee is no need in imposing the boundy condition u = nd it is sufficient to equie egulity of solutions of the full dil eqution. But it ( ) seems tht in this pticul cse when the substitution (6) is pplied, this equiement is equivlent to ou estiction (5). It is wothwhile to emphsize one impotnt notion: Of couse, to mke the substitution (6) is not necessy t ll. One cn use othe substitutions in couse of solution of Eq. (7). In this cse the conclusion of [7] becomes moe tnspent nd led to new unexpected esult. Let discuss this viewpoint in cse of Yukw potentil, i.e. of the Helmholz eqution (7), ewiting it fo the spheiclly symmetic solution s d d d + d ϕ μ ϕ = (3) nd insted of (6) mke following substitution χ ( ) ϕ = (33) Denoting z = μ we obtin n eqution d χ dχ + + χ = (34) dz z dz 4z The genel solution of it is expessed in tems of modified Bessel functions [3] χ z = I z + bk z z > ( ) ( ) ( ); / / (35) Let us emembe the symptotic behvio fo lge nd smll guments discussed bove z z e I( z), I ( ) z z z Γ ( + ) π z (36) z Γ( ) π z K( z), K ( ) z z e z z We conclude tht the second solution must be chosen owing the flling behvio t lge vlues of gument. Theefoe the solution of Eq. (3) is / ϕ = b K/ ( μ) (37) But [3] π z K/ ( z) = e (38) z o

21 μ e ϕ = c (39) ( ) i.e. the Yukw potentil gin. Howeve, s we sw bove, unfotuntely this is not the solution eveywhee, becuse of singulity t the denominto. It seems tht this fct hs vey f eching consequences. Nmely, it tun out tht the second function K ( z) is not solution of the Bessel eqution in spite of widesped belief. Actully stightfowd tnsition of one-dimensionl esults of mthemticl physics (theoy of specil functions, whee the Lplcin is pesent) does not give necessily the sme things in thee o moe dimensions. 5. CONCLUSIONS We hve found singulity like the Dic delt function in pocess of eduction the Lplce eqution in spheicl pol coodintes, which ws not mentioned elie. The conestone in ou considetion ws equiement of Dic tht the solution of the dil eqution t the sme time must be solution of the full 3-dimensionl eqution. On the bsis of this obsevtion we hve poved tht fo emoving this ext tem fom the dil eqution it is necessy nd sufficient to impose the educed dil wve function by definite estiction, which hs fom of the boundy condition t the oigin, eq. (5). Moeove this condition is independent of whethe the potentil in the Schodinge eqution is egul o singul. The singul potentil influences only the chcte of tuning to zeo of the dil function t the oigin. As egds of the fu ll dil function R() its eqution is comptible with the pimy (3-, dimensionl) eqution () if the estiction (43) is stisfied. Theefoe, to void the misundestndings, it is pefeble to wok by the eqution (4) in noneltivistic nd () in eltivistic (Klein-Godon eqution) cses, coespondingly. Moeove only nonsingul solutions of full dil eqution must be tken into ccount, only they e comptible with the full 3- dimensionl equtions. The substitution (6) is convenient becuse the poblem educes to one dimensionl one on the semi-xis. The el pictue is s follows: ticle in pinciple is ble to move on the whole xis, but the effective potentil is infinite fo ll negtive vlues of gument. In this cse the wve function is identiclly zeo on the whole negtive xis. The condition u ( ) = guntees continuity of the wve function t =. This povides the comptibility wit h the full eqution nd the equivlence to one-dimensionl poblem [4]. Above descibed sitution tkes plce in spces with dimensions thee nd moe. Theefoe in ll equtions of mthemticl physics, whee the Lplcin is involved, fte the seption of ngul vibles the singul solutions, genelly speking, would not be the solutions of the pimy equtions. If we shut eyes to tem with the delt function nd fomlly use the educed dil eqution, then ll esults deived till now with the id of this eqution fo egul potentils with egul boundy condition t the oigin, emin vlid. It is not n insignificnt esult fom pcticl point of view. Howeve, when one consides singul potentils the use of eqution fo the full dil function R ( ) in pllel with the SAE pocedue of the full dil Hmiltonin is necessy,. The ppopite exmples, consideed bove, elucidte this sttement.

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