Completeness of scattering states of the Dirac Hamiltonian with a step potential

Transcription

1 Journal of Physcs Councatons PAPR OPN ACCSS Coltnss of scattrng stats of th Drac Haltonan wth a st otntal To ct ths artcl: M Ocha and H Naaato 18 J. Phys. Coun. 156 Vw th artcl onln for udats and nhancnts. Ths contnt was downloadd fro IP addrss on 9/3/19 at :38

2 J. Phys. Coun. ( htts://do.org/1.188/399658/aa9fc OPN ACCSS RCIVD 4 Octobr 17 RVISD 16 Novbr 17 ACCPTD FOR PUBLICATION 7 Dcbr 17 PAPR Coltnss of scattrng stats of th Drac Haltonan wth a st otntal M Ocha and H Naaato Dartnt of Physcs, Wasda Unvrsty, Toyo , Jaan Kywords: coltnss, Drac Haltonan, st otntal PUBLISHD 1 January 18 Orgnal contnt fro ths wor ay b usd undr th trs of th Cratv Coons Attrbuton 3. lcnc. Any furthr dstrbuton of ths wor ust antan attrbuton to th author(s and th ttl of th wor, journal ctaton and DOI. Abstract Th coltnss, togthr wth th orthonoralty, of th gnfunctons of th Drac Haltonan wth a st otntal s shown xlctly. Ths gnfunctons dscrb th scattrng rocss of a rlatvstc fron off th st otntal and th rsoluton of th dntty n trs of th (coltnss s shown by xlctly sung th u, whr arorat tratnts of th ontu ntgratons ar crucal. Th rsult would brng about a bass on whch a fld thortcal tratnt for such a syst can b dvlod. 1. Introducton In uantu chancs, a hyscal obsrvabl s rrsntd by a slfadjont orator actng on a Hlbrt sac. Its gnvalus ar all ralvalud, corrsondng to ts asurd valus n xrnts, and th gnfunctons blongng to dffrnt gnvalus ar orthogonal to ach othr, whch s n accord wth th hyscal stuaton that th two vnts wth dffrnt outcos ar utually xclusv and nvr han sultanously. Ths statnts ar asly rovd and th roofs ar found n any txtboo of uantu chancs, n whch, howvr, w always fnd anothr statnt accoand, clang that th gnfunctons for a colt st. Th statnt s hyscally sound bcaus f w asur a hyscal uantty n any stat, w would and should hav a rsult, th obtand valu of whch s on of th gnvalus of th corrsondng orator and ths ls that any stat can b xandd n trs of th gnfunctons. vn though w would all adt ts ncssty for a hyscal ground, th roof of coltnss of gnfunctons would not so asly b don onc thy consttut a contnuous sctru. Ths consttuts a strng contrast to th cas whr only a fnt nubr of dscrt stats xst, for n such a cas on can convnc onslf that a fnt nubr of utually ndndnt vctors ar nough to xrss any stat, thus th roof of thr coltnss s trval. Whn a contnuous sctru s nvolvd, only xcton whr on can asly dscuss and confr th coltnss of th gnfunctons would b th fr cas. In ths cas, th lan wavs f ( x,.., th gnfunctons of th fr Haltonan H for a colt orthonoral st n th sns ( 1 d 3f ( x f* ( x d3( x x, d 3xf* ( x f ( x d3(, f ( x 1 x, ( 1 ( 3 whch ar ssntally th (nvrs Fourr transfors of Drac s dlta functon, or th rsoluton of th dntty. What s not trval s to show ths rlatons whn th lan wav solutons ar rlacd wth th scattrng wav functons for a statonary otntal robl. As a attr of fact, vn though th asytotc coltnss of th scattrng stats s rovd wthn th frawor of foral scattrng thory (s,.g. [1] and thr xsts a classc rvw on th analytcal rorts of radal wav functons by Nwton [], th coltnss of, say, th Coulob scattrng wav functons s shown on th bass of Nwton s thod rathr rcntly [3]. Ths s bcaus th longrang otntals wr not covrd by thr tratnts and to show t on th bass of Nwton s thod, on has to valuat rathr nvolvd ntgrals n th colx lan. Surrsngly nough, th stuaton s not at all bttr for cass wth sl otntals, say, th st or wll otntals n on dnson. In fact, th roof of th coltnss of ondnsonal scattrng wav functons off a st otntal aard only n th lat 8s [4] and a athatcally or rgorous tratnt for st and wll otntals has to b watd untl 1 [5]. 18 Th Author(s. Publshd by IOP Publshng Ltd

3 J. Phys. Coun. ( M Ocha and H Naaato Aarntly ths otntals do not satsfy so of th rss assud for th gnral roofs and on has to wor out such cass saratly vn though th otntals ar vry sl and consdrd rathr lntary. In ths ar, w shall donstrat that th scattrng wav functons of a Drac Haltonan for a syst wth a st otntal for a colt orthonoral st, along th sa ln of thought as [4], by sung u all of th. Though th coltnss of th n th sa syst s alrady shown n [6] n a athatcally rgorous way, t s stll of ntrst and nstructv to s xlctly that th gnfunctons of an ntractng Drac Haltonan can rlac th lan wavs n (1 to lay th sa rol. Th rocdur tan hr s ntutv, straghtforward, though uch nvolvd and n accord wth what hyscsts would xct. Th ar s organd as follows. Aftr th ntroducton of basc ngrdnts for th fr Drac Haltonan n th nxt scton, th scattrng wav functons of th Drac Haltonan for th syst wth a fnt st otntal, rsonsbl for an ulsv forc at th orgn, ar rsntd n scton 3. Thy ar catgord nto svral cass dndng on th boundary condtons,.., f a Drac fron wth a ostv (or ngatv fruncy s ncdnt fro th lft or th rght of th st otntal, and on th valu of ncdnt nrgy. Th coltnss condton xrssd n th coordnat sac rfrs to two satal onts and snc th otntal tas dffrnt valus at th lft and th rght of th otntal, t s xand for thr dffrnt cass n scton 4.1. Th orthogonalty btwn th lftncdnt and rghtncdnt wav functons s shown xlctly for a artcular cas as an llustraton n scton 4.. Th last scton 5 s dvotd to a suary and rosct.. Solutons of fr Drac uaton For latr convnnc, w frst fx th notatons and noralatons of fourcoonnt snors of th fr Drac Haltonan. Th statonary lan wav solutons for th fr Drac Haltonan ar H s a,, b a b s ( u, s s s t, H u, s u, s x, 3 for a ostv fruncy and v, s s x s t, H v, s v, s, 4 for a ngatv fruncy, whr s s th thrd Paul atrx, th unt atrx and x( sa twocoonnt snor. Not that snc w wll consdr a st otntal, w stu th axs at rght angl wth th boundary of th otntal and th trval dndnc on th othr coordnats x and y s surssd fro th bgnnng and s gnord coltly. Th robl s ssntally n on satal dnson, though th snors lv n fourdnsonal sact and hav four coonnts. Ths snors ar norald and for a colt orthonoral st u (, su (, s dss,, v (, sv (, s dss,, u (, su (, s v (, sv (, s dss,, ( 5 å[ u (, su (, s v (, sv (, s] 4 4. ( 6 s Now, f a constant otntal s addd to th abov Haltonan n all sac H H V a b V, (< <, ( 7 th corrsondng lan wav solutons ar ( for a ostv fruncy V V and u, s s s V t x ( 8 v, s s x s V t ( 9 for a ngatv (rlatv to V fruncy V V. Notc that th fruncy V can b ostv for sall onta < V, vn though t s to b assocatd wth th snor v. Not also that th

4 J. Phys. Coun. ( M Ocha and H Naaato Fgur 1. Scattrng stats ar catgord accordng to whthr a artcl s ncdnt fro th lft ( A: y or rght ( B: f of th st otntal and to th rangs of thr nrgs (1 4. No lft(rghtncdnt scattrng soluton xsts for th nrgy rang < ( V <. abov xlct fors of snors ar not unu snc s s coutabl wth th abov H and H and thrfor,.g., s u(, scan b usd nstad of u(, s. 3. Scattrng stats A scattrng robl of a Drac fron wth ass off a st otntal V( ( V, V >, ( 1 whr s th Havsd st functon, s asly solvd. (For slcty and dfntnss, only thos cass whr th otntal barrr s hghr than ar consdrd, s fgur 1. Th statonary solutons for th lftand rghtncdnt scattrng robls, dscrbd by th total Haltonan H H V( a b ( V ( 11 ar gvn by th gnstats of th Haltonan H and ar charactrd by thr boundary condtons,.., lftncdnt (ψ or rghtncdnt (f, and th (rang of valu of thr gnvalu Lftncdnt cass y ( (, t A1: for > V ( V and thrfor, ( V, th lftncdnt wav functon rads as 1 y ( (, t u, s Ru, s Tu, s t, 1 [ { } ] wth th rflcton and transsson coffcnts R V V, T V V V V dtrnd by th contnuty condton at. Hr and n th followngs, th dndnc on th sn s s and wll b surssd n th wav functons. Th contnuty of th currnt dfnd as j yg 3y y a y at rads as R T, ( 14 ( 13 3

5 J. Phys. Coun. ( M Ocha and H Naaato whch xrsss th consrvaton of th robablty Pr R, Pt T, P r P t 1. ( 15 A: for V > > V ( >, no oscllatng soluton xsts for th rgon > and w hav ( ( V 1 y( (, t [ ( { u(, s Ru(, s } Tu(, s ] t, ( 16 for > V and y( (, t 1 [ ( { u(, s Ru(, s } Ts v(, s ] t, ( 17 for < V, wth th rflcton and transsson coffcnts R rsultng n a total rflcton V ( V, T V ( V V ( V, ( 18 P R 1, P, ( 19 r for th currnt n th rgon > vanshs dntcally. A3: for V > >, th soluton n th rgon > s xrssd n trs of v ( V or ( V 1 y( (, t [ ( { u(, s Ru(, s } Ts v(, s ] t, ( wth t R V V, T V V V V. ( 1 Th contnuty of currnt ls th consrvaton of robablty R T Pr Pt R T 1. ( Rcall that th ordnary consrvaton law of robablty s satsfd, but a fnt and nonvanshng transsson robablty survvs vn at th nfntotntal lt V P T t ¹. ( 3 Ths hnonon s nown as th Kln tunnlng [7]. Notc that no lftncdnt artcl s allowd for > >. A4: for >, th gnstats ar gvn by th ngatvfruncy solutons ( V wth y 1 (, t v, s Rv, s Tv, s, 4 [ { } ] ( t R V V, T V V V V. ( 5 Th currnt contnuty and th consrvaton of robablty ar th sa as n (. 3.. Rghtncdnt cass f ( (, t B1: for > V, V (or, ( V and th rghtncdnt wav functon s gvn by 4

6 J. Phys. Coun. ( M Ocha and H Naaato wth 1 f ( (, t Tu, s u, s Ru, s t, 6 V [ { }] R V V, T V V V V. ( 7 Th contnuty of currnt and th consrvaton of robablty rad as as wth T R, Pr Pt R T 1. ( 8 Obsrv that no rghtncdnt artcl s allowd for V > > V. B: for V > > ( V or ( V, th soluton ay b wrttn 1 f ( (, t Tsu, s v, s Rv, s t, 9 V [ { }] R V V, T V V V V. ( 3 Th contnuty of currnt and th consrvaton of robablty ar th sa as n (8. B3: for > >, no oscllatng soluton xsts for th rgon < and w hav ( 1 f ( (, t Ts u, s v, s Rv, s t, 31 V [ { }] for > and 1 f ( (, t Tv, s v, s Rv, s t, 3 V [ { }] for <, wth R V V, T V V V V In ths cas, all artcls ncdnt fro th rght ar rflctd. ( 33 wth P, P R 1. ( 34 t B4: for >, V, and w hav (wth ostv and f r 1 (, t [ ( Tv(, s { v(, s Rv(, s }], ( 35 V ( t R V V, T V V V V. ( 36 In ths cas, th sa currnt contnuty rlaton and robablty consrvaton as n (8 follow. 4. Orthonoralty and coltnss rlaton Ths gnfunctons for a colt orthonoral st. Frst, th orthogonalty of th abov scattrng sats s du to th gnral argunt for slfadjont Haltonans, that s, th gnfunctons blongng to dffrnt gnvalus ar orthogonal to ach othr. Two ψʼs wth dffrnt nrgs ar orthogonal and norald as 5

7 J. Phys. Coun. ( M Ocha and H Naaato d y( y( ( ( d(, ( 37 whr,. (Th factor d ss, rrsntng th orthogonalty n sn sac s and wll b surssd but undrstood on th rghthand sd and whr t s ncssary. Slarly, w should hav d f( f( ( [( V( V] d(, ( 38 whr V, V and d y( f ( (. ( 39 Snc th last orthogonalty rlaton s not drvd fro th abov argunt for both y ( and f ( can han to blong to th sa nrgy, ts valdty has to b xlctly shown. Incdntally, th fact that th abov wav functons n th rcdng scton ar rorly norald can b shown xlctly by calculatng th lfthand sds of (37 and (38 for ach cas, howvr, w ay confr t ndrctly whn thy ar shown to satsfy th coltnss rlaton wth th rght factors. Th orthonoralty condtons ly that th followng for of coltnss rlaton holds å sr, r r ( ( å r r sr, d y y ( d f f ( d(, ( 4 whr th unt atrx n snor sac 4 4 has bn surssd on th rghthand sd and w hav ntroducd ( shorthand notatons y y( ( and f f( V. (Th dndnc of th gnfunctons on th sn s s surssd as bfor vn though th suaton ovr sn s xlct hr Proof of th coltnss rlaton: ontu ntgratons As dscrbd n Introducton, though t s gnrally tan for grantd that gnfunctons of a Haltonan consttut a colt orthonoral st, whch can b shown xlctly whn t has only a fnt nubr of dscrt gnstats, to rov t for a Haltonan ndowd wth a contnuous sctru can b anothr, nontrval tas. In th rsnt cas, w nd to show that th lfthand sd of (4 s dagonal n both th coordnat and snor sacs, whch would a th roof or nvolvd than th nonrlatvstc cass [4]. Snc th st otntal tas dffrnt valus dndng on th sgn of th coordnat, w nd to consdr thr cass, that s,, <, <, and < <, saratly to rov (4. Cas 1:, < Whn both and ar ngatv,.., on th lft of th otntal, th lfthand sd of th coltnss rlaton (4 s xlctly wrttn down as d [ u ( R u ( ][ u ( R u ( d v R v v R v [ ( ][ ( d Tu Tu ( [ ( ( V d T u T u s ( [ s ( V d T u T u s ( [ s ( ( V ( V d Tv Tv [ V d Tv Tv ( [ (, ( 41 ( V whr t s undrstood that th suaton ovr sn s has to b tan (though not xlctly wrttn down and th subscrts and ar usd to dstngush uantts assocatd wth th lftncdnt ( cas and th rghtncdnt ( cas (wth th sa nrgy. It s to b notcd that th rflcton altud for th lftncdnt cas R can b xrssd as 6

8 J. Phys. Coun. ( M Ocha and H Naaato R ( 4 for all by a ror analytcal contnuaton fro a larg ( V (or V. Slarly th transsson altud for th rghtncdnt cas T can b undrstood as a ror analytc contnuaton of T. ( 43 Aftr th chang of varabls fro to (or κ whn V V, th abov xrsson s slfd as d {( u( u( R v( v( ( v( v( R u( u( ( Ruu * ( Rv( v ( ( Ru( u Rvv * ( ( } d T u ( u ( ( ( d T u u T v v ( ( ( d T v v ( ( (, ( ( ( 44 whr and th fact that s s coutabl wth both snors u and v has bn usd. W hav to ay du attnton to th fact that, vn though not xlctly wrttn, th valus of (and thrfor thos of ar dffrnt whn t aars n assocaton wth th snor u or v, whch, of cours, als also to R and T n th abov. (Ths s th rason why aarntly th sa uantts hav not bn ut togthr n th last but on ln. Obsrv that thr ar trs of functons of th dffrnc and th su and thy ar aarntly ndndnt of ach othr. Frst, all trs that ar functons of ar collctd to yld d u u R v v T v v v v R u u T u u. ( 45 W not that th transsson robablty for th rghtncdnt cas T s th sa as that for th lftncdnt cas T, whch can b xlctly shown by drct calculaton, statng that th rcrocal rlaton n uantu chancs also holds n ths cas. Fnally, th consrvaton of robablty R T 1 gratly slfs th abov to rach d ( u ( u ( v ( v ( ( d g g3 g g3 g g ( d(, ( 46 whr at th frst ualty th sn su has xlctly bn tan and th unt atrx 4 4 s surssd on th rghtost hand. Scond, th ranng trs that ar functons of ar shown to cancl to ach othr. W undrstand that th contrbutons cong fro th ntgral 7

9 J. Phys. Coun. ( M Ocha and H Naaato d Ruu Rv v Ru u Rvv {( * ( ( ( ( ( * ( ( } ( 47 can b valuatd on th colx lan, bcaus th cobnaton n th xonnts s ngatv dfnt, th rflcton altud dcays at last l 1 for larg, assurng th convrgnc of th ntgrands at and th ntgrands hav no sngularts othr than th svral cuts on th ral and agnary axs. Th ntgrand roortonal to ( can b valuatd on th ngatv agnary axs, whl th othr on roortonal to ( s to b valuatd on th ostv agnary axs, whr n both cass. On th ngatv agnary axs, w st and u( u( ( u( u( (, v( v( ( v( v( (, ( 48 whl on th ostv agnary axs, w hav u( u( ( u( u( (, v( v( ( v( v( (. ( 49 For sall, s a ral nubr sallr than,, and th rflcton altud assocatd wth snor u s analytcally contnud as R* V V V V V V V V R V V V V V V V V, ( 5 bcaus th urly agnary uantty V has th sa has as that of on th agnary axs, for V ( V V V, ( 51 whr ( V s a ral nubr and locatd on th ral axs. Slarly, for th rflcton altud assocatd wth snor v, w obtan R V V V V V V V V R* V V V V V V V V. ( 5 All contrbutons cong fro sall κ on th agnary axs ar ut togthr to yld d T R R u u T R R * * v v. ( 53 Th coffcnt of snor u vanshs, bcaus V V I V V V V V V V V (, ( 54 8

10 J. Phys. Coun. ( M Ocha and H Naaato whr w hav usd th rlatons and ( V wth. Slarly, th coffcnt of snor v, snc n ths cas ( V wth, also vanshs V V V V V V ( I V V V V. ( 55 Thus what s rand s th contrbutons arsng fro larg, whr bcos urly agnary. Frst w not that th functon gans th sa has as that of on th agnary axs,.., º for. Fro th drct calculatons, w undrstand that th rflcton altud for snor u s rlacd by th followng colx nubrs on th agnary axs V V R Ru(, R* Ru(, ( 56 V V V V V V whl for snor v, V V R Rv (, R* Rv (. ( 57 V V V V V V Obsrv that th followng ualts hold R ( R (, R ( R (. ( 58 v u v u Th snor arts for u and v ar xlctly word out, aftr sung ovr sn dgrs frdo, to b 1 s u u s 1, 1 s (, s ( s s v( v(,1 1 1 s (, ( 59 s ( whr th rlaton has bn usd, showng xlctly that thy ar actually th sa. Th ranng ntgratons ovr κ ar now wrttn as d R u u R v v [ u( ( ( v( ] ( d R u u u R v v v [ ( ( ] (, ( 6 whch vanshs owng to th abov rlatons (58 and (59. Ths colts th roof of th coltnss rlaton (4 for, <. Cas :, > 9

11 J. Phys. Coun. ( M Ocha and H Naaato As n th rvous cas, th lfthand sd of (4 s xlctly wrttn down d Tu ( [ Tu ( ( V ( V d Tu Tu ( [ ( V V T v T v s ( [ s ( ( V ( V d T v T v s ( [ s ( d Tv Tv ( [ ( d u R u u R u [ ( ][ ( d v R v v R v [ ( ][ (, ( 61 whr s dfnd, whn,as. Chang of varabls (or rducs ths to d T R uu v v ( ( T R vv uu ( ( ( Ru * ( u Rvv ( ( ( Ruu ( Rv * (v ( d T u u T v v. ( 6 Th rcrocal rlaton T T also holds tru n ths cas, whch can b rovn by drct calculaton, and th robablty consrvaton R T 1 brngs about th dlta functon d ( fro thos trs that dnd on th dffrnc n th abov (6. Th ranng ntgratons ovr ar valuatd on th agnary axs, whr snors bco u( u( and v( v( and th rflcton altud R for snor u s rrsntd as V V R for, V V V V V V and for v, V V V V R ( for,, ( 63 u V V R for, V V V V V V V V V V R ( for, ( 64 v and R *ʼs as th colx conjugats of th corrsondng ons. Thus th contrbutons of trs that ar functons of ar 1

12 J. Phys. Coun. ( M Ocha and H Naaato d T R R ( * u ( u ( T R R * v v d R u u u R v v v [ ( ( ( ( ( ( ] ( d R u u u R v v v [ ( ( ( ( ( ( ] (, ( 65 whch can b shown to vansh on th bass of th slar argunts as bfor. Ths colts th roof for, >. Cas 3: < < If th lfthand sd of (4 s xlctly wrttn down n ths cas, thr ar ght (actually tn dffrnt trs xstng, corrsondng to ght dffrnt cass A1 A4 and B1 B4. It can b shown, howvr, that ths trs ar catgord nto four grous, accordng to thr nrgy gnvalus, (, ( > >, ( > > and (v. ( : Consdr frst th contrbutons arsng fro thos wav functons that ar blongng to gnvalus (nrgs gratr than V. Thy ar xrssd as d [ u ( R u ( ][ T u ( d Tu u R u ( [ ( d T* u u RT* u u ( ( ( V ( Tu (u ( TRu * (u. ( 66 V Obsrv that n ths nrgy rang th rflcton and transsson altuds ar all ralvalud and, orovr, th followng rlatons hold T T ( 67 ( and ( R T * R T *. ( 68 ( Thus th abov xrsson (66 s slfd to ( V d ( T * u ( u ( T u ( u (. ( 69 As a attr of fact, t can b confrd that th contrbutons cong fro th wav functons blongng to nrgy gnvalus gratr than or ual to,, ar consstntly wrttn n ths for wth th lowr lt rlacd wth. Whn ( V > V, n ordr to a th ntgrand convrgnt at, th frst tr n (69 s dfnd by whl th scond tr s dfnd as T* u( u( T* u( u(, ( 7 Tu( u ( Tu ( u (. ( 71 Notc that hr on th lfthand sd, th varabl s to b rlacd wth that scfd at vrtcal bar or n th uantts whch ar consdrd as functons of, whl on th rghthand sd, conjugat oratons ar to b tan for functons of scfd varabls. For xal, u ( u ( and u ( u(. Snc, n th scond tr of (69, w analytcally contnu, th tr V s rlacd wth V n th altuds T, whch rsults n th rlaton T R T. (Ths rlaton also 11 *

13 J. Phys. Coun. ( M Ocha and H Naaato holds whn V > V. Th su of ths two trs s just th contrbuton fro th nrgy rang V < V. In th nrgy rang V < V, w not that ( V ( V and V s 1 u( 1, ( V V, s V 1 ( s V, V s V s,1s v( s. ( 7 V Thrfor, th ntgrand n th arnthss n (69 bcos nothng but ( T* u( v( s ( R T* u( v( s, ( 73 whch s th contrbuton fro ths nrgy rang V < V. Fnally n th nrgy rang < V, th varabl s a ostv nubr and aars n assocaton wth th snor v as v ( or v(, whch ls that th tr V has to b rlacd wth V n th altud T and wth V n T * n (69. Not that n thr cas, s to b rlacd wth. It s straghtforward to confr that th followng rlatons hold, for th lftncdnt contrbuton, ( V d [ u ( R u ( ][ T s v ( ( V d u R u T u [ ( ][, ( 74 and for th rghtncdnt contrbuton, V V ( V d T s u ( [ v ( R v ( ( V d Tu u Ru ( [ (. ( 75 V V Th scond trs on th rghthand sds of ths rlatons (.., thos roortonal to th roduct of rflcton and transsson altuds ar cancld to ach othr and fnally w arrv at th concluson that th contrbutons cong fro th nrgy rang ar consstntly wrttn down as d ( T * u ( u ( T u ( u (. ( 76 ( > > : A straghtforward calculaton ylds V ( V d T s u ( [ v ( R v ( ] d T u v T u v ( s ( ( * s ( d Tu u ( ( d( T * u ( u (, ( 77 whr TR * T* has bn usd n th frst ualty. Actually n th frst tr of th last ln, T s valuatd by rlacng V by V n th orgnal on n (13 and t s confrd that t concds wth T. Undr ths rlacnts togthr wth, w can show that 1

14 J. Phys. Coun. ( M Ocha and H Naaato u( V V 1, s V V V 1, s V V V s,1 s v s. V ( 78 (Rbr that undr ths rlacnts, s to b rlacd wth and w gt th rght xonntal factors. A slar argunt s ald to th scond tr. Th contrbuton s now suard as d T u u d Tu u *, 79 ( ( whr th ntgraton contours ar tan along th agnary axs. ( and (v :Wfrst consdr th rgon (v whr. Th contrbuton cong fro ths nrgy rang s d [ v ( R v ( ][ T v ( d Tv v R v ( [ ( ( V d T * v v T v v ( ( (. ( 8 Snc n th contrbuton arsng fro th rgon ( > > ( V V th rlaton TR* T* holds, w can rwrt ths as d T ( v ( [ v ( R v ( ( 81 d Tv v T v v ( ( *. ( 8 Furthror, w confr that th transsson altud n ths nrgy rang (5 s analytcally contnud for as < T T T, * T *, ( 83 V V V V so that th contrbuton cong fro rgon (v s xrssd as ntgratons along agnary axs d Tv v d ( ( T * v v. ( 84 Puttng all contrbutons ( (v togthr, w undrstand that th lfthand sd of (4 s convnntly wrttn as d ( T * u ( u ( T v ( v ( d Tu u T* v v ( ( (. ( 85 Obsrv that th xonntal factors a th ntgrands convrgnt at n th lowrhalf and urhalf lans for th frst and scond lns, rsctvly, f ror branchs hav bn chosn for ontu. Snc sngularts only aar on th ral and agnary axss as branch cuts, th ntgrals ar valuatd on th agnary axs, fro to for th frst ntgral and fro to for th scond. (Rbr that hr w ar consdrng th cas whr < and >. W not, howvr, that th ontu for snor u, ( V, and that for snor v, ( V, ar dffrnt uantts, though xrssd wth th sa sybol for notatonal slcty, and w hav to choos a ror has whn thy ar analytcally contnud. 13

15 J. Phys. Coun. ( M Ocha and H Naaato Actually, f w ut ( and thus º on th agnary axs, w hav, for snor u, ( V ( V ( V º, ( 86 whch ar dffrnt fro thos for snor v ( V ( V ( V ( V ( V. ( 87 W wll ut togthr thos trs that hav th sa (, dndnc. Whn analytcally contnud on th agnary axs, th frst tr and th last tr n (85 hav th sa (, dndnc. Th frst tr s valuatd as d( T u u * ( (, ( 88 whl th last tr as d T v v * (. ( 89 Obsrv that th altud T * s actually th sa uantty for both cass, for T* ( for snor u T* ( for snor v. V V V V ( 9 Aftr th sn su, th snor arts ar xlctly wrttn down as u( u( º u( u( 1 ( ( V s ( ( V s ( ( V ( ( V ( 91 and 1 ( ( V s ( ( V v( v( º v( v(. s ( ( V ( ( V ( 9 W carfully xan th hass of th suar roots and undrstand that ths two snor arts hav oost sgns and ar cancld to ach othr. Ths ans that th frst tr and th last tr n (85 ar cancld to gv vanshng contrbuton. A slar argunt shows that th ranng two trs, th scond and th thrd trs n (85, cancl to ach othr and w can conclud that (85 vanshs dntcally. Ths colts th roof of (4 for th cas of < <, whr ts rghthand sd vanshs. To suar, Cass 1 3 colt th roof of coltnss rlaton ( Orthogonalty btwn y ( and f ( As s alrady statd, only nontrval rlaton s th orthogonalty rlaton btwn y and f ( (39. For dfntnss, th orthogonalty shall b donstratd hr only for th cas of, >. In ths cas, th nnr roduct btwn y ( and f ( s xlctly calculatd as (agan a trval factor d ss, arsng fro th snor nnr roduct shall b surssd dy( ( f ( ( µ d[ u( R u( ] T u( dtu [ ] [ u( R u ( ] T u ( u ( T R u ( u ( T u u ( u u ( TR. ( 93 Th scond and th last trs contan trs roortonal to dlta functons T R u u T R u u TR TR ( ( d( ( d( d(, ( 94 14

16 J. Phys. Coun. ( M Ocha and H Naaato whch vanshs bcaus of th rlaton TR TR. Th ranng arts ar collctd to yld T ( ( ( R (( ( R ( ( T ( ( R { ( ( R. ( 95 ( ( W not that and and furthror R R R ( ( 1 ( 1 T, R ( T, R ( T, R ( T. 96 Substtuton of ths xrssons nto (95 ylds, aftr a straghtforward calculaton, TT ( ( ( ( ( TT ( ( ( ( ( TT 1 1 ( ( ( ( (, ( 97 whch s ro bcaus V and V. Thus th rghthand sd of (93 s shown to vansh, whch ans that th wav functon for lftncdnt cas y ( and that for rghtncdnt cas f ( ar orthogonal whn, >. Th abov argunt would asly b xtndd to othr nrgy rangs to show th orthogonalty btwn y and f ( ",. 5. Suary and rosct Th scattrng stats of th Drac Haltonan dscrbng fron s dynacs undr th st otntal ar shown to for a colt st by drctly valuatng th ontu ntgrals. Ths ndrctly justfs that thy ar rorly norald and thy consttut an orthonoral st. Though t has bn xctd fro a hyscal ground, to show that th scattrng stats, that s, th gnstats of a Haltonan blongng to contnuous gnvalus for a colt st (whn no bound stats ar allowd s not at all trval and actually has rurd carful analyss on thr analytc rorts, as xoundd hr n dtal. It ay b strssd that n ths rlatvstc cas, du attnton has to b ayd also on th snors, whch has ad ths ssu or nvolvd than n th nonrlatvstc cass. W obsrv that rrlvant trs, that do not contrbut to th dlta functon d ( and thus hav to b cancld n th coltnss rlaton, ar concsly xrssd as (47, (65 or (85, whch would b a rflcton of th xstnc of a or foral tratnt n th rlatvstc cas [6], l [5] n th nonrlatvstc cas. Th roof of th asytotc coltnss n [6] s basd on th fact that th scattrng stats of a artcular rang of nrgs (gnvalus consttut a rojcton orator on that nrgy rang, whch s drvd fro th constructon of th rsolvnt of th Haltonan n trs of th scattrng stats. Th tratnt s athatcally rgorous and lgant, howvr, t ght s a bt tchncal for hyscally ntutv ys. Th rsntaton hr, though rathr nvolvd and not lgant, would b just what such ol s loong for. Actually, just as n [4] for th nonrlatvstc cass, th ntgratons ovr ontu hav bn carrd out 15

17 J. Phys. Coun. ( M Ocha and H Naaato straghtforwardly and xlctly, rsultng n th dlta functon that rrsnts th coltnss rlaton (4.It would b ntrstng and nstructv to s how such an ntutv aroach dos wor vn for ths rlatvstc cas. Ndlss to say, th rsnt ssu s closly rlatd to th socalld Kln tunnlng (or Kln aradox [7]. In ths rsct, t s worth strssng that th oscllatng soluton of th Drac uaton n th otntal rgon > whn th lftncdnt nrgy s blow V s gvn by a ngatv, rlatv to V, fruncy soluton µ Tv (, s (. Ths s th rght soluton that blongs to th corrct gnvalu V V and satsfs th corrct boundary condton,.., dscrbng a ostv currnt j ya y T corrsondng to a transttd wav. Th corrctnss would also b justfd by th fact that t s on of th corrct lnts of th colt st. Othr chocs would rsult n th volaton of th vry gnvalu robl and/or th boundary condton. Thrfor as alrady statd n 3, thr s no anoaly n th consrvaton of robablty vn n ths cas, but th transsson robablty rans nonvanshng n th nfnt otntal lt V (3, whch s nown as th Kln tunnlng. Th colt orthonoral st obtand n ths ar can b usd as a bass for furthr xloratons of such a syst of fron undr th st otntal wthn th frawor of uantu fld thory. Wor n such a drcton s n rogrss and wll b rortd lswhr. Acnowldgnts HN acnowldgs frutful and ncouragng dscussons wth Savro Pascaso and Paolo Facch. ORCID Ds M Ocha htts: /orcd.org/ Rfrncs [1] Rd M and Son B 1979 Mthods of Modrn Mathatcal Physcs vol 3 (Nw Yor: Acadc [] Nwton R G 196 J. Math. Phys [3] Muhadhanov A M and An M 8 ur. Phys. J. A [4] Trott M, Trott S and Schnttlr Ch 1989 Phys. Status Sold b 151 K13 [5] Pala G, Prado H and Rys G 1 J. Phys. A: Math. Thor [6] Rujsnaars S N M and Bongaarts P J M 1977 Ann. Inst. Hnr Poncaré A 6 1 [7] S, for a rvw,.g. Doby N and Calogracos A 1999 Phys. R