— — — — — —__——- 133 020 THS SQME pmmmwss (2? THE LAME mmmmm "i‘hesis For 9in Dagme a? M. 3-. MICHEGAN STA‘E’E UNWERSH'Y Ezaasifi C‘ Haikécfies E96? IIIIIIIIIIIIIIIIIIIIIIIIIIII 3 1293 01704 0050 r‘ LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MTE DUE DATE DUE DATE DUE 11% WNW.“ BMW ormrmépomrm hail C. 331316.. AME meted to Michigan But. University mwmnn-utotwmnmu forthodogmof MASERGFSCIENCE Wt of Mains 1961 ACKNOVLEDGMENT The author wishes to express his gratitude to Dr. A. Leitner for his considerable amount of help end guidance in this work. The author would elso like to thank Dr. R. D. spence for his suggestions. ABSTRACT Solution of the Schroedinger equation for s particle in s central rum in sphere-cone]. coordinates results in len‘ functions. The properties of these functions studied in this thesis ere, existence of polynonisl solutions, orthogonality, relationship to sphericel hsrmonics, normelizstion, end nsture of the eigenvnlues. TAM OF CONTENTS htmmuMOOOOOOOOOOOOO meoordineteSysten........ MPolynaIiQSOlutions. . . . . . . WMWOOOOOOOOOOOI AnAdditionTheom......... Reduction of sphere oconel into Sphericel Ween/890........... Homlizetion.............. Eigemmlm............... Ethic l. Ethic 2. Tihle 3. Table h. List of Eigenvalues. . . . . . . . . Listofligenrunctions. . . . . . . Examples of the Addition meoren . . Reaction of Sphero-conal Harmonics. Page 19 B FIGURES 1’16. Figure l. coordinate surfaces for Sphsro-ccnsl Coordinates ......... .. ..... 15 11:11:02. beentcrsctthsnplaneccnespcnding «4:0 man-=0 ......... 5 m3.,§(fl‘2)/=3 .............. 10-6 I. IBTRODIETION This thesis my be considered a departure from a paper by Professor R. D. Spencel, and some of the mtcrial in the first two sections following the introduction is merely a verification of results which he obtained, shown in greater detail. These results are then used in subsequent work. Upon investigating the orthogonality property, we find the rather mexpected result that these Motions are not completely orthogonal. or special interest to us is the confluence that results when ve vary certain parameters, and causes the sphere- conal coordinate syste- to becom a spherical coordinate system his confluence he been studied in sou detail. sauncumneonntimornnépomanmmvem obtained by Arscottz, end a general introductim to an; auction: canbefoundin3. 'l'heequivalemebetveenthefomofhne’h equatimuedbyArscottandthefomusedinthisvorkeanbe shovnbythetransforntionéachZ) . Siniliarworkhas mommwmu". II. was WINNIE SYB'IEM me sphero-conal coordinates,/r , é , andT] are defined by the following eqmtions: 2 ’/ , “5% + €2)(fi2- 772)] 2 <1» y 365—8 £71 ’/ (lb) ]2 2 2 2 2 2=t_/L[(a-§)(c(+n) (1.) CI mamfimcmtentforeeyparticularcecrdinatesysten, Mahayaeatisfytheconditiona ”+3 .I, 'nserangesof C and TI willbediscmeedlater. Thesystenosnbeshowntobeorthogoml,withthereuouing netricooefficients: 55/ h‘ [ 9;“; )J/1 1‘3" ”[tfiiilgfilr where k). and ‘13 are the metrics associated with C and 71 respectively. 3 Fran equations (In), (lb), and (lo) we can obtain, by eliminating two of the sphero-conal coordinates in each case, .2 .2 Y“_ __ M”? 5%” if- .25? _ + 7? B‘-n°"‘0 i: <3. Cf-r- 7? Xd+ Y2+ Zi= ”ta“ 'i'hese eqmtions show the surfaces C a constant and T] - constant to be elliptic cones about the Z and X axis respectively, and the surface A - constant a sphere. These are sketched in Figure l. Clearlysettinggorn eqmltoseroputsmonthexsplans. Ir €=0,then x“ (aunt) - 2‘13”?) = 0 which, for a given TI is the eqmtion of two straight lines. iii/g X=i[gj+—%] Z Aswevaryn fraOtofB ,theslopeoftheselinesgoesfron i’fl/(X too, givingusasectorofthexsplane. If a TI 0, V. Obmn +[ 01+ ‘1 Vi X align—La 'C‘] 2 Meetwolineshavelepewaryingfron/B/Qatg'40to 00 as €"’C( giving the remaining sector of the xx plane. This is sketched infigm'ez. Figure 1. Coordinate Surfaces for Sphere-cons]. coordinates \n ,3 I I O /’/ /// /\ (b/(f ,x/’ n /’ ,1 / .b ’ /:,, Z ’2’??? Figure 2. The sectors of the x2 plane corresponding to C1110 and q :1 0 6 Whenbothc andTI areequaltozero,weobtaintheeqmtionof thelineewhichborderthesectors, X=:18/C(z . Howat €= 0 , X=x‘((/8:€;L)ZL and Z: A {a;‘§;)fi ,1 a. .2. .9. therefore, Xa+ Z 7-4 ((1 +18 } . This shows that the restriction .1 .1. (1+3 =/ ifnecessaryif/‘l istohavetheneaningof "distance from the origin". To include all points in space, X and 2 artist be defined with a I sign before the radical (cf. eqmtions la and lo). his creates anonunimsencsswherebytoagivcn/‘t , C, and 7? there correspond four points in space. In addition to this, a given point in space my be described by two sets of sphero-conal coordinates, nanely x‘z , ~§ , ~7Zmd A. C . 7? . Theformrproblencannotbeavoided. nu latter multiplicity can be resolved as follow. We can restrict c to OfCSCX andallownsoaewhatgreaterfreedon‘fisnffl . umsnegativevalms of T] willtakecareofpointsononesideofthe are plans, and positive 72's the other side. The choice of restricting 4 my seem.arbitraryz however, as we shall see later, in the limit of ,8 going to zero, 4 because sin 9 , «47768 goestoaingb. Since 9 varies from 0 to 77' andqb rm 0 to 273311;...” logicaltolinit C,snanat 7?, tononmgative values. IfweletCK (or/8)gotosero, oursphero-conalcoordinate systen degenerates to a spherical system with the x and z axis, respectively. 7 beisgthepolarsns. Poe-emu, ifweletfigotosero,this forcesrl togotoseroalso, forequatin(ls)tellsusthatif wedonotplacetherestrictim -,8=77:[3 ,wethenallow X totahoni-ginaryvaluss. Eqmticn (lc) givesusthereduction 2/2- §= sine [8+0 andeeuations (la)aad(lb)tellustutas7] and/Bactonm,we (2) nest have //z'm 7%} 3 5m ¢ (3) 3-0 his reduction brings us to the ecoventimal spherical coordinate systel. Ii'welet IB—e-I (or-*0 ),thereductionhecaes /ém 7? '5 5m 6' (5) fi-/ .123”? 4/02 = sm #5, (5) whichleadstcthesphericaleoordinatesystuwithxaepolaruis. / I 'nseangleeisneasmdfronthesaxis,aad¢isanasi-sthal sngleintheysplsne. Theseredncticnswillbeofvalnelater. 8 The tine independent Bchroedinger equation for a central field is sphero-cmal coordinates is [n+4 ) (orf—c“)C; ,. ESL“ HM W0 77‘) 394 (3+4 + (aini)fi[18&- ”4-),“ c) 02’ 2,)21 a1— slat/I'M ”Jug-+72") MUG”? 18-72 J72+V(/‘)WT‘E‘70 (5) Bessemeasolntionocfthstype LP=PM)Z(§) H {77) Beparatingtheter-sdependingoné and” fruthosedepsnding m/‘i gives [locust :1] 3% Z (4‘1 72“) AH Lk i + 'L ’ ‘- "' L?! (24$? J” ‘ Jfl‘afln‘m'” ”£31324le (7) 9 vaenoveepontothotemsdependingoné fromthoecdependingonn we introduce e not eeparation oonotont, A . ' "/.L t i .1 - A .,. A a; ; 184%Wg (fig—[(13+C lat; )] jg+j(x/+IJ€=A (8o) V). .1 9. 5: a. a. 4K2 ,2. (01+ H (fa-[WWW] fidw/mq It in obvioue frul thee. two equetione that w diocuuion of mo also (3b) eppnee to the other. In feet, H(rz)=cZ{in) I (9) be eolutiue of the“ two eqmtione are called hn‘ function. We chooeo to diam eqmtion (8a), vhich becomes " (14.3:451 ’ 1M4» ) 91A Z+ +Q‘xa‘f-f) Z + wag/2%.?) Z = 0 z"+Ptc)z'+%(4)z = 0 Vominldietolythetbothp 11:qu hevopoleeoforderlat€=ia unfit if} ,nkingtheeopointo regulertinguhrpointeofthe (8.) ‘10 dirtemntiel eqmtion. The point at infinity also turns out to he s resale: singular point. The indicicl roots st the four finite sing-uln- points ere O and 1/2 ineechcese. At infinity, theyere [/2 t‘\/I4+€(€H) . As e check, we may note that ell the indicinl roots, including the two at infinity, odd to e. total of 3 (3, p. 203, footnote). Of the various possible combinstions of indicinl roots st each singularity, we choose to elininete some for convenience. We write the solutions to equstion (8e) es follows: 00 Zm = (aicw‘éwfig‘f‘i 62M 4““ m J's-O K , Kz,cnd KJceneechheloro. ’mis fornelininetes, for example, solutions in which C+£f5 and 44p eppeer explicitly in different powers . Substitution of this solution into eqmtion (8e) yields the recursion relstion :: O (n) 63627+K3+Z + J .ZJHK3 + ejésza-z CI =Q$fiz(3j+K3+2}(2]+K5+l) A]. 7— (2-14- 5)“ 2f +K3- l )(a“/3‘)+(2Kt+\)sz-"(2K|tl)fi‘] +K2C(: K’ {3171 QJ: (2J+K3'2)(23'+K3+' ZK‘ + 2Kl-‘ ) + 2K1+8Ka +2KIK2- _,€(£#/) III. m moan. amass Referring to cqustion (ll), we now proceed to show that, by virtue of the special form of 643. and e 3-, these recursions permit certain terminated solutions. These then lead to sphero-conol functims (10) conteining polynanisls . Inthereduction [3:0 (CHI) , or [3 =/ {C(=0) the sphero-coml coordinates lead to the spherical coordinates. It is the shove referred to polanmisl solutions of the spheroocoml open-star which seduce to the spherical herncnics. He will cell them sphero- cmsl hernonics. Clcerly, for the series in eqmtim (10) to terninete, two con- secutive (5 's mast vsnish. Let us cell these two 64,7413 {.2, and 6;,HK3 +4 , thus raking (5.227%K3 the lest non-sero coefficient. Two conditions must be sstisfied for this to heppen. first, to insure 6&n4K3447-0, weneed Q52. Ea. - + 6 an .m+-K3 an .mwg-a =fl I sndsecond, for Onn+K5+§l tovenishslso,venusthsve Cn-H “ . II 12 Vecsnmltiplyosrtthecnssdcnfl, beceusethsyerenotsglel touroexceptvhena or fl is zero. Kegstive vdaesothotconrse, sreofnosigniricenceinourihylorseries. We emails condition II first. (em/401M +/)+ K3+ol][o2.(n +/)+/<3+ ”2K,+’27(; /]+ “QR,“ZKJ’ 02K, K2:— [1’ 1+ /} =.- 0 mismatimisqmdrsticinn. Hebepinnindthetthemssbern villgovsrnthedegrseorthepolmisl, i.e., 0277+,(3 . Sinceell ottheK;ereoneorsero,vecenusethefectthst Kids/ff . For “mince,“shellxintheqnbolg to K,+KJ-+K3 . #n‘z+[40’+2)n +2(I< lgHg/{s-t/{I KA+U)-j([+/): 0 l7 _ 4403-2) i 2Vl+4/M+/1 _ 8 \ necesseeheremtherirstsemstiucmstutmcslled lit/+4 insteedoi’sispl: C. herons/{Ada neededtorornepertect squere. hisaivesusthefollwingexwessimfor n: n: //2{/-O” (12) histellsusthetfortheseuestoheemel,veuetheve 13 themtrietionthst/(heesinteger. Infect,/-U’Inetbeeneven C‘ + : O n 6JH+K3 er; 61n+K3-a. Prathenetureofthedn noenseethetthisconditimplscese restrictiona A. Itgsneretessiaenvelmsin A . Sousa-pies othowthistekesplsoei'onov. I: /=Ovo seetht K,:K;=K3=Oua n:- 0 . Condition: um simply d060=0 , end sinceéo is es erbitrery ccnstent 30:0. his givesus A: 0 . ItL-Jusnn netheermhutnovueofthe K, uttei. (i) If K325 “13/1230 dad/=0 do gnaw -.- a A za‘iflil—efi” , 0L /\:C(OL=/'/8 lf/r-elueoenhm deitherQorO. HU:0,/7:/ , thsconditim for =0 I7nbecass§l=-§1_ . 2.7+K3 (so I 1h butve know fronthe recursion that C06¢=CJO (30 , so .. + ‘ __ ‘6 d1 ‘ #aZ/QIJ‘A .~ Q/QJ-IB‘I—A no "antic equtiu m; solutions f’éfl-A : 5%: A = n[aifia':W] Ird=2wogotamnn=0,orsosotonpoumsiuthtu condition do=0 . hishecass yieldethem gnaw/10614.2 K, + NE] + Kga‘i K; 8'2— A = 0 Menthreemstohsveo’zoz. (.) K3=0 A” =/<_2=/ /l=/-l)8; “) K120 /(°Z=K3 =/ A :4- 5/82 (c) fizzy /g=/{3=/ A=/«5/3'L VehsvethmohtsinedSnluestor/\ M/:£,3musfor_.1:/, sndlveluefor jzfl. 15 The general condition restricting eigenvalues is obtained as follows: Consider the following forms of the recursion. 6 - flats L+K3 CO [(3 C i 5 ._..___,_5 is, =[_éi+2la_2_]o ¥+K3 (’l 44K; C, K3 a, a! O —Z+‘f3 e " 3’ €- _ €13. '_J-_2ZSI_C " “FE-6 “—4- ) ~§ Caz — '+e’ °§64+K‘ b+K3 -,2. y+K3 4. J+K3 5;- 5:3; 3 clearlyvecenalveysvriteé intern-sot andefinite .2 3+5; 214K34- continned fraction. he condition for termination of the series st 42/1+K3 ave, m (5 M3 :3 :1 (42.”4K3'Q- n and application of this condition to the finite contimsed fraction giving 6.2m “<3 in terms 0: “HQ"- results in en eqmtion in- volflngmlythecj , the dj, andthe eJ . m. equtionwill alsebeinthefornofsfinite contimsedfraction, anditvillbe frus this eqmtim that we can obtain eigenvelue solutions for A . l6 .— — C/fl—/ _ (En—l CO ’7’/ C)}../ __ 2/ .. c _ C—A—L __ d ; fl—L “IV-2’ r" —— ‘J‘_‘ — _ (VI-x3 ”is. \Z/ - ELL. ._ 6/ C/ (I / C To solve this equation fork , we may recall that A appears linearly in each of the 09 . If we put the right side over a common denominator, there will be a term which is a product of " of the c!)- , i.e., which will contain the n th power or A . Cross multiplication by 61/“ raises this to the (nH ) at power, A . thus giving us '14—! values for Now let us look at what happens for fixed / . For convenience, we separate the case where / is odd from the case where f. is even. Hf") is odd, then CT must be odd. There are four possible combinations of the K '5 resulting in odd 0 . 17 s) b) c) d) POOH HOPO HPOO misgimus3caseswith OE/andlcssevithO'rB. Theresa VH/ solutionsror/lforeachcase. ihusthetotaleuberof differenteigenvaluesfor 1'de willbe,usingeqmtim(12), 3 [/Z/L/fi/J + / [/5 /é—3)+/] =21+/ Hf iseven,sgsinhcastisatimotthe K 'saxepeesible, thistiuforevea C7 . K, ’3} K} a) O O 0 b) 1 1 o e) l 0 l a) o 1 1 Vehaveaceseewith 03$ andlcssewith J:O . WWW «eigenvalueswillbe 3% aw + / [WW] ,/ 18 The conclusion is that for any/ there will be Z/+/eigenvalues of A . It should be emphasized that each of these eigenvalues is a function offlor of f3 . Either of these parameters depends on the other through z r” the relation )1 r-lc’ ;/ and only one of them is, therefore, significant. We will choose A as the variable. - r We can now write these eigenvalues as AZ (/8) where (will order the Z/J/eigenvalues associated with each / We shall let f’take on integer values from -/ to /Z. Upon considering/1 as a function ofB , we find that no two A '3 associated with a singlejcross, except at the end points,fl=0 and ,8 = / . We can thus order (by the sizes of the eigenvalues them- selves taken at any point except/3 :0 or [B .-:= / . This means that the largest A; will have 7’; ,5 , the next largest fzjv/ etc. down to [2%. Some of these eigenvalues are now listed in table 1. f To each eigenvalue Al there corresponds aéz’ in the form of equation (10), since a givean has definite K's associated with it. A list of some of the eigenfunctions is given in table 2. It should be noted that until we decide upon a norm for these functions, bothéO and (3/ are arbitrary constants, one corresponding to the odd and one to the even polynomials . 19 Table l. List of Eigenvalues i0 =0 if = 43” A0 .-.-. /, M” A: swflp - X: = 2(/—-2/3y——\/"/3~+/34) X/ =/—Sflp A: :/ ‘2/32/ A; 94-5/3” . Z, :m-mW—BZB“) <2 A.3 A: 5 r. 2431’4 VIP-[3:34 _ -.: 540/3”- 2\/4-,7flz’+4/34 =4 -8/3" A} = 2 —- walla/”[5195" A: = 5 -— mg“? 2\/4 —/3”+ /34 9C z.— 5 —‘7/3 ’1. #4432434 20 Table 2. List of Eigenfunctions Z7:- /O<:€ 965., 2/0: 56/ y/fv 2}: (Big) a "é = X” C” 2’“ “SJ’W‘LFUEJ Z =,/o/g+n)]sz3=—_o (7, “,- - / L4“ -04 7: 7‘: = [ti-£5? +£15€7fi +/(/* 19 +77 )1Zj H1 t-Wtz #34372 ] Again, we maupu the first «nation by Z; l‘ Zr HT and subtract (7) 7" ’*,thouomdby ,5 ‘ T__ T T—IT" W 5, new; H/ 2m 44—4-] -"[/(/'—+/) 1{/+/>]Z:3 H; E H; :0 ///+J'/L[+/J fl ‘2 T ' f = i 5/; (4 ”)ZZ'H/ZJ'HI T f ‘— - ' - H " T 7:- + T’ T T [32H] dd; 5W(Z1, Z3”) 2/32], fifiWA—inj) W hero signifies the “when determinant. 26 Integrating over the sphere, we obtain ?' /30 37W [2(Z’LOCZ/z/r01JfZ/{ZZ’ [/gy‘oja/éo/q 91A,.“ {X . T _O(,B 3, - flCX £W(ZZ; f.LL/:#?Jq+;w/MT31;Z__£ Q ( 1v £4," ) may be identified as i111. 7C! {2. 3sz We discussed the conditions necessary for a bracket like the one appearing above to vanish at the limits -CX andCX in the earlier part of this section. Here we need to have the two brackets vanish simultaneously. Therefore, orthogonality over the sphere exists between Sphero- conal harmonics of different/1(whenever the associated 1:;are unlike, or, if they are alike, whenever both pairs of/éj and,4;:are also alike. V. AN ADDIT ION THEOREM 7.“. Now since the Sphere-cone. harmonics Z/ fikf and the Spherical (5 ' harmonics f0 c are solutions to the same partial differential equation f 7’ in difierent coordinate systems, we expect that Z’? #1 may be expressible as a linear combination of spherical harmonics with the same 47 . We ESSUEEB ,[ 5 7d xi“ 5' [5 ' - .N f / Z/(sMQ a/= ; a P c t // s I s=~£ Let us take the right side of equation (11+) and substitute it for Z in equation (Be). It should satisfy the equation, since 2? is a solution, ’\ and /47/is a constant with respect to the .37; operator. ”El 5 , ‘95 I ' (a \‘ [5q:) 1 S (JTQD fij‘fiéz :52 (fix/DC +[Zflfiyét’A]:/€ 4556 =0 ,/ ‘ 5 . V af[7fl 12L li+1(/*/)€" ‘AI QOCZJ®~O 1...... é) ' ‘- .5:—/[ 4 94 To evaluate (75$ we set it equal to a]?! I CO5 9 3”?"' 6/ ‘* __ fl: / r 5-0327?" J/“A’Je / .21 5f/ 5' <. .__._.i(__.. , -/~, {6/0 2(’+/J(/_ij%é2 ~/- 4’" 5/45/74 7‘:'7‘ “”47"“ This may be rewritten as Z 425 {/8 L434, — [4/j 7: A)’2/32/<51;/6Z f/)> 1175 7— B ”(A/«277722722 2 272 2. M2 l' 0 and finally 2/ /3 g,- {4z’2ti)—.2,/3”[5"+2/222)] “'2 +/2¢‘«5—//‘// _.2)/;2'—’22~2/)//22 7'3/"'32+22:0 (18) giving us the desired relation between coefficients. This relation has been tested and verified for all cases for which T f 75(5V/9' é; (a? was worked out independently, and expressed as a series Offiggf ‘. Some of these are listed in table 3. ( It will be noted that certain of the 1?? are absent in these . ' 5 series. In particular, for a giveneAK: all theié? withts even, or .3; .2, ,2. ha all the\“'8-’< 7* 'flm Jr“. 2 C 3—3—83 I l . ,2 2 ' - ' I z 1 2" l ”I 3‘1.” -1: ‘ ”1‘ +14 l-- 2/31 + V713 rs‘Ma +\;:~,s +53,“ 52?, e .4131qu l 32 J One can, without reference to the recursion, predict which “5?? will be absent. It is a matter of symmetry. Consider the form of the Legendre polynomials. {J J . g" (as c'fijzfi'm (9) (polynomial of degree/Z-s“ in (as 6) Now 5m 9 is even about the equator,9:—- fil/Z , whereas (ad 9 is odd. Thusg?’ is odd or even about the equator according to whether ,4(.s_r is odd or even. ,1 n/ L The sphero—conal harmonics possess like symmetry. The functions ;€;'/91 .4 are odd or even about the equator according to whether/g is l or O. This can be seen from the following arguments: the xy plane corresponds to(:::CX_ in the sphero-conal system, cf. equations (lc) and (15c), due V¥v£ . to the factor(Ex-w>‘) . This factor is present in the Sphero-conal harmonic wheneverfi§:-/’, and absent when &;:.é3, cf. equation (10). A 1: sign must be attached before this radical, respectively, according to whether 9 g 7%. Therefore, depending on the choice of,Z¢, and depending on the value of/C , evenfonly, or oddS only will be present in the addition theorem. VI. REDUCTION OF SPHERO-CONfiL INTO SPHERICAL HARMONICS AS £3-—6>C3 In the limit£§_a¢)and in the limit/3—e>/ the sphero-conal coordinates reduce to spherical coordinates, as has been mentioned in the first section of this thesis. We may expect in these limits that each of the sphero-conal harmnics é7(€)//22}77) and the associated eigenvalues ,AjE'CEL) reduce to a single Spherical harmonic and eigenvalue, respectively. This is indeed the case, as we will show in the present section. Moreover, we shall choose - arbitrarily - to normalize the sphero-conal harmonics in such a manner that they reduce exactly to their associated limiting Spherical harmonic as ,£;'9'C3 . Recalling equations (2), (3), (h), and (S), the confluence of our coordinates into spherical coordinates in the limits/?—912 / respectively takes the form [($22 C : 5Z2; 6 (2) ,890 17(7),! __—; ”5/ ' (3) 39(2 % 70‘) xéfc2z7 7? :;_5(§2 69/ (h) 37/ [’(I/y/ cg :2 58/? Q5/ (5) Is>/ The first system, (/2’ , (9 , (p ), corresponds to polar axis Z , / / the second, (A; ,¢9 , ¢5 ), to polar axis Xi.- 33 3h When we letgi—919, equation (18) becomes 7/ 3' '2 .. 4(5 "11)?“ 0 Indicating that all the as are equal to zero except the two for which T' t. _ /}[:5 . This value of J we will call “1 . (19) M in) ¢ _M 'm’ z . /,.0-M is A" The two surviving terms involve/6 :2 and/£7J (:1 s An linearly dependent upon‘ég . The two surviving terms combine into In AV one, which can be expressed asfié? 5nw7n7qb or i2? (795 A” 95 . This can easily be verified. In the case/3'rffgéf /Zé reduce to a single {é?(?;::b”§0 ,' but not the same 5 as the reduction when/3+ O. This value of 5 we / shall refer to as rn . We choose a normalization for the 21's (and.//as), such that in 1’ 7’ m the limit —»0 /—/, will 0 to . or ‘ ”7 ( )8 )Z,{ i 3 if) ova/77¢) /i,. (as mm with coefficient 1. We can, at the same time, include the proper power ofC) , so that ’2’ z" 235 24/' will remain finite in the CX—e>5) reduction as well, even though, as we find upon detailed calculation, we cannot simultaneously make the coefficient I for both reductions. We will choose the ,L5‘*CD reduction, simply because this takes us to a more familiar coordinate system. A list of some of the sphero-conal harmonics is given in table h along with the klassociated with each, and the “by as particular/2’20; 277(1) or /) 5/4 mq‘) to which they reduce in the . A A! limitp—po . LA) \J] Table h. Reduction of Sphere-conal Harmonics Sphero-conal Harmonic Z~/fi/—/ / / ”<7 1 .L 4’? Reduction as /3 7* O 1 p0 / E's/fig) C§>Cfch5'Qb p a , )0 Two things may be noted in table b. one is the relationlfj(being / ”9/” O or 1) to .rmfly in the 2:? being even or odd, as was predicted earlier. The other is the fact that thecfas /W§t> reduction seems to be related to K3: 5) , and th 5/4 ”NZ; reduction to K; -'= / . This latter fact will be discussed further in the next section. VII. NORMALIZATION As indicated in the foregoing section, we choose, for convenience, . a normalization factor for any given sphero-conal harmonic ééiqgj/29777YJ as follows: In the limitfl-a O , the sphero-conal coordinates flow into the spherical coordinates whose polar axis is if. we now cause 7'Ieégr in the limit /3-> C? to reduce exactly to its associated spherical harmonic. To achieve the normalization, we study the form of the sphero-conal harmonics as obtained from equations (9) and (10). We keep in mind that we are considering the polynomial solutions, i.e. the summation goes to r), not to infinity, We also set (3 of equation (9) equal to l for reasons of symmetry. . K/ (3/), (I n J 4:2?65: [fix-{1d +7] /] [CB+§7(/8‘7( Jj (CU?) wém:€z ) (26M 50),]? (20) 7: £230 g)? :75 ((0.5 QJ/Szflé’j ((5.54)) (ow/(b) (9(26/0;)[0V CS (Xe/men gag». m6 .2222} . m (21) We know that this expression in equation (21) must be Ci; cfca§ /ngb I or (Z? ”5/4 MCfi . Thus we can match up the 6 and QD dependence. We start with the 95 dependence. 37 38 (0.5 Inc'bwith even ”fie an even polynomial of degree /)7 in (5-5 (p or an even polynomial of degree /77 in .574 d) (05 Wfibvith odd /)7 is an odd polynomial of degree fly in (.11; ¢ or is 505$ (even polynomial of degree //7-/ in 5/0 95) 5/1/777prith even /7/ = 320$ (a 54) (even polynomial of degree 0-2 inCuQ) :j//;¢ (05¢) (even polynomial of degreefli-J in 57/705) 5/!) MG) with odd/7 =—>’//)q> (even polynomial of degree ”7—/ 11160.5 Q5 ) = odd polynomial of degree /77 in 5/4 <25 ”- ( 7 Looking at the form of the g A? , it seems better to choose the polynomials in 5/}; Q‘) for our representation. In equation (21), the summation can only contribute an even poly- nomial in 5/4 ¢> . This leavesé05@)fizb®j, and results in the following four possibilities: a) If both/(land (jars zero, we have an even polynomial in 52/? (Z), and the reduction must be to (6.5”) Qb with ”7 even. b) If K2,:0 and KJ=/ we get an odd polynomial inS/O (Z; , or reduction to 5/0 mg) , with W odd. c) If K}:/ and KJ'JO we have (45¢ (even polynomial in 510$ ) meaning a reduction to Ca; m¢Jwith 07 odd. (1) Finally, if both KJ/and KJ are equal to 1 we get 501(1) (054/) (even polynomial in5z'4¢)), i.e. reduction to 5/0 ”7?) n7 even. 39 Summarizing, if 1‘3:- 0, the reduction is always to (as fl)¢". If () L in the limit [8"0 will break off at the point where the c[ of order/8 appears in theéjfl‘ . The last non-zero coefficient will then be c 3 ‘~"'"._‘.‘_2:_‘I‘.2. , corresponding to 2 3’ = m“ K;K . This will contain 61H , 1/ but not 65/,- , which is of order 5 . The highest power of sx'flCb will be /)7—-/<’;(J , in agreement with the conclusion reached on the previous page. We also find that multiplication by a power of )8 is not needed for the 17 series. The series in 4 have coefficients CS which behave like U-fk the ones in the 17 series; however 4 reduces to 107$ (whereas 7 reduced tofiS/kmp) and thus multiplication by some power of/3 will be necessary for finite, non-zero reduction. 1+1 m Let us consider, once again, the 9 dependence of the Pi . P2 = 5m 9 (a polynomial of degree/(Ca, in (45 9) This must be equal to the t9 dependence of the right side of equation (21) i.e. I? if k /d , 43 +6 . ’ / r ’/~ , ’ ((65 9) (”M / 6/] (Z (””9) ’6‘“)? 61.74%?) J30 3’90 The lowest surviving power ofsm (9 must be M , and so the lowest surviving power in the summation must be fl/«lé-‘L'réj . The highest surviving power must bej—m ~43 (always an even number) higher than that. This means that the highest surviving power must be of degree ' [*(j-éJf/{J in 5M 9 . By equation (12), this is equal to 30 , the original highest power of J in the summation. The power of/3 necessary to send the powers of Sin Elower than ”Pg-k.) in the summation to zero is then, clearly, the negative of the power of/g) in (S wich :fl- -/€ . .z:7’+/<’3 I J g 3 N \ "1“; {5 N (‘1 )ngi’ dmitgcfi,“ d.Clo 2 cm-.. -K -1 mpg (3) The 01 of order [3" is not contained here. Thus ( ,Hpos-sesses as its .. M‘ch highest negative power offs ’3‘ ,0 and the series has to be m-K-w multiplied by (3 ‘ 5. h2 K rK In addition, there appears in equation (21) a factor [.3 ‘ 3 . c—(‘I— K This requires multiplication by [3 3 , making the full power of /3 m—ZK‘—-2KJ . necessary for normalization f3 . We can achieve this by Hr %-K‘.{’ 1133’? A completely analogous investigation gives the not too surprising multiplying each of the ,ZQf and m' ~K'-K result that a factor of <1 A 3is required for finite, non-zero reduction of the sphero-conal harmonics in the limit an . m' is defined by t |> It“) /l (p) :oirv‘!) [5.9, f ._ ‘. This leaves us only to consider the factor necessary to make the "‘ (fl 3 . coefficient of Pg COS (EKymmequal to l in the [3’90 reduction. From equation (21) and subsequent work, we see that 7‘ Z -' ’ ’ Z/I/v Z! /4// _;_— 5' 65/409 (def/6:9 (05’6pr 5/.”A/J¢> 6k+0(51.n”¢) .6 £45 ‘ . 3 ‘ /)/—;(;—'<‘j J [/3 (S + (9 (5/4 79) ”7—5 which for small values of 9 and 91) becomes 7 ' -t 42 ZgiJ/IOA, e 5"”[j d) 6k I80 A J ém~£ J 3’ whereas the spherical harmonies for smallé? and(¢)take the following forms: h3 vim . P” A. '7‘} {UN m cos m? ~ m '-—-— Sm J g a m! {vi/rig m LI)!” 1 Pg an cf ~ 1 7,, its;- “1? Sm j 8 m! {é-- m,’ . “71' . 1: Clearly the factor by which 4-1, [—12 needs to be multiplied (let I ‘C' 2. us call it’\. I’ )is (’ t '-,z__ lflljm£+m1-'flr?3 , _ 1 ”ex '" I, , ,I 3- “J ”‘3 Mrs v . 2 m. (I m). m c}? L 0153 [3 d m-Kz fir U mjl —-v—- --—.-—-—.~—-—--....._.- . .-___.___ 2M1m-K ‘1 (t’m 1 'K3 Or Meg-«'3 \ 3"{4 1" L K3/3 Jam—Ks. J wax-m The only surviving term in M“ will be the one containing {3 3 1» cf. equation (23). Let us examine this term. I O Kim fl m4 .30 @ "1323};4 , . . C0 C0 From equation (22) we can see that the product of the d 's can be written as [(m—af— M; i 55!. N i 3 hh . «5+2 iKétl I From equation (11) we can write the C s as m K WM 1 so that Z‘ m k-K3 f __ ._ M :5“; arm-2 "'2 (gm- 4 f4” , .‘KfistM/«gkgka-Wl} W fl QM, - '2 .. . . (”0 am‘Kz},m"‘z“. Meats”? :3“) ("“4143 (2rn—2,{|2m~42. . , , I‘vYHKthjf \ I f! _' 'l: ' \ .. gf 1.3 ‘33. .. tm—-K‘,_.—!+K}} It will be recalled that Mama-K3 is always even, ,’ ,t ‘-_2 {It {(+migl}.‘31{5},, W1 KfH-KJ} 1A a :- -. .. \ , —-— (21+) \ [.1 2 {anKayh’e-Yni! (.ng \rr\+K+K3 erK3 +K+2}. ”:2m'2; Now if we define N: by .- 1 (Am N: ’ZHK: =Pm Casi-ilkrmcp; (5-90 1‘ {4 I -\ E2 -K «K3 (25) 75 c I \ : //" I; - K2: K3 _ 1 \Je /‘\Q' (3 (I ' EIGENVALUES we discussed briefly how the eigenvalues r);Q3) were obtained in section III, and a partial list was given in table 1. In figure 3, the 7 eigenvalues associated with [:3 have been sketched. It may be noted from figure 3 that the eigenvalues come together in pairs at both ends of the range of fi3(0 and l), where they take on values of squares of integers. Only the value 0 is taken on by a single eigenvalue. These values can be easily seen from equation (19) to be rn‘at(3=0(analo- gously (m‘)3 at {3‘4 ). These degeneracies correspond to the degeneracy of the eigenvalues m m A . of the pair of spherical harmonies F; Ufi>QJanmfand IZ<¢J‘9/Cos~?when 3-49, '7‘. , I I . I , zmdoftmagnrtgfiwad snvn? mm FyYUMQ)C35M? mmn ,5.al , as was already shown on pp. 33 and 3h. ( We find, empirically, that the/dz M correspond to P, C05 «1? (and <3 =0 ), whereas the smaller T with the larger 1: will will correSpond to the F?“ St“ nu? reduction (and ‘5331 ). This enables us to write an empirical formula for if. It will be recalled that the ”('8 are ordered by the size of the corresponding /l: , and that 1' takes on integer values from —£ to ,6. We can.order the C 's, then, in terms of the values of the ii; whenIB=C>(i.e. by ordering the vn's), and resolve the duplicity resulting from two ): having the same value, of; by using the empirical result Just mentioned, and distinguish the two by their ‘§3 's. The expression we want then is: ’t : Einq~—E —,<é t . Figure-5, MP),3=3- 1+7 It would be desirable to prove the empirical fact that, of the two T ,, ,%£ (X?) which reduce to the same A7 , the one with.F<3 ==69 is the larger. In order to show this, we embarked on a calculation of a series ?’ >' 2/ expansion for A2? (12’) in the powers of 18’ . It is permissible to assume that éijzmvk A, 31;" A” [3): , , , (.26) where 0 f M f j , and. where to each value of”) are associated two eigenvalues of different 3'. The values of./\} can be found by successive approximations of the finite continued fraction determining the eigenvalues, as given on p. 16. It is interesting to note that for the pairs of eigenvalues P reducing to the same/W asfi§*>-o , Jaj’ and JAuL’ come out to be the same. Fbr example A / ”i [Ma/[([wA “To 11 KB— [2491)? h=O -/\2’ also does not differ for the pair of eigenvalues, except for 'similar special cases. We suSpect strongly that a difference will appear in JAE , but the calculations for higher coefficients becomes extremely laborious, and no results have yet been obtained for this case. BIBLIOGRAPHY 1. R. D. Spence, Am. J. Hum, Vol. 27, No. 5, 329-335, (1959). 2. r. H. Arscott, J. London Math. Soc. 31, 360361., (1956). 3. Hhittaker and Hatsm, A Course in Modern Analysis, hth edition, Cambridge University Press (1958). h. 0. Kraus, mean 11.8., Michigan State University, (1952). HICHIGQN STQTE UNIV LIBRQRIES 293017040050