Nonsingular spiked harmonic oscillator V. C. AguileraNavarro and R. Guardiola Citation: Journal of Mathematical Physics 32, 2135 (1991); doi: 10.1063/1.529186 View online: http://dx.doi.org/10.1063/1.529186 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/32/8?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:00:15 http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/1774531301/x01/AIP-PT/JMP_ArticleDL_022614/aipToCAlerts_Large.png/5532386d4f314a53757a6b4144615953?x http://scitation.aip.org/search?value1=V.+C.+AguileraNavarro&option1=author http://scitation.aip.org/search?value1=R.+Guardiola&option1=author http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov http://dx.doi.org/10.1063/1.529186 http://scitation.aip.org/content/aip/journal/jmp/32/8?ver=pdfcov http://scitation.aip.org/content/aip?ver=pdfcov Nonsingular spiked harmonic oscillator V. C. Aguilera-Navarro Institute de fisica Tebrica-UNESP, 01405 Sao Paula, SP, Brazil FL Guardiola Departamento de FIsica Atbmica y Nuclear, Avda. Dr. Moliner SO, 46IOO-Burjasot, Valencia, Spain (Received 11 December 1990; accepted for publication 3 April 199 1) A perturbative study of a class of nonsingular spiked harmonic oscillators defined by the Hamiltonian H = - d 2/dr2 + r 2 + ;1 /r cI in the domain [ 0, CO ] is carried out, in the two extremes of a weak coupling and a strong coupling regimes. A path has been found to connect both expansions for a near 2. I. INTRODUCTION The spiked harmonic oscillator system is defined by the quantum Hamiltonian , , drL ra defined in the one-dimensional half space [ 0, to 1, the eigen- functions obeying Dirichlet boundary conditions. Its name comes from the graphical shape of the full potential, which shows a pronounced peak near the origin for /2 > 0. We will assume this condition through the paper. This is a two parameter problem, R and a, with physical relevance but also with fascinating properties from the point of view of mathematical physics. First of all, there is no dominance of any of the two terms of the interaction poten- tial r 2 and /2 /r n for extreme values of /2. So, in the case of A -0, the l/r” adds an infinite repulsive barrier near the origin. On the other hand, in the /2+ 00 limit one cannot neglect the r 2 term, the potential being in this case like a wide valley extending from 0 to co. Potentials of this kind are of relevance in a wide range of physical situations, in chemical physics, nuclear physics, and particle physics. However, we will pay more attention to their mathematical physics interest.‘-’ Detwiler and Klauder2 realized that normal perturbation theory could not be applied for values of a)5/2, and they were able to predict the kind of dependence of the ground state energy for small values of /2. Afterwards Harrel13 developed a special perturbation theory, called singularperturbation theory, and obtained the first terms of the small /z expansion, which turned out to be a nonpower series expansion. Special meth- ods have been designed to compute the eigenvalues with high precision. 4*5 EstCvez and the present authors6 managed to obtain a strong coupling expansion for the ground state ener- gy- Our interest in this problem was motivated by Harrell’s result for the ground state energy corresponding to a = 5/2, E(&,A)=3+ ‘r’:;;;; A 16 +- I-( l/2) ;1*lnil+O(i12) (2) and the close analogy with the low-density expansion of the energy of a many-body boson system at zero temperature’ E/N= (2&‘/m)pa[ 1 + C, (pa3)“* + C2pa3 ln(pa3) + C3pa3 + ***I. (3) We intended to apply extrapolant methods to Eq. (2) analo- gous to those so successfully used in the case of the low- density expansion’ to extend the range of applicability of Eq. (2) to higher values of R. We did not find the appropriate extrapolant, even if the information coming from the strong coupling expansion was used. So we come back again to this problem, i.e., to find a bridge connecting values of the ground state energy at small il with values at large il, this time focusing the simpler, non- singular cases corresponding to a < 5/2. The paper is organized as follows. In Sec. II we obtain the weak coupling expansion using standard perturbation theory up to second order. In Sec. III we extend the algebraic calculations of our previous work6 so as to compute up to the tenth order of the strong perturbation expansion. In Sec. IV we try to get a path from the weak coupling regime to the strong coupling one. This turns out to be quite simple for a = 2, but again we do not find the way of dealing with the other cases. However, we have found this path for values ofil in the vicinity of 2, as we show in Sec. V. Finally, in Sec. VI we present a summary of the problem, as it was treated here. II. THE WEAK COUPLING PERTURBATION EXPANSION ww Let us split the Hamiltonian ( 1) as usual in an H,, part H,= -$+r’, r>O and a perturbation HI =R/ra. (5) The eigenstates of Ho are the odd-parity solutions of the one- dimensional harmonic oscillator satisfying the Dirichlet boundary condition g(O) = 0. We will represent these states by In), n = 0,1,2 . . . . their unperturbed energy being E (‘) = 3 + 4n. n The relevant matrix elements for the perturbative calcu- lation up to second order are6 (Olr-“In) = I?[ (3 - a)/21 ( - 2)“(a/2), dmr(3/2) ’ (6) 2135 J. Math. Phys. 32 (a), August 1991 0022-2488191 I0821 3507$03.00 @ 1991 American institute of Physics 2135 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:00:15 where (a) n is the Pochhammer symbol. The WCP expan- sion for the ground state energy results Eo=E~oP’+S,R+S,A2+~~*, (7) with s, = I-[ (3 - @)/21/I-(3/2) and (8) s = _ rW3 -d/21 c 2*"(a/2)f 2 r73/2) nZ~ 4n(2n + l)! ’ (9) provided the relevant integrals are defined, i.e., only valid for the nonsingular cases where a < 5/2. In the case a = 5/2 the correction S, is well defined, but the sum appearing in Eq. (9) is infinity.6 An alternative way of writing S, is s, = - rw3- a)/21 c (a/2)2, r73/2) n+~ 4n(3/2),n! * (10) We have not found a simpler form for S, , except for the case a = 2, to be discussed later. The sum appearing in Eq. ( 10) is very slowly convergent, so that to have several digits of precision it will be necessary to consider a huge amount of terms. For practical use, it is convenient to use one of the stan- dard methods for summing up convergent series. Let us make a comparison with another exactly summable series, Consider the sum F= c (a/2): n#o 4(n + 1) (3/2),n! (11) obtained by replacing the quantity n of the denominator of Eq. (10) by (n + 1). Let (12) be the sum we are interested in. Obviously, for n large enough the terms of F and G will be very close. These series are connected by the expression G=F+ c (a/2)2, +O 4n(n + 1)(3/2),n! ’ (13) The correction series in this equation will require less terms to be computed. Note finally that Fmay be written in terms of a special form of the hypergeometric function F= l+l (14) whereas the correcting series is evaluated by direct summa- tion. In this way one obtains E,(a=t)=3+1.022766/2-0.0137842A2++*~, E,(a = 1) = 3 + 1.1283792 - 0.0778902,X 2 + ++A, &(a = 3) = 3 + 1.382735/2 - 0.334783/2 2 + *a+, Eo(a=2)=3+2A--2A2+***. (15) We have also included the term corresponding to a = 2 in the above list of results, but this term cannot be obtained by the above described technique. Actually, in this case Eq. ( 14) has no sense. However, this case may be evaluated by carrying out transformations on Eq. (9) : use the duplication formula of the gamma function9 to obtain (2n -+- l)!= (2rr)-122”+3’21’(12+ l)I’(n + l/2) andrea- lize that the resulting sum is equivalent to S,(a=2) = -j,F,(l,l;$;l) = -2. (16) It is not a surprise to obtain simple numbers for the perturbative expansion in the case a = 2. Actually, the Schriidinger equation may be exactly solved in this case, as found in some textbooks on quantum mechanics.1o The unnormalized eigenstates of the full Hamiltonian are given by $,(a = 2) = rPexp( - r2/2)M( - n,p,r2), (17) with p = [ 1 -t j-1 /2 and M is the confluent hyper- geometric function that degenerates in this case to a polyno- mial in r ‘. The corresponding eigenvalues are E,(a=2)=2+4n+dw. (181 From this closed form for the energy one can easily obtain both a weak coupling expansion and a strong coupling ex- pansion, namely, E,(a=2) =3-t-U--2A2+4A3- 10it4+ -10 E,(a=2)=2+2@ 1+--&A 128R.’ + 1 5 -- *‘* 10242 3 32 768/i 4 + I ’ (19) the first expansion being valid for /,l ] < l/4 whereas the sec- ond converges for ]A I> l/4. Before ending this section we would like to mention a question regarding these cases of nonsingular perturbation theory. The problem is that whereas the resulting expansion for the energy is neatly obtained (even more, it agrees with the numerically determined eigenvalues for small ;1), the perturbed wave functions cannot be always obtained in the same way. Dealing specifically with the a = 2 case we have for the first correction to the wave function tihlj(a = 21 = dT%w( - f-‘/2) 4I’(3/2) xc (j- l)!( - l)QYZj+, (r) (2j-l- l)! ’ (201 &O which may be rearranged in a power series in r by using the explicit form of the Hermite polynomials. It results tlAIj(a = 2j = “‘“‘“,‘,4;p:, r2/2) xc u- wm$, ;.;);:;;2;;;, . if0 (21) By collecting the terms ofeach given power of r in the double sum of Eq. (2 1) it turns out that each coefficient is infinity. For example, the r term of this double sum has the coeffi- cient 2X,?= , l/j, i.e., the harmonic series. This is a quite sur- prising result, having a properly defined expansion for the 2136 J. Math. Phys., Vol. 32, No. 8, August 1991 V. C. Aguilera-Navarro and FL Guardiota 2136 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:00:15 energy eigenvalues but an inappropriate expansion for the wave function. In other words, at least for a = 2 the stan- dard perturbation theory is in fact a singular perturbation theory in Harrell’s sense (2). This result for the wave function is easily understood if one expands the exact wave function in powers ofR. One has to use the wave function ( 17) normalized, because the norm depends also on L For n = 0 we have tio(a=2) = ( 2 > l/2 r[ (2 + JGm/2] Xr r2 (I + YT’m/2eXp _ 1 ( > and the first-order perturbative correction is (22) ‘I%” = 2r-“4/2r[1n r- $+J ($)]exp + , ( > (23) the coefficient of the exp term is no longer a polynomial in r. This explains the strange result of the perturbation expan- sion for the wave function, but nevertheless does not clarify why the energy has a regular expansion. It would be interest- ing to find a special summation formula for Eq. (2 1) which could produce the nonpower series expansion of Eq. (23). Ill. THE STRONG COUPLING PERTURBATION EXPANSION (SCP) Probably the most popular case of an SCP expansion is the anharmonic oscillator x2 + Rx4, which is also described in quantum mechanics textbooks. When ;1 is very large, the x2 term can be neglected with respect to the perturbation ,Xx4, and a scaling in the coordinate leads to the conclusion that the energy in the L + m limit is given by nil “3, where E is the eigenvalue of the operator - d 2/dx2 + x4.” In our case, however, there is not such a simple transi- tion. Even for large values of the coupling constant both terms of the interaction x2 and A /x a are important. A spe- I E _ * = w2/(w2 - 2), E, =co, E, = ( - co4 -I- 1 1w2 - 10)/72, E4 = (w8 - low6 - 39w4 + 3200’ - 272)/17280, E6 = ( - 317~” + 2847~” + 29 6610~ - 162 779~~ - 793 236~~ + 4 130 832~’ - 3 207 008)/9 331 200w2, Es = (2177~‘~ - 8692~‘~ - 551 5780” + 2 902 556~” + 14 552 8970~ - 56 629 936~~ - 324 369 568~~ + 1 337 123 072~~ - 973 020 928)/895 795 200w3, E,, = ( - 624 689~~’ - 5 466 679~‘~ + 363 722 7060’~ - 1 117 832 430~‘~ - 22 741915 965~” + 104 066 243 541~” + 265 411 385 292~~ - 1017 106 275 9360~ - 5 381 642 164 608~~ +2O15151O2743O4w2-l4O98737345536)/316O3654656OOw4. (27) t In Table I we show the values of these coefficients for taking a quick look at the tendency of these coefficients. So, some particular values of a which correspond to nonsingular in Table II we have included the values of the same set of perturbation cases. These values are of interest for further coefficients, this time computed in FORTRAN with a floating- analytic study of the series, but it does not help very much point representation. The table extends up to E,, . Note that 2137 J. Math. Phys., Vol. 32, No. 8, August 1991 V. C. Aguilera-Navarro and R. Guardiola 2137 cial way of dealing with the strong coupling limit of the spiked oscillators was presented in full detail in our previous work.6 The procedure is based in an idea borrowed from Witten’s study of hydrogenic atoms in the so called l/N expansion,‘* where N is the number of dimensions of the space. The system is described starting from the classical equilibrium point, i.e., the minimum of the full potential, and afterwards the quantum corrections and the residual interaction are added up. In practice this means that there are a base or zero energy given by the value of the potential at the minimum V( rmi, ), a zero-order Hamiltonian Ho = - d 2/dx2 + w22 of harmonic type, w being related to the second derivative of the potential at the minimum, and finally anharmonic corrections 23, z4,... . In this description z is the distance measured from the classical equilibrium point, z = r - rmin. In this way there results an expansion of the energy in powers of the quantity p = (2/Aa)“(a+2) (24) of the form E(p)=E-,/-‘+EO +E~P~+E~P~+***, (25) i.e., involving nonintegral powers of the coupling constant il. The way of computing the coefficients E, as well as some of them (up to E4) was presented in Ref. 6. The calculation was greatly reduced by using computer algebraic manipula- tion codes, but, perhaps due to our lack of experience with this technology, we were only able to compute up to E4 be- fore running out of memory. Some improvements have per- mitted us to arrive up to E,, . The results are simpler to display (and compute) in terms of the parameter W=Jm, (26) and are listed below: This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:00:15 TABLE I. The coefficients of the SCP expansion up to order 10 in rational (or irrational) form for several values of the exponent a from l/2 up to 5/2. a l/2 1 312 2 5/2 E 5 3 7 -* -T 2 9 3 EO lr 5 2 8 J 7 r 2 $ E* 5 7 65 1 73 32 36 288 4 288 4 99 37 1465 - 0 1967 512m 4326 13 824,~% 41 472J2 85 877 2573 - & 29 921 -1 - 46 748 387 2 048 000 139 968 I1 943936 xi- 2 687 385 600 20 240 451 168 233 - E* 73 238 695 0 25567506209 13 1072 OOO,t% 67184644 16 052 649 984$4 1 547 934 105 600@ 40 1673843203 11 834297 - 1085434210835 1 1483965461593 20971 520000 725594112 1585744976019456 512 1337415067238400 with the exception of the case a = 2, the coefficients behave rather erratically with the order. It turns out that this perturbation expansion is abnor- mal. The eigenvalues obtained with this method, as we have shown in Ref. 6, do actually give a very good value of the ground state energies for very large A, but the radius of con- vergence of the SCP expansions is not known (obviously, with the exception of the case a = 2). On the other hand, we know that the expansion for the wave function is not conver- gent because the successive corrections extend also into the interval [ - CO ,O]. Note that we are dealing in this SCP ex- pansion with a one-dimensional harmonic oscillator prob- lem extending in all the space. So we are again facing an expansion reasonable for the energy but inappropriate for the wave function. IV. IS THERE A PATH FROM I- 00 TO It+O? The content of this section is quite speculative. We know few terms of the WCP expansion and also several terms of the SCP expansion. On the other hand, except for the case a = 2, we do not know the radii of convergence of these expansions. The question is if it is possible to find a construc- tive way of connecting both expansions and, still better, to get an estimate (even if approximate) of the ground state energy valid for all regimes of the coupling constant. The case of a = 2 is quite illuminating because in this case there is a simple and obvious way of carrying out this connection. Going back to the second of Eqs. ( 19)) which is a particular way of writing down the SCP expansion, we square the series in brackets and take the square root of the result: E,(a=2)=2+2& l+l-- 1 8;1 128/l’ f 1 5 2 l/2 ~- 1024A 3 + 32768A4 “* 11 ’ (28) Computing the square of the inner bracket produces, within the known coefficients, the very simple result of 1 f l/4/2 so that one ends up with Eo(a=2)=2+2fl(1+ l/4/%)“’ (29) which is the exact result quoted in ( 18). Of course, the ma- TABLE II. The coefficients of the SCP expansion up to order 20 in floating form for several values of the exponent a from i/2 up to 5/2. All digits are exact but the last, which may have been rounded. N a= l/2 a=1 a = 3/2 a=2 a = 5/Z -2 5.00000000 3.ooOOOOOo 2.33333333 2*oOOOOOtM 0 1.58113883 1.73205081 1.87082869 2.00000@00 2 0.15625000 0.19444444 0.22569444 0.25000000 4 0.06114560 0.04944898 0.02832304 0 6 0.04193213 0.01838277 - 0.00250512 - 0.01562500 8 0.04883265 0.01445708 - 0.00121935 0 10 0.07981506 0.01630980 - 0.00068449 0.00195312 12 0.16418623 0.02190383 - 0.00131199 0 14 0.45470360 0.03492499 - 0.00116044 - 0.000305 18 16 I. 16029548 0.06508549 - oJnM44473 0 18 3.78818681 0.13837039 0.00005696 0.0000534 1 20 13.86291828 0.32974546 - O.COOO2870 0 1.80000000 2.12132034 0.26736111 - 0.03353779 - 0.01739549 0.01167941 0.00110958 - 0.00682551 0.00254260 0.00817874 - 0.01109314 - 0.01442769 2138 J. Math. Phys., Vol. 32, No. 8, August 1991 V. C. Aguiiera-Navarro and R. Guardiola 2138 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:00:15 nipulation of this series may seem speculative, but it is a common useful practice in the field of series extrapolation. Actually there are two steps involved in this procedure. First, the explicit presence of nonintegral powers is removed by using the trick of rising the series to some adequate power, and maintaining frozen (i.e., not expanding) the compensat- ing inverse power. The second step, which was not used here, is to construct Pad& or similar extrapolants to the trans- formed series, so as to have still control of the (known) do- minating power in the limit of interest. This technique has been largely applied in different physical systems as two- species-fermion hard-spheres,r3 fully interacting fermion gases, I4 and Lennard-Jones boson fluids.” We now try to apply the same procedure to other cases. The SCP series has only even powers of the effective cou- pling constant p = (2/&x) “(a + ‘) so the simpler case will correspond to a = 1. The actual series for the energy has the form E(a= I> (30) with En given in Tables I and II. The most immediate way of getting rid of the nonintegral powers parallels the a = 2 case. Consider the dominant term with coefficient E _ 2. The next exponent differing from 2/3 in an integer is the term E4, the next one is E,, , and so on. So we may extract a subseries from Eq. (30) : ($)='3[E-2 +E4($)=+E,,($‘++ -] (31) and the way of having only integral powers of/2 is to raise the square bracket to the power 3 and later take the cubic root, or some other combination like 2/3 and 3/2. Something similar may be done with the other two sub- series contained in Eq. (30)) namely the series containing Eo,Et,,.‘,,... and theseries E2, Es,... . Unfortunately, the meth- od does not work. In fact, at the end there will result expres- sions depending upon/z * that will not be able to generate the WCP expansion which contains also odd powers of il. In addition, this mechanism of extracting subseries will depend critically on the value of a and on the congruencies of the exponents 2n/ (a + 2). In conclusion, some new kind of extrapolation mecha- nism should be devised to deal with this analytic continu- ation. Certainly, there remains still the question of the exis- tence of such a path connecting the SCP and the WCP expansion. This question is analyzed to some extent in the next section. V. A TWO-PARAMETER PERTURBATION EXPANSION The key point of this section is that because we do not know what to do with one-parameter expansion, let us jump forward and add an extra parameter. Again we are inspired by the many-body problem, specifically in the so-called van der Waals or quantum thermodynamic perturbation theory (also applied in the study of classical fluids), in which the low-density expansion of the energy per particle is expanded again by splitting the interaction into a strongly repulsive part and an attractive part switched on by means of the new expansion parameter. ‘G-‘~ The resulting double series is af- terwards analyzed by constructive methods. Our idea is to expand the perturbative series around the values a = 2, where we know the way of connecting the two perturbation regimes. So consider a = 2 + 77 and expand both the SCP and the WCP expansion up to first order in 7. From Eq. (24) we readily get ,~==;l-“~[l+(lnR/16-~)~+**~], (32) and the expansions (27) of the E, in terms of 7 turn out to be E-,=2-7,7/2+..., E,=2+~/4+..., E, = l/4 + $24 + . . . . E4 = -17/16-t .*., (33) E6 = - l/64 - 61~/3840 + ‘0.) E8 =7/64+ a.., E,, = l/512 + 527~/64512 + a.* . From here we may obtain the new SCP expansion, which with some special ordering of its terms turns out to be ... I (34) I valid for small 17 and large R. The second term appearing in -$~lnil(il/&iZX), (35) the rhs of Eq. (34) is our previous series (28), so that, for /2 > - l/4, its value is dm. The series of the third term which we guess to be valid for il > 0. Something analogous happens with the fourth term, its value being $ /( 1 + 4/2). can be recognized as - l/q=, so that the third Unfortunately neither chance nor intuition suggest any term becomes closed form for the last series. 2139 J. Math. Phys., Vol. 32, No. 8, August 1991 V. C. Aguilera-Navarro and R. Guardiola 2139 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:00:15 There is, however, an alternate approach to this two- parameter expansion problem, which consists in using stan- dard perturbation theory to compute E(2 + r,~), at least up to first order in 7 and valid for allphysical values of ;1. Take now as the unperturbed Hamiltonian Ho= -$+r=+$ (36) and use as perturbation H, = -/277ru21nr, (37) whichistheo(v) termofil(l/rZf~- l/r’).Notethatdue to the particular dependence on ;Z of the perturbation ( 37)) which is linked to 7 in the form /2~, a perturbative calcula- tion at first order in v will be valid for all values of R, pro- vided the integrals are well defined. The final result will be valid for/z > - l/4, as it will be shown. One can compute the 0( 7) correction by direct integra- tion ofH, times the square of the ground state wave function mentioned in Sec. II, Eq. (22). However, the presence of the logarithm in H, invites us to search for an alternate proce- dure. It turns out to be much simpler to get the first-order correction by expanding in powers of 7 the expectation value of the full perturbation ( l/r2 + R - l/r ‘). This is a valid procedure for the first-order correction, but it will not be appropriate for higher orders. One obtains in this way r([(4-==~)/21 -1 (38) I and the energy is given by E=2+Jm- ,,A ?b( Jy) + O(rl21, , (39) where $ is the digamma function.’ The importance of the above equation is that it is the link between the SCP regime and the WCP regime, at least in some region around of = 2. From Eq. (39) one can obtain the new form of the WCP expansion corresponding to Eqs. ( 15), with the result a2 + rl) = 3 + a (2 - wqf)) +a “( - 2 4 27&Q) - v&)) + **I. (40) It is quite easy to check that the coefficient of il coincides with the expansion of S, , Eq. (8). However, it is quite cum- bersome to obtain directly the coefficient of /z 2 from the valueof&, Eq. (9). More interesting is the case of large /i, i.e., the SCP re- gime. It is not easy to obtain the expansion of Eq. (39) for large /2. Terms like jm are transformed as Ji-Fiz=2Jx- ( > I,--$ I’= =2J;i- 1+&--&+&+*** ( > ’ (41) resulting in an expansion in nonintegral powers of R. The trouble comes with the digamma function. It is convenient to use the asymptotic expansion’ f/i(z) =lnz--&---&+---&-l+ ***, 252~~ (42) with z=jX-(l+ 1/42)“2 (43) expanded as above, and reexpand each of the terms of the digamma asymptotic expansion. The final result of all this procedure is, as expected, exactly Eq. (34). So, to conclude this section, we have at least found an expansion of the ground state energy, given in Eq. (39), which serves as a bridge between the K --+ 0 and the R -+ 00 regimes. This representation, however, is only valid in some region around LZ = 2. One then may guess that there must exist a constructive method to go from the SCP to the WCP regimes. To find it makes up an interesting challenge. VI. SUMMARY In this work we have obtained several results concerning the family of nonsingular spiked harmonic oscillators, as de- scribed below. ( 1) The weak coupling expansions for several values of the exponent a of the repulsive barrier, Eqs. ( 15). (2) The strong coupling expansion up to tenth order, in algebraic form. (3) A constructive method to connect both expansions for arbitrary coupling constant/z in a region around Q = 2. However some new questions have arisen regarding this family of potentials, and we will single out two of them. ( 1) The abnormal behavior of the standard, weak cou- pling perturbation theory, when trying to get the first correc- tion to the wave function, and (2) our failure in obtaining a general constructive method to connect the two perturbative regimes for general values of the exponent a of the perturba- tion. We hope our work will encourage further research, not only in the simpler case of nonsingular interactions, but also in the more challenging situation of singular perturbative cases. ACKNOWLEDGMENTS The authors are grateful to Manuel de Llano for his advice and stimulus along the development of this work. The patience of computer analysts of various computing centers (Granada, Valencia, Sevilla, and SBo Paulo) and their allowance to use extensively computer resources is fully ac- knowledged. One of us (R. G. ) is grateful to the Instituto de Fisica Te6rica (UNESP) for its warm hospitality, and to the Junta de Andalucia for supporting the travel expenses. R. G. was supported by CICyT, Spain under Grant 0969/87. ’ R. M. Spector, J. Math. Phys. 8,2357 ( 1967). IL. C. Detwiler and J. R. Klauder, Phys. Rev. D 11, 1436 ( 1975). ‘E. M. Harrell, Ann. Phys. (NY) 105,379 ( 1977) and references therein. 4M. Znojil, J. Math. Phys. 30,23 ( 1989). ‘M. Znojil, J. Math. Phys. 31, 108 (1990). 6V. C. 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