Quantum equivalent of the Bertrand's theorem

A procedure for constructing bound state potentials is given. We show that, under the natural conditions imposed on a radial eigenvalue problem, the <span>*only*</span> special cases of the general central potential, which are exactly solvable and have infinite number of energy eigenvalues, are the Coulomb and harmonic oscillator potentials.

The celebrated Bertrand's theorem in classical mechanics states that <span>*the only central forces that result in closed orbits for all bound particles are the inverse square law and Hooke's law*</span> @ber [@gold]. The extension of the Bertrand's theorem to spherical geometry led again to two potentials for which closed orbits exist @higgs. In the limit of vanishing curvature, these potentials reduced to the oscillator and Coulomb potentials. As is well known, these two potentials are also special in quantum mechanics. These are the only ones, which can be solved exactly in arbitrary dimensions and have infinite number of bound states. From the symmetry point of view, the degeneracy structure of these potentials admit a Lie group larger than the $SO(d)$ group for a non-relativistic particle moving in $d$-dimensional Euclidean space @bac [@ste]. For the Coulomb problem, the group is $SO(d+1)$ for bound states and $SO(d, 1)$ for scattering states. For harmonic oscillator, the corresponding group is $SU(d)$. In case of spherical geometry, the eigenvalue problem can be solved exactly only for the two, above mentioned potentials @higgs [@lak]. These are the only central potentials for which the corresponding Schrödinger equations can be factorized to yield both the energy and angular momentum raising and lowering operators @sch [@pr; @fac1; @fac2]. These two potentials are also special from the semi-classical point of view @iva. Recently, it has been shown that, starting from a suitably gauged action, these two potentials can be derived in lower dimensions by appropriate gauge fixing @bars. However, unlike the classical case where a general central potential, under the twin constraints of bound and closed orbits, led to the Coulomb and oscillator potentials; in quantum mechanics, the natural conditions imposed on a radial eigenvalue problem, in a given dimension, have not yielded these two potentials as unique choices.

In this paper, we construct a family of potentials, dependent on a parameter $\alpha$, keeping in mind the two important features of the quantum mechanical bound state problems, <span>*i.e.*</span>, the discreteness of their eigenspectra and the polynomial nature of the corresponding normalizable wavefunctions. Interestingly, under these general conditions, there are only two central potentials which are exactly solvable and have infinite number of bound states. These two unique potentials, appearing for the two values of $\alpha$, one and two, are the Coulomb and the harmonic oscillator potentials, respectively. This quantum mechanical situation is an exact analogue of the Bertrand's theorem in classical mechanics.

We begin with a Fock space spanned by the monomials, $\rho^n$; where, $n =
0,1, 2, \cdots$. The most general diagonal operator which has a well defined action on this Fock space can be written as $\sum_{k = - \infty}^\infty C_k
D^k$; here, $D \equiv \rho \frac{d}{d \rho}$ and $C_k$'s are constants. The two special cases corresponding to, (i) all the $C_k$'s being zero except $C_1 = 1$ and $C_0 = \epsilon$ (modulo an overall scaling factor) and (ii) all the $C_k$'s being zero, except $C_2 = 1$, $C_1 = \beta$, and $C_0 =
\delta$, can be unambiguously mapped to the Schrödinger equation by a series of similarity transformations. The first case leads to a class of potentials, dependent on a parameter $\alpha$. It is explicitly shown that, the only two exactly solvable radial potentials, having infinite number of energy eigenvalues, are the harmonic and Coulomb potentials. Our method naturally gives their respective eigenfunctions. Further possibilities of solvable potentials are analyzed by performing a point canonical transformation. It is found that, no other exactly solvable central potential results and Morse potential arises as a conditionally exactly solvable case of this general potential. We then construct the second class of potentials and point out that, unlike the first, this case can not be solved as a radial eigenvalue problem, for infinite number of levels, for all the allowed values of $\alpha$ and for the various possible choices of the other parameters appearing in it.

The choice (i), mentioned above, yields, $$\label{eu}
\left(\rho \frac{d}{d \rho} + \epsilon \right) \eta(\rho) = 0 \qquad;$$ where, $\eta(\rho) = \rho^{- \epsilon}$.

Performing a similarity transformation (ST): $\phi(\rho) =
\exp\{\hat{A}/\alpha\} \eta(\rho)$, where, $\hat{A} \equiv a \rho^{2 -
\alpha} \frac{d^2}{d \rho^2} + b  \rho^{1 - \alpha} \frac{d}{d \rho} + c
\rho^{- \alpha}$, $\alpha$, $a, b$ and $c$ being arbitrary parameters, one gets $$\label{bc}
(\rho \frac{d}{d \rho} + a \rho^{2 - \alpha} \frac{d^2}{d \rho^2} 
+ b \rho^{1 - \alpha} \frac{d}{d \rho} + c \rho^{- \alpha} + \epsilon) \phi =
0 \qquad.$$ The above choice of the ST is motivated from our desire to map (\[eu\]) to the Schrödinger equation and also to have a closed form expression for the second order differential equation.

Choosing $\phi = \rho^{\alpha - 2} \chi$, it is straightforward to check that, $$\begin{aligned}
 \label{chi}
\frac{d^2 \chi}{d \rho^2}&& + \frac{1}{a} \left(\rho^{\alpha - 1} + (b - 4 a
+ 2 a \alpha) \rho^{-1}\right) \frac{d \chi}{d \rho} \nonumber\\
&&+ \frac{1}{a}
\left((\epsilon + \alpha - 2) \rho^{\alpha - 2} + (a \alpha^2 + (b - 5 a)
\alpha + 6 a - 2 b + c) \rho^{- 2}\right) \chi = 0 \qquad.\end{aligned}$$ For the purpose of clarity, we will cast (\[chi\]) as a radial eigenvalue equation in three dimensions; it can be easily seen that our method generalizes to $d$-dimensions. Taking $\psi(\rho) = S(\rho)
\chi(\rho)$, one gets $$\label{sch}
\left(\frac{d^2}{d r^2} + \frac{2}{r} \frac{d}{d r} - \frac{l(l + 
1)}{r^2} + \frac{2 m}{{\hbar}^2} (E - V(r)) \right)
\psi = 0 \qquad,$$ where, the radial coordinate $r = \lambda \rho$, $\lambda$ being an appropriate length scale. Here, $S(\rho) = \rho^{A_0} \exp\{\rho^\alpha/(2 a
\alpha)\}$, $A_0 \equiv (b - 6 a + 2 a \alpha)/(2 a)$ and $l$ is the conventional angular momentum quantum number.

The explicit form of the potential is $$\label{gcp}
V(r) = E + g_1 r^{2 (\alpha - 1)} + g_2 r^{\alpha - 2} + g_3 r^{- 2} \qquad,$$ where, $g_1 = \tilde{g}_1 \lambda^{2 (1 - \alpha)}$, $g_2 = \tilde{g}_2
\lambda^{2 - \alpha}$, $g_3 = \tilde{g}_3 \lambda^2$, $\tilde{g}_1 =
{\hbar}^2/(8 m a^2 \lambda^2)$, $\tilde{g}_2 = {\hbar}^2 (2 A_0 - \alpha + 5
-2 \epsilon)/(4 m a \lambda^2)$ and $\tilde{g}_3 = {\hbar}^2 \{A_0 (A_0 + 1) -
l (l + 1) - [a (2 - \alpha) (3 - \alpha) + b \alpha - 2 b + c]/a\}/(2 m
\lambda^2)$.

The corresponding unnormalized eigenfunctions, obtained through inverse similarity transformations, are $$\psi(\rho) = \rho^{(\pm \sqrt{\Delta} - 1)/2} \exp\left\{\rho^{\alpha}/(2 a
\alpha) \right\} L_n^{\pm \sqrt{\Delta}/\alpha}\left(- \rho^{\alpha}/(a \alpha)
\right) \qquad,$$ where, $\Delta \equiv (1 - b/a)^2 - 4 c/a$. Sign of $\sqrt{\Delta}$ should be chosen such that the resulting wavefunctions are normalizable. The Laguerre polynomial, $L_n^{\pm \sqrt{\Delta}/\alpha}$, is obtained by demanding that the $(n + 1)th$ term in $\exp\{\hat{A}/\alpha\} \rho^{- \epsilon_n}$ is equal to zero@ak; <span>*i.e.*</span>, $\hat{A}^{n + 1} \rho^{- \epsilon_n} =
0$. This gives $$\label{eev}
\epsilon_n^{\pm} = - \alpha n - (1/2) (1 - b/a) \pm
(1/2)\sqrt{\Delta}\qquad.$$

Before proceeding to study the exact solvability of the special cases of the bound state problem obtained in (\[gcp\]), it is worth repeating the corresponding scenario in classical mechanics. In classical case, one demands the bounded motion of a test particle, subjected to a general central force of the form $F(r) = - \kappa r^{\beta^2 - 3}$, to trace a <span>*closed orbit*</span>; this leads to two possible values for $\beta$, one and vknight@example.net. These values give rise to the well known inverse-square and Hooke's laws, respectively. Interestingly, our general construction leads to a central potential, governed by a real parameter $\alpha$. Much akin to the classical case, <span>*only*</span> for $\alpha = 1$ and $2$, as will be shown below, this potential can be solved for the complete set of energy eigenvalues. This is a quantum analogue of the classical Bertrand's theorem.

The general central potential in (\[gcp\]) has free parameters $g_1$ or $g_2$ for $\alpha 
= 1$ or $2$: one can then impose $E + g_i = 0$, for $i = 1, 2$, in order that the potential is independent of constant terms and the corresponding Schrödinger equation is exactly solvable for all the energy eigenvalues.

<span>*Case I*</span>: $\alpha = 1$.

To cast the eigenvalue problem in the standard radial form, <span>*i.e.*</span>, to make $g_3 = 0$, we choose $a = 1/(2 \sigma)$, $b = 2/\sigma$ and $c = [2 -
l(l + 1)]/(2 \sigma)$ with $\sigma^2 = - 2 m \lambda^2 E/{\hbar}^2$. We now demand that $E +g_1 = 0$. (\[gcp\]) then reduces to the Coulomb potential, $$V(r) = \frac{g_2}{r} \qquad.$$

Choosing $g_2 = - e^2/(4 \pi \epsilon_0)$, $\epsilon_n^-$ in (\[eev\]) gives the energy eigenvalues as $$E = -  g_1 = - \frac{m e^4}{32 \pi^2 \epsilon_0^2 {\hbar}^2} \frac{1}{(n +
l + 1)^2} \qquad.$$

<span>*Case II*</span>: $\alpha = 2$.

Akin to the previous case, choosing $a = 1/(2 \sigma)$, $b = 1/\sigma$ and $c
= - l (l + 1)/(2 \sigma)$ with $\sigma = - m \omega \lambda^2/\hbar$, one can check that $g_3 = 0$. Further demanding $E + g_2 = 0$, the potential reduces to the harmonic oscillator potential: $$V(r) = g_1 r^2 \qquad.$$

For, $g_1 = m \omega^2/2$, the eigenspectra is $$E = -  g_2 = \hbar \omega (2 n + l + 3/2) \qquad.$$ It is worth pointing out that only for these two values of $\alpha$, one can make the potential independent of constants and hence obtain the full spectrum. We would like to add that, the above two cases can also be solved for $g_3 \ne 0$@lan [@mpl; @fac3]. For values of $\alpha$ different from one and two, the potential can be solved for $E = 0$, but for different values of the angular momentum quantum number l. This situation is similar to the classical scenario, where, some values of $\beta$ other than $1$ and $2$ may give rise to bounded motion, which is not closed.

At this moment, one is naturally curious to see whether there exists the possibility of exactly solvable models emerging, under a point canonical transformation (PCT) $\rho \rightarrow f(\rho)$ when $\alpha \ne 1,
2$@de [@pkp]. Performing the PCT, (\[bc\]) can be brought to the form (\[sch\]), with a potential given by $$\begin{aligned}
 \label{pct}
\frac{2 m \lambda^2}{{\hbar}^2} V(\lambda \rho) &=& \frac{2 m
\lambda^2}{{\hbar}^2} E + \frac{1}{4 a^2} (f^\prime f^{\alpha - 1})^2 
+ \frac{1}{2 a^2} [b + a (\alpha - 1 - 2 \epsilon)]
(f^\prime)^2 f^{\alpha - 2} \nonumber \\
&&+ \left[\frac{b}{2 a} \left(\frac{b}{2 a} -
1\right) - \frac{c}{a}\right] \left(\frac{f^\prime}{f}\right)^2 - \frac{l (l +
1)}{\rho^2} \nonumber\\
&&+ \frac{3}{4} \left(\frac{f^{\prime \prime}}{f}\right)^2 
- \frac{1}{2} \frac{f^{\prime \prime \prime}}{f^\prime}
- 2 \frac{f^{\prime \prime}}{f f^\prime} 
+ (2 - \alpha) \frac{f^\prime f^{\prime
\prime}}{f^2} + 2 \left(\frac{f^{\prime \prime}}{f^\prime}\right)^2 \qquad,\end{aligned}$$ where, $f^\prime \equiv df/d\rho$.

Different choices of $f$ will give rise to different potentials; however, all of them may not be exactly solvable. For example the potential in (\[pct\]) can be made independent of constants, even for continuous values of $\alpha$, if one chooses $f(\rho) = e^\rho$: $$\begin{aligned}
\frac{2 m \lambda^2}{{\hbar}^2} V(\lambda \rho) = \frac{1}{4 a^2} e^{ 2
\alpha \rho} + \frac{1}{2 a^2} [b + a (\alpha - 1 - 2 \epsilon)] e^{\alpha
\rho} - 2 e^{- \rho} - \frac{l (l + 1)}{\rho^2} \qquad.\end{aligned}$$ Since, the above potential contains the term '$- l (l + 1) \rho^{- 2}$', one can only solve the eigenvalue equation for a fixed '$l$'. This is the case of the conditionally exactly solvable potential encountered in Ref.. Physically this implies that, the potential is exactly solvable only in one dimension, <span>*i.e*</span>, without the centrifugal term. The corresponding energy eigenvalues are independent of $l$; they can be obtained from $$\frac{2 m \lambda^2}{{\hbar}^2} E +
\frac{b}{2 a} \left(\frac{b}{2 a} - 1\right) - \frac{c}{a} - \alpha + 17/4 =
0 \qquad,$$ and the unnormalized eigenfunctions are $$\psi = \frac{1}{\rho} f^{(\pm \sqrt{\Delta} + 1)/2}
\exp\left\{f^{\alpha}/(2 a \alpha)\right\} L_n^{\pm
\sqrt{\Delta}/\alpha}\left(- f^{\alpha}/(a \alpha)\right) \qquad.$$ One can easily see that, the Morse potential emerges as a special case for $\alpha = - 1$. Now, we proceed to find the second class of potentials starting from $$\left(D^2 + \beta D + \delta \right) \bar{\eta}(\rho)  = 0 \qquad.$$ Writing $\bar{\chi}(\rho) = \exp\{- \gamma \hat{O}\} \bar{\eta}(\rho)$, where $\hat{O} \equiv a \rho^\alpha \frac{d}{d \rho} + b \rho^{\alpha - 1}$, one gets $$\label{F}
\left(F_1(\rho) \frac{d^2}{d \rho^2} + F_2(\rho) \frac{d}{d \rho} + F_3(\rho) 
+ \delta\right)\bar{\chi} = 0 \quad,$$ where, $F_1(\rho) = A_1 \rho^2 (\rho^{\alpha - 1} + A_2)^2$, $F_2(\rho) = B_1
\rho (\rho^{\alpha - 1} + B_2)^2 + B_3$ and $F_3(\rho) = C_1 (\rho^{\alpha
- 1} + C_2)^2 + C_3$. The coefficients are given by $A_1 = A_2^{- 2}$, $A_2^{-1} = (\alpha - 1) \gamma a$, $B_1 = (\alpha a + 2 b)/(a A_2^2)$, $B_2
= [2 b + a (3 + \beta - \gamma)]/(2 a A_2 B_1)$, $B_3 = 1 + \beta - B_1
B_2^2$, $C_1 = b (a \alpha - a + b)/(a^2 A_2^2)$, $C_2 = b (\beta - \gamma +
1)/(2 a A_2 C_1)$ and $C_3 = - C_1 C_2^2/b$. Here, if one chooses $\hat{A}$ instead of $\hat{O}$, then the resulting equation will contain still more higher derivative terms and can not be straightforwardly mapped to a Schrödinger equation.

Now, writing $\bar{\chi}(\rho) = \left(F_1(\rho) S(\rho)\right)^{-1}
\bar{\psi}(\rho)$; where, $S(\rho)$ can be obtained from $$\frac{1}{S} \frac{d S}{d \rho} = \frac{F_2(\rho) - 2 F_1^{\prime}(\rho)}{2
F_1(\rho)} - \frac{1}{\rho} \qquad,$$ it can be checked that, $\bar{\psi}$ obeys the Schrödinger equation with a potential, $$\begin{aligned}
\tilde{V}(\lambda \rho) &=& \tilde{E} + \frac{[B_1 \rho (\rho^{\alpha - 1} +
B_2)^2 + B_3][D_1 \rho (\rho^{\alpha - 1} + D_2)^2 + D_3]}{4 A_1^2 \rho^4
(\rho^{\alpha - 1} + A_2)^4} \nonumber\\
&&+ \frac{D_1 (\rho^{\alpha - 1} + D_2) [(2 \alpha - 1) \rho^{\alpha - 1} +
D_2]}{A_1 \rho^2 (\rho^{\alpha - 1} + A_2)^2} \nonumber\\
&&+ \frac{8}{\rho^2} \left(\frac{\alpha \rho^{\alpha - 1} + A_2}{\rho^{\alpha -
1} + A_2}\right)^2 \left(\frac{1}{\rho} + \frac{\alpha - 1}{\rho^{\alpha - 1}
+ A_2} + \frac{\alpha (\alpha - 1)}{\alpha \rho^{\alpha - 1} + A_2} \right)
\nonumber\\ 
&&- \frac{[C_1 (\rho^{\alpha - 1} + A_2)^2 + C_3 + \delta]}{A_1 \rho^2
(\rho^{\alpha - 1} + A_2)^2} - \frac{l (l + 1)}{\rho^2} \qquad.\end{aligned}$$ Here, $\tilde{V}(\rho ) \equiv \frac{2 m \lambda^2}{{\hbar}^2} V(\lambda
\rho)$, $\tilde E \equiv \frac{2 m \lambda^2}{{\hbar}^2} E$, $D_1 = (2 b - 3
\alpha a)/(a A_2^2)$, $D_2 = [2 b + a (\beta - \gamma - 4 \alpha - 1)]/(2 A_2
D_1)$ and $D_3 = 1 + \beta - 4/D_1 - D_1 D_2^2$.

Unlike the first class, these potentials can not be made independent of constants as a general radial problem, for any special values of $\alpha.$ Hence, the possibility of obtaining infinite number of levels, in arbitrary dimensions, does not arise here. However, it can be solved for the $E=0$ state, for different values of $l$.

At this point, we would like to add that, there have been many works in the literature, where point canonical transformation is used after writing $\psi(x)=f(x)F[g(x)] $, for mapping the Schrödinger equation to the confluent hypergeometric or hypergeometric equations @ecg [@nat; @gin; @khare]. Hence, from the known solutions of these equations, one is able to obtain solvable potentials in the Schrödinger eigenvalue problem, after imposing certain restrictions on f and g. In our case, we have started with the diagonal operators $D \equiv \rho \frac{d}{d \rho}$ and $D^{2}+\alpha{D}$ acting on the space of monomials $\rho^n.$ By inverse transformation, we, not only obtained the Schrödinger eigenvalue equation, but also the connection of the respective eigenfunctions with the monomials. In this approach, the special nature of the Coulomb and oscillator potentials, <span>*i.e.*</span>, exact solvability with infinite number of levels in arbitrary dimensions, came out naturally without taking recourse to PCT.

We would also like to point out that, a number of Calogero-Sutherland type interacting, many-body Hamiltonians, both in one and higher dimensions, can be mapped to the Euler operator $\sum x_i\frac{\partial}{\partial{x_i}}+c$, through similarity vknight@example.net. In fact, our partial motivation behind this work is due to a theorem established in Ref. :

All $D$ dimensional $N$ particle Hamiltonians, which can be brought through a suitable transformation to the generalized form: $\tilde H = \sum_{l=1}^D
\sum_{i=1}^N x_i^{(l)} \frac{\partial}{\partial x_i^{(l)}} + \epsilon + \hat
A$ can also be mapped to $\sum_{l=1}^D \sum_{i=1}^N x_i^{(l)}
\frac{\partial}{\partial x_i^{(l)}} + \epsilon$ by $\exp\{- d^{-1} \hat A\}$; where, the operator $\hat A$ is any homogeneous function of $\frac{\partial}{\partial x_i^{(l)}}$ and $x_i^{(l)}$ with degree $d$ and $\epsilon$ is a constant. For the normalizability of the wave functions, one needs to check that the action of $\exp\{- d^{-1} \hat A\}$ on an appropriate linear combination of the eigenstates of $\sum_{l=1}^D \sum_{i=1}^N x_i^{(l)}
\frac{\partial}{\partial x_i^{(l)}}$ yields a polynomial solution. It will be of great interest to apply the method employed here to the many-body systems and examine carefully, how far the results obtained for the single particle case generalizes to the N-particle systems. It is worth mentioning here that, apart from the Cologero-Sutherland model, which has recently been mapped to harmonic oscillators@prb, the other many-particle exactly solvable system is of Coulombic vknight@example.net.

The authors would like to acknowledge useful discussions with Profs. V. Srinivasan and S. Chaturvedi. N.G thanks U.G.C (India) for the financial support through the S.R.F scheme.

<span>99</span>

J. Bertrand, Comptes Rendus <span>**77**</span>, 849 (1873).

H. Goldstein, Classical Mechanics, 2nd ed., (Addison-Wesley, New York, 1980).

P.W. Higgs, J. Phys. <span>**A**</span>: Math. Gen., <span>**12**</span>, 309 (1979).

H. Bacry, H. Ruegg and J.M. Souriau, Commun. Math. Phys. <span>**3**</span>, 323 (1966).

P. Stehle and M.Y. Han, Phys. Rev. <span>**159**</span>, 1076 (1967).

M. Lakshmanan and K. Eswaran, J. Phys. <span>**A**</span>: Math. Gen. <span>**8**</span>, 1658 (1975).

E. Schrödinger, Proc. R. Ir. Acad. <span>**46 A**</span>, 9 (1940).

F. Cooper, A. Khare and U. Sukhatme, Phys. Rep. <span>**251**</span>, 268 (1995) and references therein.

J.D. Newmarch and R.M. Golding, Am. J. Phys. <span>**46**</span>, 658 (1978).

Y.F. Liu, Y.A. Lei and J.Y. Zeng, Phys. Lett. <span>**A 231**</span>, 9 (1997).

I.A. Ivanov, J. Phys. <span>**A**</span>: Math. Gen., <span>**29**</span>, 3203 (1996).

I. Bars, pre-print no. USC-98/HEP-B2, hep-th/9804028.

N. Gurappa, A. Khare and P.K. Panigrahi, Phys. Lett. <span>**A 224**</span>, 467 (1998).

L. Landau and E.M. Lifshitz, Quantum Mechanica: Non-Relativistic Theory (Oxford Univ. Press, 1975).

N. Gurappa, C.N. Kumar and P.K. Panigrahi, Mod. Phys. Lett. <span>**A 11**</span>, 1737 (1996).

Z.B. Wu and J.Y. Zeng, J. Math. Phys. <span>**39**</span>, 5253 (1998).

R. De, R. Dutt and U. Sukhatme, J. Phys. <span>**A**</span>: Math. Gen. <span>**25**</span>, L843 (1992).

A. Gangopadhyaya, P. Panigrahi and U. Sukhatme, Helv. Phys. Acta. <span>**67**</span>, 363 (1994).

A. Bhattacharjie and E.C.G. Sudarshan, Nuovo Cimento <span>**25**</span>, 864 (1962).

G.A. Natanzon, Teor. Mat. Fiz. <span>**38**</span>, 146 (1979).

J.N. Ginocchio, Ann. Phys. (N.Y.) <span>**152**</span>, 203 (1984).

F. Cooper, A. Khare and U. Sukhatme, Phys. Rep. <span>**251**</span>, 267 (1995), and references therein.

N. Gurappa and P.K. Panigrahi, Phys. Rev. <span>**B**</span>, R2490 (1999), and references therein.

A. Khare, J. Phys. <span>**A**</span>: Math. Gen. <span>**29**</span>, L45 (1996).

{'source': 'arXiv_out/quant-ph9905011'}