1. Introduction

Interest in the bounded quartic oscillator started with the pioneering work of Barakat and Rosner [1] who employed a power series method to obtain numerical values of the eigenvalues through an iteration scheme. Researchers have since continued to investigate the bounded quartic oscillator and related systems, using various techniques (see references [2, 3, 4, 5] and the references in them).

The bounded quartic oscillator is described by the Hamiltonian

(1)

where is the mass of the oscillator and is the coupling constant. The Hamiltonian lives in a Hilbert space with inner product between any two real-valued functions and in defined by , where the functions and and indeed all vectors of are required to vanish at the boundaries

About three and a half decades ago, using their hypervirial pertubative method (HPM), Fernández and Castro [2] derived the following expression (their equation (22) in our notation) for the eigenvalues of the bounded quartic oscillator:

(2)

for quantum numbers

In this paper we show that the standard Rayleigh-Schrödinger (RS) perturbation theory with as the perturbation parameter gives the same result for as given in (2). As a matter of fact we stumbled upon the work of Fernández and Castro only after we had obtained our result for . The Computer Algebra System Waterloo Maple came to our aid in simplifying the resulting perturbation sums and finding their closed form.

2. Basis Functions and the Matrix Elements of H

Since , the eigenstates of have definite parity. For the complete orthonormal functions where,

constitute a suitable set of basis functions in for a matrix representation of the bounded quartic oscillator Hamiltonian , since they also satisfy the boundary conditions .

The identities

and

for and the repeated application of Leibnitz rule for differentiating an integral allow to calculate the matrix elements of as

(3)

provided that and have the same parity, and otherwise.

The matrix elements facilitate the direct diagonalization of the bounded quartic oscillator. The energy eigenvalues can be made arbitrarily accurate by increasing the dimension of the Hamiltonian matrix used; the eigenvalues obtained can therefore be considered exact. We are, however, not concerned here with exact diagonalization but we need the matrix elements for our perturbation calculations.

For sufficiently small (see [3] for a rigorous discussion of the convergence criteria), the oscillator potential may be treated as a perturbation of the unperturbed Hamiltonian (the free particle in a box Hamiltonian).

In the standard Rayleigh-Schrödinger perturbation theory for non-degenerate states, the approximate energy eigenvalues of , to second order in , are to be calculated from . We have immediately that

(4)

and

(5)

where

The second order correction to the energy of the bounded quartic oscillator, , is given by

(6)

where

(7)

so that

(8)

Since is a real symmetric matrix, (6) is simply

(9)

We note that the matrix elements occuring in (9) are necessarily off-diagonal (since ). Furthermore the only surviving elements , according to (3), are those for which and are both odd or both even.

It therefore follows from (3) that

(10)

where

Taking (8) into account, the summand in (9) is therefore

(11)

The sum in (9) is easier to evaluate if the energy eigenvalues are grouped by parity:

and

for quantum number

Using the summation identity (equation 2.6 of [6])

where denotes the floor of , that is, the greatest integer less than or equal to , the above sums can be expressed as

(12)

and

(13)

Maple is able to evaluate the sums in (12) and (13), with the appropriate summand in each case obtained from (11), and we have (see the Maple code in the appendix)

and

from which it follows that

(14)

Adding (4), (5) and (14), we finally obtain

as an approximate expression for the eigenvalues of the bounded quartic oscillator.

3. Summary and Conclusions

Using the Rayleigh-Schrödinger perturbation theory and with the aid of a summation identity and the Computer Algebra System Maple, we have derived an approximate expression for the energy eigenvalues of the bounded quartic oscillator. This is the same expression that was obtained much earlier in reference [2] through a more complicated approach. Similar results to ours are also contained in reference [3] where exact diagonalization was done and perturbative series up to the third order in were also developed for the energy levels. However, the RS sums were determined numerically in that paper as, apparently, closed form could not be found for them; and furthermore only results for the ten lowest eigenvalues were computed.

Appendix

Maple code to evaluate

>V:=(r,s)->(-1)^(r+s)*c1*(1/(r-s)^2-1/(r+s+2)^2) -(-1)^(r+s)*c2*(1/(r-s)^4-1/(r+s+2)^4):

>epsilon:=(r,s)->epsilon*(r+s+2)*(r-s):

>simplify(expand(eval(V(r,s)^2/epsilon(r,s),[r=2*r,s=2*s]))) assuming r,posint,s,posint,c1>0,c2>0:

> summand:=convert(

>S1:=sum(summand,s=0..r-1) assuming r,posint,s,posint,c1>0,c2>0:

>S2:=sum(summand,s=r+1..infinity) assuming r,posint,s,posint,c1>0,c2>0:

> ssum:=simplify(S1+S2):

>fsum:=expand(eval(ssum,[epsilon=Pi^2*h^2/8/m/a^2, c1=16*lambda*a^4/Pi^2,c2=16*24*lambda*a^4/Pi^4])):

>arranged:=collect(fsum,Pi):

>terrms:=[seq(factor(op(i,arranged)),i=1..nops(arranged))]:

> add(L,L=terms);