The energy eigenvalue equation for the linear potential

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Abstract
Recently the solution of the linear potential problem for any value of the orbital quantum number l was given via a new class of combinatorics functions. This paper deals with the eigenvalue problem and presents a further reduction of the energy eigenvalue equation leading to a general method for obtaining numerical results. Originally the eigenvalue equation was obtained by requiring the vanishing of an infinite order polynomial, Hl(t) =J∞m=02(l +1) Km(l) tm−1, that should reduce to the Airy function, Ai(−t), in the case l=0. The expansion coefficients Km(l) were in turn given as the limit for infinitely large integer k values of some other coefficients K(k)m(l). These latter coefficients were given explicitly in terms of combinatorics functions, σl(k,p), that individually diverge in the limit as k goes to infinity. No real attempt was made to show that H0(t) ≡Ai (−t), to calculate the limit as k→∞, or even prove the cancellation of infinities. This work has three objectives: (i) Show the complete equivalence between the combinatorics function solution and the Airy function at l=0; (ii) Obtain a closed form expression for the expansion coefficients 2(l+1) Kp(l) by relating them in a very simple manner to the finite part fp(l) of σl(k,p) in the limit as k→∞; (iii) Calculate a few lower order coefficients fp(l) and obtain some numerical results.

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