arXiv:math-ph/0612053v1 16 Dec 2006 Green’s operator for Hamiltonians with Coulomb plus polynomial potentials E. Kelbert, A. Hyder, F. Demir, Z. T. Hlousek and Z. Papp Department of Physics and Astronomy, California State University, Long Beach, California 90840 (Dated: February 7, 2008) The Hamiltonian of a Coulomb plus polynomial potential on the Coulomb-Sturmian basis has an infinite sym- metric band-matrix structure. A band matrix can always be considered as a block-tridiagonal matrix. So, the corresponding Green’s operator can be given as a matrix-valued continued fraction. As examples, we calculate the Green’s operator for the Coulomb plus linear and quadratic potential problems and determine the energy levels. PACS numbers: 03.65.Nk I. INTRODUCTION Coulomb plus polynomial potentials, v(r)= k  i=−1 a i r i = a −1 /r + a 0 + a 1 r + a 2 r 2 + ··· , (1) are often used to model various physical phenomena. The Coulomb potential, v(r)= a −1 /r, describes the interaction between charged particles. A Coulomb plus linear potential, v(r)= a −1 /r + a 1 r, also known as Cornell potential, is the most common potential for modeling the confining quark in- teraction. It is also used in atomic physics to describe the Stark effect which occurs when the hydrogen atom is placed in an electric field. The two dimensional Coulomb plus quadratic potential, v(r)= a −1 /r + a 2 r 2 , is related to the Zeeman effect; the hydrogen atom in magnetic field. The quartic har- monic oscillator potential, v(r)= a 2 r 2 +a 4 r 4 , is used in field theory to model the spontaneous breaking of symmetry. It is evident that there is a great deal of physics that depends on the precise knowledge of the dynamics of the Coulomb potential with various polynomial interactions. Over the years, several approaches have been developed to study some special cases of this problem (for a recent review see Ref. [1]). To the best of our knowledge, no method has been proposed yet that could treat this problem with arbitrary potential strength a i and arbitrarily high power of k. In this work we calculate the Green’s function of the non- relativistic quantum Hamiltonian with a Coulomb plus poly- nomial potential. The Green’s function of polynomial poten- tials on the harmonic oscillator basis has been calculated be- fore [2, 3]. The use of the Coulomb-Sturmian basis will allow us to incorporate the Coulomb potential. If we know the Green’s operator, then we have complete knowledge of the physical system. For example, the eigenval- ues of the Hamiltonian coincide with the poles of the Green’s operator. The corresponding eigenvectors can be determined from the relation |ψ n 〉〈ψ n | = 1 2πi  C G(z )dz, (2) where C encircles the eigenvalue E n in a counterclockwise direction without incorporating other poles. In our previous works, Refs. [4, 5], the Coulomb Green’s operator was calculated in the Coulomb-Sturmian basis. In this basis, the operator J = z − H has an infinite sym- metric tridiagonal, i.e. Jacobi, or J-matrix structure, where z is a complex number. We have shown that the G(z )= (z − H ) −1 = J −1 Green’s operator can be calculated in terms of continued fractions. This Coulomb Green’s operator was used to solve the Faddeev integral equations of the three-body Coulomb problem [6]. In the Coulomb-Sturmian basis, the Hamiltonian with the potential (1) is represented by an infinite symmetric band ma- trix. An infinite band matrix can always be considered as a block-Jacobi matrix with m × m blocks, where m is finite. Thus the continued fraction becomes a matrix-valued contin- ued fraction. This paper is organized as follows. In Section II we in- troduce the D-dimensional Coulomb-Sturmian basis. In Sec- tion III we calculate the infinite band matrix representation of the Hamiltonian. In Section IV we derive the matrix continued fraction for the Green’s operator. Some applica- tions are considered in Section V. First, to demonstrate the power of the method we show an analytically known case, the harmonic oscillator in two and three dimensions. Then we consider the Coulomb plus linear confinement potential v(r)= a −1 /r + a 1 r in three dimensions and the Coulomb plus quadratic confinement potential, v(r)= a −1 /r + a 2 r 2 , in two dimensions. II. THE COULOMB-STURMIAN BASIS The kinetic energy operator in D-dimension, with D ≥ 2, is given by H 0 = − 1 2  d 2 dr 2 − L(L + 1) r 2  , (3) where L = l +(D − 3)/2. The Coulomb-Sturmian (CS) functions are the solutions of the Sturm-Liuoville problem of the Coulomb Hamiltonian [7]  H 0 − λ r  〈r|n〉 = − b 2 2 〈r|n〉, (4) where b> 0 is a parameter, n is the radial quantum number, n =0, 1, ··· , and λ =(n + L + 1)b. In coordinate space, the