A study of new solvable few body problems

Abstract

We study new solvable few body problems consisting of generalizations of the Calogero and the Calogero–Marchioro–Wolfes three-body problems, by introducing non-translationally invariant three-body potentials. After separating the radial and angular variables by appropriate coordinate transformations, we provide eigensolutions of the Schrödinger equation with the corresponding energy spectrum. We found a domain of the coupling constant for which the irregular solutions are square integrable.

Lagrange-mesh solution of the Schrödinger equation in generalized spherical coordinates

“…where c k are constants. These constants are determined consistent with the general Pöschl-Teller Hamiltonian [19]. The eigenvalues corresponding to (19) are…”

“…All the differential equations (13) have a form that is consistent with the Pöschl-Teller Hamiltonian [19]. Such a Hamiltonian admits exact solutions that involve Gegenbauer functions, which are a special case of the Jacobi functions.…”

“…Because of this symmetry, the equations may be solved considering only one segment of each domain and then adapt the results to the entire domain. Such a treatment is also necessary when solving (13) with potentials that have singularities that segment the angular domains [19,20]. This approach leads to general solutions of the form P sin cos cos 2 19…”

“…] . Therefore, solutions to (1) with general potentials may involve combinations of (16) and (19). However, the relation (16) may be more suited for domains in which the potential is even.…”

“…Illustrative applications of the two solutions discussed above can be found in the literature. The D=3 case, the solution of which involve Θ 2 (θ 2 )Θ 1 (θ 1 ) usually referred to as the ring-shaped functions, is discussed in [21], while several D3 cases can be found in [19,20]. The construction of the numerical solutions Θ k (θ k ) to (13) in the Lagrange-mesh method is discussed in the next section.…”