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Journal of Mathematical Physics

Partially solvable quantum many-body problems in D-dimensional space (D=1,2,3,…)

Francesco Calogero

Published Sep 1, 1999• 8 citations
Publisher

Abstract

A simple technique employed almost three decades ago to manufacture partially solvable quantum many-body problems is revisited. [A quantum problem is “partially solvable” if (only) some of its eigenvalues and eigenfunctions can be exhibited]. The models thereby generated are characterized by Hamiltonians of normal form, i.e., standard kinetic plus momentum-independent potential energy; in most cases the latter features three-body, in addition to two-body and one-body, interactions. The setting refers to D-dimensional space; the examples focus on D=1, D=2, and D⩾2, and include generalizations of, and additional results on, cases recently discussed in the literature, as well as new models.

DOI
10.1063/1.532961

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A simple technique employed almost three decades ago to manufacture partially solvable quantum many-body problems is revisited. [A quantum problem is “partially solvable” if (only) some of its eigenvalues and eigenfunctions can be exhibited]. The models thereby generated are characterized by Hamiltonians of normal form, i.e., standard kinetic plus momentum-independent potential energy; in most cases the latter features three-body, in addition to two-body and one-body, interactions. The setting refers to D-dimensional space; the examples focus on D=1, D=2, and D⩾2, and include generalizations of, and additional results on, cases recently discussed in the literature, as well as new models.
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We introduce a family of many-body systems of distinguishable continuous-variable particles in which interparticle interactions are set by the adjacency matrix of a graph. The ground-state wave function of such systems is of a generalized Jastrow form involving the product of pair-correlation functions over the edge set of the graph. These systems describe quantum fluids when the graph is complete, and the pair function has a well-defined permutation symmetry. In general, they provide the continuous-variable generalization of spin systems on graphs, with broken permutation symmetry. The corresponding parent Hamiltonian is shown to include (a) two-body interactions determined by the graph adjacency matrix and (b) three-body interactions over all possible 2-paths on the graph. Employing elements of graph theory, we chart the landscape of models, recovering known instances in the literature and providing numerous new examples of ground-state solvable models for which the system Hamiltonian, ground-state wave function, and corresponding energy eigenvalue are specified.
Abstract We employ generalized Euler coordinates for the n body system in dimensional space, which consists of the centre-of-mass vector, relative (mutual) mass-independent distances r ij and angles as remaining coordinates. We prove that the kinetic energy of the quantum n -body problem for can be written as the sum of three terms: (i) kinetic energy of centre-of-mass, (ii) the second order differential operator which depends on relative distances alone and (iii) the differential operator which annihilates any angle-independent function. The operator has a large reflection symmetry group and in variables is an algebraic operator, which can be written in terms of generators of the hidden algebra . Thus, makes sense of the Hamiltonian of a quantum Euler–Arnold top in a constant magnetic field. It is conjectured that for any n , the similarity-transformed is the Laplace–Beltrami operator plus (effective) potential; thus, it describes a -dimensional quantum particle in curved space. This was verified for . After de-quantization the similarity-transformed becomes the Hamiltonian of the classical top with variable tensor of inertia in an external potential. This approach allows a reduction of the dn -dimensional spectral problem to a -dimensional spectral problem if the eigenfunctions depend only on relative distances. We prove that the ground state function of the n body problem depends on relative distances alone.