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Register a new userMultichannel coupling with supersymmetric quantum mechanics and exactly-solvable model for Feshbach resonance
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Added by
Jean-Marc Sparenberg
Publication date
2006
fields
Physics
and research's language is
English
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A new type of supersymmetric transformations of the coupled-channel radial Schroedinger equation is introduced, which do not conserve the vanishing behavior of solutions at the origin. Contrary to usual transformations, these ``non-conservative transformations allow, in the presence of thresholds, the construction of potentials with coupled scattering matrices from uncoupled potentials. As an example, an exactly-solvable potential matrix is obtained which provides a very simple model of Feshbach-resonance phenomenon.
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Starting from a system of $N$ radial Schrodinger equations with a vanishing potential and finite threshold differences between the channels, a coupled $N times N$ exactly-solvable potential model is obtained with the help of a single non-conservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as well as its Jost matrix. It depends on $N (N+1)/2$ unconstrained parameters and on one upper-bounded parameter, the factorization energy. A detailed study of the model is done for the $2times 2$ case: a geometrical analysis of the zeros of the Jost-matrix determinant shows that the model has 0, 1 or 2 bound states, and 0 or 1 resonance; the potential parameters are explicitly expressed in terms of its bound-state energies, of its resonance energy and width, or of the open-channel scattering length, which solves schematic inverse problems. As a first physical application, exactly-solvable $2times 2$ atom-atom interaction potentials are constructed, for cases where a magnetic Feshbach resonance interplays with a bound or virtual state close to threshold, which results in a large background scattering length.
We study in detail the relationship between the Tavis-Cummings Hamiltonian of quantum optics and a family of quasi-exactly solvable Schrodinger equations. The connection between them is stablished through the biconfluent Heun equation. We found that each invariant $n$-dimensional subspace of Tavis-Cummings Hamiltonian corresponds either to $n$ potentials, each with one known solution, or to one potential with $n$-known solutions. Among these Schrodinger potentials appear the quarkonium and the sextic oscillator.
We extend the class of QM problems which permit for quasi-exact solutions. Specifically, we consider planar motion of two interacting charges in a constant uniform magnetic field. While Turbiner and Escobar-Ruiz (2013) addressed the case of the Coulomb interaction between the particles, we explore three other potentials. We do this by reducing the appropriate Hamiltonians to the second-order polynomials in the generators of the representation of $SL(2,C)$ group in the differential form. This allows us to perform partial diagonalisation of the Hamiltonian, and to reduce the search for the first few energies and the corresponding wave functions to an algebraic procedure.
Relativistic arbitrary spin Hamiltonians are shown to obey the algebraic structure of supersymmetric quantum system if their odd and even parts commute. This condition is identical to that required for the exactness of the Foldy-Wouthuysen transformation. Applied to a massive charged spin-$1$ particle in a constant magnetic field, supersymmetric quantum mechanics necessarily requires a gyromagnetic factor $g=2$.
We consider the self-adjoint extensions (SAE) of the symmetric supercharges and Hamiltonian for a model of SUSY Quantum Mechanics in $mathbb{R}^+$ with a singular superpotential. We show that only for two particular SAE, whose domains are scale invariant, the algebra of N=2 SUSY is realized, one with manifest SUSY and the other with spontaneously broken SUSY. Otherwise, only the N=1 SUSY algebra is obtained, with spontaneously broken SUSY and non degenerate energy spectrum.
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