We present a barrier potential with bound states that is exactly solvable and determine the eigenfunctions and eigenvalues of the Hamiltonian. The equilibrium density matrix of a particle moving at temperature T in this nonlinear barrier potential field is determined. The exact density matrix is compared with the result of the path integral approach in the semiclassical approximation. For opaque barriers the simple semiclassical approximation is found to be sufficient at high temperatures while at low temperatures the fluctuation paths may have a caustic depending on temperature and endpoints. Near the caustics the divergence of the simple semiclassical approximation of the density matrix is removed by a nonlinear fluctuation potential. For opaque barriers the improved semiclassical approximation is again in agreement with the exact result. In particular, bound states and the form of resonance states are described accurately by the semiclassical approach.We present a barrier potential with bound states that is exactly solvable and determine the eigenfunctions and eigenvalues of the Hamiltonian. The equilibrium density matrix of a particle moving at temperature T in this nonlinear barrier potential field is determined. The exact density matrix is compared with the result of the path integral approach in the semiclassical approximation. For opaque barriers the simple semiclassical approximation is found to be sufficient at high temperatures while at low temperatures the fluctuation paths may have a caustic depending on temperature and endpoints. Near the caustics the divergence of the simple semiclassical approximation of the density matrix is removed by a nonlinear fluctuation potential. For opaque barriers the improved semiclassical approximation is again in agreement with the exact result. In particular, bound states and the form of resonance states are described accurately by the semiclassical approach.

arXiv:chem-ph/9602002The Journal of Chemical PhysicsMay 15, 19966 citations

Approximation Methods for Bound States

WORLD SCIENTIFIC eBooksMar 1, 20130 citations

Approximation Methods for Bound States

WORLD SCIENTIFIC eBooksSep 23, 20140 citations

Approximations to the Ground State

Tim Joachim Zuehlsdorff

Springer thesesJan 1, 20150 citations

s -wave scattering length of a Gaussian potential

Péter Jeszenszki, Alexander Yu. Cherny, Joachim Brand

We provide accurate expressions for the $s$ -wave scattering length for a Gaussian potential well in one, two, and three spatial dimensions. The Gaussian potential is widely used as a pseudopotential in the theoretical description of ultracold-atomic gases, where the $s$ -wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the $s$ -wave scattering length from the parameters of the Gaussian, but find that its accuracy is limited in the vicinity of singularities that result from the formation of new bound states. We then derive simple analytical expressions that capture the correct asymptotic behavior of the $s$ -wave scattering length near the bound states. Expressions that are increasingly accurate in wide parameter regimes are found by a hierarchy of approximations that capture an increasing number of bound states. The small number of numerical coefficients that enter these expressions is determined from accurate numerical calculations. The approximate formulas combine the advantages of the numerical and approximate expressions, yielding an accurate and simple description from the weakly to the strongly interacting limit.We provide accurate expressions for the $s$-wave scattering length for a Gaussian potential well in one, two, and three spatial dimensions. The Gaussian potential is widely used as a pseudopotential in the theoretical description of ultracold-atomic gases, where the $s$-wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the $s$-wave scattering length from the parameters of the Gaussian, but find that its accuracy is limited in the vicinity of singularities that result from the formation of new bound states. We then derive simple analytical expressions that capture the correct asymptotic behavior of the $s$-wave scattering length near the bound states. Expressions that are increasingly accurate in wide parameter regimes are found by a hierarchy of approximations that capture an increasing number of bound states. The small number of numerical coefficients that enter these expressions is determined from accurate numerical calculations. The approximate formulas combine the advantages of the numerical and approximate expressions, yielding an accurate and simple description from the weakly to the strongly interacting limit.

arXiv:1802.07063Physical Review AApr 23, 201829 citations

We provide accurate expressions for the

s

-wave scattering length for a Gaussian potential well in one, two, and three spatial dimensions. The Gaussian potential is widely used as a pseudopotential in the theoretical description of ultracold-atomic gases, where the

s

-wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the

s

-wave scattering length from the parameters of the Gaussian, but find that its accuracy is limited in the vicinity of singularities that result from the formation of new bound states. We then derive simple analytical expressions that capture the correct asymptotic behavior of the

s

-wave scattering length near the bound states. Expressions that are increasingly accurate in wide parameter regimes are found by a hierarchy of approximations that capture an increasing number of bound states. The small number of numerical coefficients that enter these expressions is determined from accurate numerical calculations. The approximate formulas combine the advantages of the numerical and approximate expressions, yielding an accurate and simple description from the weakly to the strongly interacting limit.