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A basis for variational calculations in $d$ dimensions. (English) Zbl 1064.81023
Summary: We derive expressions for matrix elements $(\phi_i,H\phi_j)$ for the Hamiltonian $H=-\Delta+\sum_qa(q)r^q$ in $d\geq2$ dimensions. The basis functions in each angular momentum subspace are of the form $\phi_i(r)=r^{i+1+(t-d)/2}e^{-r^p/2},\ i\geq0, p>0, t>0$. The matrix elements are given in terms of the gamma function for all $d$. The significance of the parameters $t$ and $p$ and scale $s$ are discussed. Applications to a variety of potentials are presented, including potentials with singular repulsive terms of the form $\beta/r^\alpha, \alpha,\beta>0$, perturbed Coulomb potentials $-D/r+Br+Ar^2$, and potentials with weak repulsive terms, such as $-\gamma r^2+r^4, \gamma>0$.

##### MSC:
 81Q05 Closed and approximate solutions to quantum-mechanical equations 49S05 Variational principles of physics
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