Speaker
Description
In many applications one has to deal with functions that depend on two directions. In this case
a convenient basis for function expansion is provided by bipolar harmonics that are given by irreducible
tensor product of the spherical functions with different arguments. The basis of biharmonic functions is
overcomplete for a fixed total angular momentum and for arbitrary internal angular momenta.
Very often bipolar harmonics with a small rank of total momentum enter the final results while the ranks
of the internal tensors can run over a wide (or infinite) range. But it is possible to decompose
the bipolar harmonic using the smallest set of internal orbital momenta for a fixed total momentum.
Here we apply the new method for calculations of decomposition coefficients at low values
of total angular momenta and arbitrary values of internal momenta, which is not related with the
special choice of a coordinate system. Then the basis functions from the minimal set are modified
in two respects. 1) All dependence on angle between two directions in bipolar harmonica will be contained
only in expansion coefficients and basis functions are independent of this angle. 2) The new basis will
be orthogonal and have the same normalization. We can call these tensors as the normalized orthogonal
bases from the minimal set of bipolar harmonics. The new basis and expansion coefficients for the low
values of total orbital momentum are presented explicitly.
