  Scattering of an electron in quasi-one dimensional quantum wires have many
unusual features, not found in one, two or three dimensions. In this work we
analyze the scattering phase shifts due to an impurity in a multi-channel
quantum wire with special emphasis on negative slopes in the scattering phase
shift versus incident energy curves and the Wigner delay time. Although at
first sight, the large number of scattering matrix elements show phase shifts
of different character and nature, it is possible to see some pattern and
understand these features. The behavior of scattering phase shifts in
one-dimension can be seen as a special case of these features observed in
quasi-one-dimensions. The negative slopes can occur at any arbitrary energy and
Friedel sum rule is completely violated in quasi-one-dimension at any arbitrary
energy and any arbitrary regime. This is in contrast to one, two or three
dimensions where such negative slopes and violation of Friedel sum rule happen
only at low energy where the incident electron feels the potential very
strongly (i.e., there is a very well defined regime, the WKB regime, where FSR
works very well). There are some novel behavior of scattering phase shifts at
the critical energies where $S$-matrix changes dimension.
