  We consider a spatially periodic (cosine) potential as a model for a
crystalline solid that interacts with a harmonically oscillating external
electric field. This problem is periodic both in space and time and can be
solved analytically using the Kramers-Henneberger co-moving frame. By analyzing
the stability of the closely related Mathieu-type differential equation, the
electronic band structure can be obtained. We demonstrate that by changing the
field intensity, the width of the zero-field band gaps can be drastically
modified, including the special case when the external field causes the band
gaps to disappear
