Cut the door in half, lengthwise. Place the door onto your workbench. Clamp the straightedge along the center line of the door and cut it lengthwise. Measure and level the center line twice before cutting once. Turn the circular saw on and cut along the straightedge guide.
Measure the width and length of the hollow cutout on the interior of the door. Use the circular saw to cut a strip off of a 2-by-4 that equals those dimensions. For example, if your hollow core cutout is 1-1/2 inches high and 36 inches long, lay the 2-by-4 flat on the workbench and cut a strip that is 1-1/2 inches wide and 36 inches long. Use your straightedge as a guide.
Place the 2-by-4 strip on the wall where you want the floating shelf to go. Have an assistant hold it in place while you check that it is level. Draw a guideline over its top edge to mark this spot on the wall. Use a stud finder to locate the studs that the 2-by-4 traverses. Mark these studs with an "x" on the wall just above the 2-by-4 and just below on the 2-by-4's front face.
Drill 1/4-inch holes at each of the stud marks.
Draw a 3-inch vertical line through the first stud hole. Use a ruler to make sure the line is straight. This vertical guideline will help you line up the stud marks on the strip with the stud holes in the wall.
Hold the 2-by-4 strip against the wall, level with and just above the guideline. Line up the stud marks on the strip with the stud holes in the wall. Check it twice to make sure that it is level and aligned.
Drill holes at each stud mark with a 1/8-inch bit.
Install lag screws into each hole with a wrench.
Remove the first 2 inches of each vertical slat inside the door's hollow core with a wood chisel. Take care not to gouge the wood. Sand away any residual glue.
Apply a line of wood glue to the top of the 2-by-4 strip. Apply a second line of glue to the bottom lip of the hollow core door. Keep each glue line at least 1/4-inch away from the edge.
Slide the door onto the strip.
Secure the strip to the door. Hammer brad nails through the top edge of the shelf and into the strip at 6-inch intervals along the wall.