TPS63001 output with less than 2.4Vin

In that case, you are exceeding the input current limit of the device.  So, the output voltage will drop if you try and draw 800 mA out of it with less than 2.4Vin.  You can order the EVM to see this behavior.

The 2.4V, it a worst case type value, so you will typically be able to get out more power.  See figure 1 in the D/S.  The TPS63010 is a slightly higher power device that can deliver more power to lower input voltages.

But what type of battery do you have that goes down to 1.8V?

Hi Chris Glaser!

         Lithium Energizer L91 AA battery(nominal voltage is 1.5V)   .Here  I m using 3xAA battey in series.    

Based on their D/S, http://data.energizer.com/PDFs/l91.pdf, it is basically used up at 1.2V.  So, your voltage range is more like 3.6V to 4.5V or so.  A simple buck converter is best is this case.  TPS62080 or TPS62085 would be my recommendation.

Hi Chris Glaser!

         Since i m using 3 AA battery connected in series , the cuttoff voltage for battery will be 2.4V . please correct me if i wrong .  

Per the graphs in their D/S, the cutoff voltage is basically 1.2V per cell, 3.6V total.  That's where the curves really start to go downhill (waterfall curve).  You are not getting hardly any extra run time by discharging them below 1.2V.