Voltage conversion +1V to -3V inside a new IC design
You are looking for a inverting boost converter. The easiest way of implementation is the usage of an IC like LT1617-1. This does exactly what you expect from the circuit. Everthing you need to run this application can be found in the datasheet.  

The min. input voltage is \$1\,V\$ for the LT1617-1.
Output voltage can be up to \$-34\,V\$ at max. \$100\,mA\$.