The simple thing first would be to try on an external power supply and/ or reset with the restart button the rear. I looked at my service manuel (that I bought when I bought my 2035 new ) and I see there is a 3volt lithium right of IC9 voltage regulator so that could be a problem also considering the age of this scanner. The scanner has five regulators in the power supply and three of those are IC's so it could be one of these have failed or even a cold solder joint on the PC board around the PS area ( especially around the IC's ). Best bet would be a trip to Rad shack but being so old they may not repair not to mention the cost against the cost of just buying a new scanner. I must admit the 2035 & 2006/ 2005 has been the better GRE built as in interference ( pagers on high band ) compared to my latest models like the 2055 & 2096.
Funny that was mentioned! I found a PRO 2011 from a flea market today and it has the same problem. Just the light for the display comes on, no digits and no noise. If you turn the volume full blast, and open the squelch, you hear some noise. I might have to check the transformer, since it is built in and uses a regular AC cord.
If it is powering up its likely the transformer is ok, but don't hurt to check it and the pins of the transformer going through the PC board. I have had those crack around the lugs before in other electronic goodies. The rectifiers ( diodes ) right off your transformer may have took a hit too from a surge.