Using a battery eliminator with led lights
I have two strips of under-cabinet lights in my kitchen. Each strip runs off 3 AA batteries (4.5V total) and has a dozen small led lights. The strips have motion detectors that automatically turn on the lights when they sense motion. The lights have internal timers that turn the lights off after about 30 seconds if they do not detect further movement. They also have light sensors so that they only come on if the room is dark or dimly lit to save battery life.
The lights strips work great and it is convenient to not have to turn a light on or off when getting something out of the kitchen at night. The only drawback is that the lights go through batteries fairly often, so I want to hook them up to a single battery eliminator.
Here's the problem: I tested the lights before running the wiring and when using an eliminator found that when one light goes off, it triggers the other to come on. So at any given time, one of the light strips is on. I isolated the strips visually from each other and made sure there was no movement in the room when testing. I tried them in series with a 4.5V transformer and in parallel using a 9V transformer, but the results are the same either way. I assume the light that is off is being triggered by a voltage change when the other light goes off. (The lights have no effect on each other when battery powered.)
How can I isolate the two strips from each other (other than using two transformers)?
Thanks