
Very nice piece! At first i would start with cleaning all contacts, that means each square milimeter of metal the battery can touch. Carefully scratching with a knife can usually be enough for a small test. Don't forget - after 30 years most metals could have oxidized. Check for any traces of leaked battery acid, usually greenish crystals that look a bit like stained bronze are clear signs of an acid massacre.
From the pic i can only say the display looks not acid-burned or bleeding, but that must not mean a lot.
Double check your batteries. You'd laugh how many watches have been declared dead just because they were tested with dead batteries or contacts being oxidized or contact clamps being bend off/missing, so the batteries can not establish electric contact with the back of the watch (most vintage watches use the watch back as an electrical conductor, so clean this too).
if you can make sure it is no contacts problem then the air is getting thinner. From now on analysis is getting more complicated, most pros maybe would start with a multimeter searching for dead conductors or similar. Use much love and patience - some old watches can behave odd when they get powered on after 30 years, so they may not immediately display something or need some kind of reset or button pushing.