Suppose we have a 2d problem (for instance a 2d box). You look for separable solutions in x and y. But it seems to me that the solutions are in a way determined by how we choose the separation constant. Do we know anything a priori that tells us if the separation constant should be negative or positive? Consider for instance the 2d box. The separation constant for the x and y solution should have opposite sign, yet the physical situation is exactly the same. What is wrong here?