I'm trying to find, How does the number of bound state change as we change the dimension of space?
In 1 & 2 dimension, finite square well we know that there always exists a bound state, as shown here. For higher dimensions, it's easy to see that, For a given value of $V_0$ (depth of the well), the number of bound states decreases. Considering this to be in general, that dimension $d$ is increasing. How I can show that this will always be true?
In $d$ dimension, there are $d$ direction, say $x_1,x_2,\cdots, x_d$ so that SE given by $$-\frac{\hbar^2}{2m}\nabla^2\psi(x_1,x_2,\ldots, x_d)+V(x_1,x_2,\ldots ,x_d)\psi(x_1,x_2,\ldots,x_d)=E\psi(x_1,x_2,\ldots,x_d), $$ where potential is $-V_0$ if $x_1^2+x_2^2+\cdots +x_d^2\leq a^2$ else $0$. I'm clueless, How to start? Any hint is appreciable.
