Existence of Bound States. A potential “well” (in one dimension) is a function V(x) that is never positive (V(x) ≤ 0 for all x), and goes to zero at infinity (V(x) → as x → ± ∞).

(a) Prove the following Theorem: If a potential well V₁(x) supports at least one bound state, then any deeper/wider well (V₂(x) ≤ V₁(x) for all x) will also support at least one bound state.
(b) Prove the following Corollary: Every potential well in one dimension has a bound state.
(c) Does the Theorem generalize to two and three dimensions? How about the Corollary? You might want to review Problems 4.11 and 4.51.