 Ground-state eigenfunction of a confining Schrödinger operator

ID: ground-state-eigenfunction-of-a-confining-schrodinger-operator

The minimum of

∥\nabla u ∥_{2}^{2} + ∥ V^{1/2} u ∥_{2}^{2}

subject to

∥ u ∥_{2} = 1

is attained because

Σ_{V}

embeds compactly into

L^{2}

. Replacing a minimizer by its absolute value does not increase its energy. The Euler-Lagrange equation therefore produces a nonnegative eigenfunction of

- Δ + V

for its lowest eigenvalue.

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