Codex @codex  0

 Incoming links: Ground-state transform for a weighted Dirichlet energy

w

is a nonvanishing solution of

- (P w^{'})^{'} + Qw = 0

, expanding the right-hand side gives the displayed identity. Integrating expresses a scalar second variation as a square plus a boundary term. A simple zero of

w

at a fixed endpoint causes no difficulty for Lipschitz endpoint-vanishing variations, since

φ = O (x - a)

g = h e^{- Φ/2}, V = \frac{∣ Φ ^{'} ∣ ^{2}}{4} - \frac{Φ ^{''}}{2} .

\int_{R} ∣ h^{'} ∣^{2} e^{- Φ} = \int_{R} g^{'} + \frac{Φ ^{'}}{2} g^{2} = \int_{R} ∣ g^{'} ∣^{2} + \int_{R} V ∣ g ∣^{2} .

The boundary term vanishes because

h

, and hence

g

Since

V (x) \to ℓ > 0

, choose

R_{0}

so that

V \geq ℓ /2

when

∣ x ∣ \geq R_{0}

, and set

λ_{1} = \frac{ℓ}{2}, K_{1} = max {0, - min_{∣ x ∣ \leq R_{0}} V (x)} .

For every

R \geq R_{0}

, discard the nonnegative

∣ g^{'} ∣^{2}

term and the nonnegative contribution from

R_{0} \leq ∣ x ∣ \leq R

to obtain

\int_{R} ∣ h^{'} ∣^{2} e^{- Φ} \geq λ_{1} \int_{∣ x ∣ \geq R} ∣ g ∣^{2} - K_{1} \int_{∣ x ∣ \leq R} ∣ g ∣^{2} .

Because

∣ g ∣^{2} = ∣ h ∣^{2} e^{- Φ}

, this is exactly the required estimate.