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Finite-subspace variational method

For an orthonormal trial set

ϕ_{1}, \dots, ϕ_{N}

, the Rayleigh quotient on their span is

\frac{α ^{†} H α}{α ^{†} α}, H_{nm} = ⟨ ϕ_{n} ∣ H ∣ ϕ_{m} ⟩ .

Its minimum is the smallest eigenvalue of

H

, which is therefore the optimal variational upper bound available in that trial subspace.

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- Two-mode variational bound for a linearly tilted square well Finite-subspace variational method 37

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