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Rayleigh energy perturbation for normalized finite states

Statement

Let $H = F^{n_{1} \times \dots \times n_{d}}$ be a finite tensor-array space over $F \in {R, C}$ , let $H : H \to H$ be self-adjoint, and let $ψ, ϕ \in H$ satisfy $∥ ψ ∥_{F} = ∥ ϕ ∥_{F} = 1$ . Then $∣ ⟨ ψ, H ψ ⟩_{F} - ⟨ ϕ, H ϕ ⟩_{F} ∣ \leq 2 ∥ H ∥_{o p} ∥ ψ - ϕ ∥_{F} .$

Facts & Assumptions

Given: A finite tensor-array space, a self-adjoint endomorphism $H$ , and unit states $ψ, ϕ$ in that space.

[F1]

Finite tensor arrays and the Frobenius inner product: the Frobenius pairing is the finite sum of conjugate-entry times entry, and its induced norm is $∥ \cdot ∥_{F}$ .

[F2]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: on a nonzero finite-dimensional domain, $∥ H ∥_{o p} = \max_{∥ x ∥ = 1} ∥ H x ∥$ .

[F3]

Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors: the absolute value of an inner product is at most the product of the two vector norms; conjugating or swapping the arguments leaves the modulus unchanged, so the inequality applies to the conjugate-first Frobenius pairing as well.

[F4]

Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space: self-adjointness means $H^{*} = H$ .

Proof

1.1givenF2algebra

For every $v \in H$ , $∥ H v ∥_{F} \leq ∥ H ∥_{o p} ∥ v ∥_{F}$ : this is immediate for $v = 0$ , and otherwise follows by applying [F2] to $v / ∥ v ∥_{F}$ and scaling back.

2.1givenF1F2F3F4step 1.1algebra∎

By [F1], expand the difference as $⟨ ψ - ϕ, H ψ ⟩_{F} + ⟨ ϕ, H (ψ - ϕ) ⟩_{F}$ . Apply [F3] to both terms, then [F2] and step 1.1 to obtain a bound of $∥ ψ - ϕ ∥_{F} ∥ H ∥_{o p} ∥ ψ ∥_{F} + ∥ ϕ ∥_{F} ∥ H ∥_{o p} ∥ ψ - ϕ ∥_{F} = 2 ∥ H ∥_{o p} ∥ ψ - ϕ ∥_{F}$ . Finally, $\overline{⟨ ψ, H ψ ⟩_{F}} = ⟨ H ψ, ψ ⟩_{F} = ⟨ ψ, H ψ ⟩_{F}$ by [F1] and $H^{*} = H$ from [F4], so the energy values are real.

Depends on

Used by

TT discarded weights bound the post-sweep DMRG energy increase Theorem

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ulrich Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326 (2011), §6.3 (standard reference, not scraped)
Grégoire Misguich, Introduction to matrix-product states and tensor networks, complete lecture notes (2026), §2.7 (standard reference, not scraped)