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An affine family of lifts and couplings

Definition

Fix a finite-dimensional complex inner-product space with a specified orthonormal basis, a real number $s_{0} > 0$ , and finitely many indices $a$ . Let $H_{0}$ and $D$ be Hermitian matrices diagonal in that basis, and let $X_{a, 0}, Y_{a}, A_{a}, B_{a}$ be arbitrary matrices of the same size. For $s \in [- s_{0}, s_{0}]$ , define $H (s) = H_{0} + s D$ , $X_{a} (s) = X_{a, 0} + s Y_{a}$ , and $L_{a} (s) = A_{a} + s B_{a}$ . Call this an affine lift, coupling, and jump-operator family. Here the $X_{a}$ are candidate system coupling matrices and the $L_{a}$ are candidate jump matrices; the definition does not assert a microscopic relation between them or that either family describes a physical device.

For a matrix $M$ , write $∥ M ∥$ for its operator norm. This family is regularity-controlled in the explicit sense that $∥ X_{a} (s) - X_{a} (t) ∥ \leq ∣ s - t ∣ ∥ Y_{a} ∥$ and $∥ L_{a} (s) - L_{a} (t) ∥ \leq ∣ s - t ∣ ∥ B_{a} ∥$ throughout the interval. Its directed transition energy from basis state $x$ to basis state $y$ is the source eigenvalue of $H (s)$ minus the destination eigenvalue.

Facts & Assumptions

[F1]

The finite operator conventions specify matrices, adjoints and orthonormal bases on a finite-dimensional complex inner-product space. Finite operator and ket conventions

[F2]

For a diagonal lift, the change in directed transition energy is the lift eigenvalue at the source minus that at the destination. Calibrated diagonal lifts and transition phase differences

[F3]

The inner-product length $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ is a norm on a real or complex inner-product space. The induced length is a norm

[F4]

Every linear map from a normed space with a specified finite ordered basis to a normed space over the same scalar field is bounded. A linear map from a finite-dimensional normed space is bounded

[F5]

For a bounded linear map $M$ between normed spaces, its operator norm is the finite number $∥ M ∥ = \sup {∥ M v ∥ : ∥ v ∥ \leq 1}$ . On the space of bounded linear maps it is a norm, so $∥ λ M ∥ = ∣ λ ∣ ∥ M ∥$ . Here the vector norm is the inner-product norm of [F3]. The operator norm as the least bound and as the unit-sphere or unit-ball supremum The operator norm is a norm on the space of bounded linear operators

Justification

Given: The matrices and interval in the definition.

Goal: Establish that the family and its stated regularity and transition energies are well defined.

1.1 Each displayed family is a finite matrix sum for every real $s$ . Since $H_{0}$ and $D$ are Hermitian and diagonal in the same basis, $H (s)$ is Hermitian and diagonal there. By [F1], every matrix here represents a linear map on the specified finite-dimensional complex inner-product space with its specified finite ordered orthonormal basis. By [F3] this is a normed space, so [F4] applies to each such map with this space as both domain and codomain. Thus every fixed matrix and every displayed family member is bounded and has finite operator norm by [F5]. [F1, F3, F4, F5, algebra]

2.1 Subtraction gives $X_{a} (s) - X_{a} (t) = (s - t) Y_{a}$ and $L_{a} (s) - L_{a} (t) = (s - t) B_{a}$ . Homogeneity of the operator norm in [F5] yields both displayed bounds. If $D$ has diagonal entries $d_{x}$ , [F2] gives the transition-energy change $s (d_{x} - d_{y})$ relative to $H_{0}$ . Thus all stated data and bounds are well defined; no relation between the two matrix families has been inferred. [F2, F5, step 1.1, algebra] ∎

Depends on

Used by · 1

Degeneracy lifting that also modifies jump operators

Dependency tree · 2 levels

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