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A two-level finite-gap model: exact eigenvalues, the second-order correction and a finite check of the cubic remainder bound

Physical scope

A conditional two-level computation inside the declared perturbed-family model with a finite gap: the exact eigenvalues of the family, its second-order coefficient, the exact quartic expansion, and a finite numerical check of the page's cubic remainder bound at half the perturbation radius. The example is a model computation in the declared finite-dimensional subspace; it asserts no physical system identification, no measured splitting and no transition.

Example

Adopt P1, P2 and P4 for a declared two-level system (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution) with state space $H = C^{2}$ in the standard orthonormal basis $(e_{1}, e_{2})$ and with the observable reading of Assigned sharp observables and the outcome law of the framework; read all energies in joules as in Dimensions, SI unit conventions and quantity types of the framework. Fix real gap and coupling parameters $g > 0$ and $c > 0$ , and declare the model's energy observable family by the perturbed-family model of The declared perturbed-family model: bounded self-adjoint perturbation, isolated gapped label and followed spectral cluster with $H_{0} : = (\begin{array}{cc} 0 & 0 \\ 0 & g \end{array}), V : = (\begin{array}{cc} 0 & c \\ c & 0 \end{array}), H (λ) = H_{0} + λ V (λ \in R) .$ Then $H_{0}$ has the isolated simple eigenvalue $E = 0$ with gap $g$ , $v = ∥ V ∥ = c$ , $ψ_{0} = e_{1}$ , and the reduced inverse of the model clause (c)(iii) is $S = diag (0, 1 / g)$ on $Q_{0} H = C e_{2}$ . Then:

(a) for every real $λ$ the matrix $H (λ)$ is Hermitian with exactly two eigenvalues $E_{\pm} (λ) = \frac{g \pm \sqrt{g^{2} + 4 λ^{2} c^{2}}}{2},$ both simple for $λ \neq 0$ and spanning the spectrum for every $λ$ ; in the model's valid contour range $2 ∣ λ ∣ c / g < 1$ , its distinguished label is $E_{λ} = E_{-} (λ)$ . The formula $E_{-}$ remains an exact continuous lower spectral branch for all real couplings, even beyond that fixed-contour range;

(b) the exact perturbation coefficients are $E^{'} (0) = 0, E^{''} (0) = - \frac{2 c^{2}}{g}, ψ^{'} (0) = - S V ψ_{0} = - \frac{c}{g} e_{2}, ψ_{0} = e_{1},$ and the exact eigenvalue has the convergent expansion $E_{-} (λ) = - \frac{λ^{2} c^{2}}{g} + \frac{λ^{4} c^{4}}{g^{3}} - \frac{2 λ^{6} c^{6}}{g^{5}} + O (λ^{8});$

(c) the cubic remainder bound of Nondegenerate stationary perturbation of a simple gapped label: first- and second-order coefficients and an explicit cubic remainder clause (e) holds in this model for every $∣ λ ∣ < R = g / (8 c)$ , and at the test point $λ = R / 2 = g / (16 c)$ the exact remainder and its bound are $E_{-} ⁣ (\frac{g}{16 c}) + \frac{g}{256} = \frac{g}{2} (1 - \frac{\sqrt{65}}{8}) + \frac{g}{256} \approx 1.51 \cdot 10^{- 5} g \leq \frac{g}{32},$ so the finite check is satisfied with a wide margin;

(d) the exact spectrum computed in (a) is a spectral statement about the declared two-level family; it identifies no physical system, asserts no measured splitting, and asserts nothing about which of the two spectral subspaces a preparation occupies (that is fixed by the preparation and the evolution of P4, not by the perturbative expansion).

Verification

Given: the declared two-level model with parameters $g, c > 0$ , matrices $H_{0}, V$ , the family $H (λ)$ , and the label data $E = 0$ , $g$ , $v = c$ , $ψ_{0} = e_{1}$ , $S = diag (0, 1 / g)$ ; the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the model's spectral suppliers (The declared perturbed-family model: bounded self-adjoint perturbation, isolated gapped label and followed spectral cluster).

[L1] The model's theorem. Under the model's hypotheses with a simple label, for every real $∣ λ ∣ < R = g / (8 v)$ the followed label $E_{λ}$ is the unique spectral value of $H (λ)$ in $∣ z - E ∣ < g / 2$ , the coefficients and remainder bound of clause (e) of the statement of Nondegenerate stationary perturbation of a simple gapped label: first- and second-order coefficients and an explicit cubic remainder hold, and $E^{'} (0) = ⟨ V ψ_{0}, ψ_{0} ⟩$ , $ψ^{'} (0) = - S V ψ_{0}$ , $E^{''} (0) = - 2 ⟨ V S V ψ_{0}, ψ_{0} ⟩$ (Nondegenerate stationary perturbation of a simple gapped label: first- and second-order coefficients and an explicit cubic remainder clauses (a)-(e), The declared perturbed-family model: bounded self-adjoint perturbation, isolated gapped label and followed spectral cluster).

[L2] Elementary algebra used. For real $u$ , $\sqrt{u^{2}} = ∣ u ∣$ ; the binomial series $\sqrt{1 + t} = 1 + \frac{1}{2} t - \frac{1}{8} t^{2} + \frac{1}{16} t^{3} + O (t^{4})$ as an identity for the analytic square root near $t = 0$ ; and for vectors $w_{\pm} = (λ c, E_{\pm})$ the inner product in the standard basis is $w_{+} \cdot \overline{w_{-}} = λ^{2} c^{2} + E_{+} E_{-}$ , together with $E_{+} + E_{-} = g$ and $E_{+} E_{-} = (g^{2} - (g^{2} + 4 λ^{2} c^{2})) / 4 = - λ^{2} c^{2}$ , all direct consequences of the displayed formulas.

[L3] Spectral reading. If a linear map on $C^{2}$ has two orthonormal eigenvectors $u_{1}, u_{2}$ with eigenvalues $μ_{1}, μ_{2}$ , then every $v = α u_{1} + β u_{2}$ satisfies $H v = α μ_{1} u_{1} + β μ_{2} u_{2}$ , so $H v = μ v$ with $v \neq 0$ forces $μ = μ_{1}$ or $μ = μ_{2}$ : the spectrum is exactly ${μ_{1}, μ_{2}}$ . This is finite-dimensional linear algebra on the declared two-dimensional space (State and composition postulate P1: Hilbert spaces, preparations and composite systems).

Proof technique: direct; exhibit an orthonormal eigenbasis of the two-by-two family, read off the spectrum, expand the displayed square root and evaluate the page's remainder bound at the test point.

Given: the data of the Example and [L1]-[L3].

1.1L2L3givenalgebra

[Hermiticity and the eigenbasis] Both $H_{0}$ and $V$ are real symmetric, hence $H (λ)$ is Hermitian for real $λ$ . Put $s : = \sqrt{g^{2} + 4 λ^{2} c^{2}} > 0, E_{\pm} : = \frac{g \pm s}{2},$ and, for $λ \neq 0$ , $w_{\pm} : = (λ c, E_{\pm}) \neq 0$ . From $E_{\pm}^{2} - g E_{\pm} - λ^{2} c^{2} = 0$ , which follows from $E_{+} + E_{-} = g$ and $E_{+} E_{-} = - λ^{2} c^{2}$ by [L2], the first component equation $λ c E_{\pm} = E_{\pm} λ c$ holds identically and the second reads $λ c \cdot λ c + g E_{\pm} = λ^{2} c^{2} + g E_{\pm} = E_{\pm}^{2}$ , so $H (λ) w_{\pm} = E_{\pm} w_{\pm}$ ; moreover $w_{+} \cdot \overline{w_{-}} = λ^{2} c^{2} + E_{+} E_{-} = 0$ by [L2], so the normalizations $u_{\pm} : = w_{\pm} / ∥ w_{\pm} ∥$ form an orthonormal pair. For $λ = 0$ the vectors $u_{1} = e_{1}, u_{2} = e_{2}$ are an orthonormal eigenbasis with eigenvalues $0 = E_{-} (0)$ and $g = E_{+} (0)$ . By [L3] the spectrum of $H (λ)$ is exactly ${E_{-} (λ), E_{+} (λ)}$ for every real $λ$ , and the eigenvalues are simple for $λ \neq 0$ because then $s > g$ and $E_{+} \neq E_{-}$ . For $2 ∣ λ ∣ c / g < 1$ , $s < \sqrt{2} g < 2 g$ , so $∣ E_{-} ∣ = (s - g) / 2 < g / 2$ while $E_{+} \geq g > g / 2$ ; thus the unique enclosed label is $E_{-}$ , proving the contour identification in (a).

1.2L1L2givenalgebra

[Coefficients] With $P_{0}$ the projection onto $C e_{1}$ and $Q_{0}$ the projection onto $C e_{2}$ , one has $V ψ_{0} = c e_{2}$ , $⟨ V ψ_{0}, ψ_{0} ⟩ = 0$ , $S V ψ_{0} = (c / g) e_{2}$ and $V S V ψ_{0} = (c^{2} / g) e_{1}$ ; hence $E^{'} (0) = 0$ , $ψ^{'} (0) = - S V ψ_{0} = - (c / g) e_{2}$ and $E^{''} (0) = - 2 ⟨ V S V ψ_{0}, ψ_{0} ⟩ = - 2 c^{2} / g$ , as displayed. Expanding $E_{-} (λ) = \frac{g}{2} (1 - \sqrt{1 + 4 λ^{2} c^{2} / g^{2}})$ by [L2] gives the displayed expansion. This is clause (b).

2.1L1L2step 1.2algebra

[Finite remainder check] By [L1], the cubic estimate holds on $∣ λ ∣ < R = g / (8 c)$ ; at $λ = R / 2 = g / (16 c)$ its bound is $R c / 4 = g / 32$ . The exact lower energy is $E_{-} = (g / 2) (1 - \sqrt{65} / 8)$ and the quadratic truncation is $- g / 256$ . Thus the exact remainder is $g (129 - 16 \sqrt{65}) / 256 = g / [256 (129 + 16 \sqrt{65})] > 0$ , since $129^{2} - (16 \sqrt{65})^{2} = 1$ . It is less than $g / 32768 < g / 32$ , because $129 + 16 \sqrt{65} > 128$ , proving (c) without an unsupported decimal or square-root bound.

3.1step 1.1step 2.1L1given∎

[Scope] The computation used only the declared two-dimensional model, its exact diagonalization and the page's theorem; the two spectral values are spectral data of $H (λ)$ . Which spectral subspace a preparation occupies is fixed by the preparation and the evolution of P4 and is not determined by the label $E_{λ}$ ; no physical system, measured splitting or transition is asserted, which is clause (d).

Remarks

Why this is a "finite-gap" check. The complementary subspace is one-dimensional with distance $g$ , so the reduced inverse is the explicit number $1 / g$ and every coefficient of the page's theorem can be compared against a closed-form eigenvalue. The second-order coefficient is strictly negative for the followed lower level, the signature of a level repulsion from the complementary state.
The cubic bound is not sharp. The exact difference from the second-order truncation is $O (λ^{4})$ , and the page's bound at the test point is more than three orders of magnitude larger; the check verifies the inequality, not optimality.

A two-level finite-gap model: exact eigenvalues, the second-order correction and a finite check of the cubic remainder bound

Example

Verification

Remarks

Depends on

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