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Harmonic rigidity at fixed spectral gap in one dimension

Arseny Pantsialei

TL;DR

This work establishes a sharp static quantum speed-limit analogue for one-dimensional confining potentials at fixed spectral gap $\Delta$, proving $\mathrm{Var}_0(x)\le \hbar^2/(2m\Delta)$ with equality iff the trap is harmonic and $\hbar\omega=\Delta$. It provides a complete rigidity picture: a deficit $\varepsilon$ linearly bounds the TRK tail through the second gap $\Gamma$ and bounds the structural deviation $V'(x)-m\omega^2x$ in $L^2(\rho_0)$, yielding explicit D1–D3 inequalities. The framework extends to magnetic settings, giving longitudinal iff results and transverse TRK identities via the guiding-center coordinate $R_u$, with quantitative rigidity tied to the projected Hessian of $V$; these results underpin sharp bounds on static polarizability and quantum metric. Collectively, the paper offers precise, experimentally actionable limits for trap design and metrology, plus a roadmap for extensions to multidimensional, many-body, and time-dependent scenarios.

Abstract

We solve the static isoperimetric problem underlying the Mandelstam-Tamm bound. Among one-dimensional confining potentials with a fixed spectral gap, we prove that the harmonic trap is the unique maximizer of the ground-state position variance. As a consequence, we obtain a sharp geometric quantum speed-limit bound on the position-position component of the quantum metric, and we give a necessary-and-sufficient condition for when the bound is saturated. Beyond the exact extremum, we establish quantitative rigidity. We control the Thomas-Reiche-Kuhn spectral tail and provide square-integrable structural stability for potentials that nearly saturate the bound. We further extend the analysis to magnetic settings, deriving a longitudinal necessary-and-sufficient characterization and transverse bounds expressed in terms of guiding-center structure. Finally, we outline applications to bounds on static polarizability, limits on the quantum metric, and benchmarking of trapping potentials.

Harmonic rigidity at fixed spectral gap in one dimension

TL;DR

This work establishes a sharp static quantum speed-limit analogue for one-dimensional confining potentials at fixed spectral gap , proving with equality iff the trap is harmonic and . It provides a complete rigidity picture: a deficit linearly bounds the TRK tail through the second gap and bounds the structural deviation in , yielding explicit D1–D3 inequalities. The framework extends to magnetic settings, giving longitudinal iff results and transverse TRK identities via the guiding-center coordinate , with quantitative rigidity tied to the projected Hessian of ; these results underpin sharp bounds on static polarizability and quantum metric. Collectively, the paper offers precise, experimentally actionable limits for trap design and metrology, plus a roadmap for extensions to multidimensional, many-body, and time-dependent scenarios.

Abstract

We solve the static isoperimetric problem underlying the Mandelstam-Tamm bound. Among one-dimensional confining potentials with a fixed spectral gap, we prove that the harmonic trap is the unique maximizer of the ground-state position variance. As a consequence, we obtain a sharp geometric quantum speed-limit bound on the position-position component of the quantum metric, and we give a necessary-and-sufficient condition for when the bound is saturated. Beyond the exact extremum, we establish quantitative rigidity. We control the Thomas-Reiche-Kuhn spectral tail and provide square-integrable structural stability for potentials that nearly saturate the bound. We further extend the analysis to magnetic settings, deriving a longitudinal necessary-and-sufficient characterization and transverse bounds expressed in terms of guiding-center structure. Finally, we outline applications to bounds on static polarizability, limits on the quantum metric, and benchmarking of trapping potentials.
Paper Structure (26 sections, 5 theorems, 132 equations, 1 figure, 1 table)

This paper contains 26 sections, 5 theorems, 132 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Setup and basic facts
  3. Main result: extremizing the variance at fixed gap
  4. Quantitative rigidity at fixed gap
  5. Statements and explicit constants
  6. Proof of D1
  7. Proof of D2
  8. Proof of D3
  9. Proof of the upper-window strengthening
  10. Remarks and corollaries
  11. Multidimensional case $d>1$ without a magnetic field
  12. Uniform magnetic field: parallel direction
  13. Across the field: guiding center $R_\perp$
  14. Quantitative rigidity (transverse).
  15. Remarks.
  16. ...and 11 more sections

Key Result

Theorem 1

The bound eq:var-par-bound holds, and equality is achieved if and only if where $\mathbf x_\perp:=\mathbf x-(\hat{\mathbf b} \cdot \mathbf x) \hat{\mathbf b}$, $W$ is a confining function of the transverse variable, and $C\in\mathbb R$.

Figures (1)

  • Figure 1: Numerical illustration for the anharmonic oscillator $V(x)=\tfrac{1}{2} x^2+\lambda x^4$ (units $\hbar=m=\omega=1$). Horizontal axis: $\lambda$. (a) Deficit $\varepsilon(\lambda)$ from the sharp bound on $\mathop{\mathrm{Var}}\nolimits_0(x)$. (b) Ground-state momentum variance $\mathop{\mathrm{Var}}\nolimits_0(p)$ compared with the theoretical bounds from Sec. 6.3. (c) Ratio of $\mathop{\mathrm{Var}}\nolimits_0(p)$ to the corresponding upper bound.

Theorems & Definitions (15)

  • Theorem 1: Longitudinal iff in a uniform field
  • Lemma 1: Equality equivalences for $x_\parallel$
  • Remark 1: On the choice of gap
  • Remark 2: TRK constant is unchanged
  • Remark 3
  • proof
  • Theorem 2: Sharp longitudinal bound and iff for $\mathbf B(\mathbf x)\parallel\hat{\mathbf b}$
  • proof
  • Remark 4: A priori non-separability
  • Remark 5: Why there is no general transverse iff for inhomogeneous $B$
  • ...and 5 more