Asymptotically Scaled Switching Families

• ASSFs are defined in two contexts: as switching functions in quantum detector theory yielding controlled responses, and as switching signals in systems ensuring global stability.
• They leverage Fourier transform scaling and Lyapunov-based analysis to derive long-time detector responses, detailed balance, and asymptotic stability criteria.
• The methodology reveals that sign changes in switching functions are crucial for low-frequency suppression, unifying experimental design and nonlinear stability theories.

Asymptotically Scaled Switching Families (ASSFs) is a term used in two distinct technical literatures. In quantum detector theory, an ASSF is a one-parameter family of switching functions for an Unruh–DeWitt detector whose Fourier transforms exhibit a specific scaling limit, allowing controlled long-time analysis of response functions and effective temperatures in circular motion (Parry et al., 8 Jul 2026). In switched-systems theory, an ASSF is a class of switching signals characterized by asymptotic switching frequency, asymptotic activation fractions, and asymptotic transition densities, with global asymptotic stability obtained from a Lyapunov-based asymptotic inequality (Kundu et al., 2013, Kundu et al., 2017). The shared terminology reflects an asymptotic-scaling viewpoint, but the mathematical objects, goals, and applications are different.

1. Terminological scope and disambiguation

The current literature uses “Asymptotically Scaled Switching Family” for two unrelated constructions.

Context	Mathematical object	Asymptotic structure
Quantum detector theory	Family $\{\chi_\lambda(\tau)\}_{\lambda>0}$ of switching functions	Scaling of $\lambda^{-1}\hat\chi_\lambda(u/\lambda)$ and a uniform Fourier-space majorant
Switched systems	Switching signal $\sigma:[0,\infty)\to P$	Long-run measures $\nu$, $\pi_i$ or $\eta_j$, and $\rho_{ij}$

In the detector-theoretic usage, the term arises in the study of the circular motion Unruh effect in $2+1$ dimensions, where suitable ASSFs were shown to recover a temperature of the order of the linear-acceleration Unruh temperature in a simultaneous long-time and small-gap limit (Parry et al., 27 Aug 2025). The 2026 analysis proves that, under certain boundedness and localisation assumptions, sign changes in the detector-field coupling are necessary for obtaining a nonvanishing limiting effective temperature (Parry et al., 8 Jul 2026).

In the switched-systems usage, ASSFs were introduced as stabilizing switching signals for continuous-time switched linear systems and later absorbed into a more general umbrella framework for nonlinear switched systems with multiple Lyapunov-like functions (Kundu et al., 2013, Kundu et al., 2017).

2. Detector-theoretic ASSFs in circular-motion Unruh analysis

For the detector problem, let $\{\chi_\lambda(\tau)\}_{\lambda>0}$ be a one-parameter family of real, absolutely integrable, $C^1$ switching functions with Fourier transforms

$\lambda^{-1}\hat\chi_\lambda(u/\lambda)$0

The family is an ASSF if two conditions hold (Parry et al., 8 Jul 2026). First, there exists $\lambda^{-1}\hat\chi_\lambda(u/\lambda)$1 with $\lambda^{-1}\hat\chi_\lambda(u/\lambda)$2 such that

$\lambda^{-1}\hat\chi_\lambda(u/\lambda)$3

almost everywhere in $\lambda^{-1}\hat\chi_\lambda(u/\lambda)$4. Second, there is $\lambda^{-1}\hat\chi_\lambda(u/\lambda)$5 and $\lambda^{-1}\hat\chi_\lambda(u/\lambda)$6 such that for all $\lambda^{-1}\hat\chi_\lambda(u/\lambda)$7,

$\lambda^{-1}\hat\chi_\lambda(u/\lambda)$8

for almost all $\lambda^{-1}\hat\chi_\lambda(u/\lambda)$9.

Under these assumptions, Proposition 3.1 establishes a long-time scaling law for the detector response: $\sigma:[0,\infty)\to P$0 where $\sigma:[0,\infty)\to P$1 is the detector-line Wightman spectrum (Parry et al., 8 Jul 2026). This identifies ASSFs as a precise mechanism for turning finite-time detector responses into a controlled spectral limit.

The paper also records explicit constructions. For adiabatic switching,

$\sigma:[0,\infty)\to P$2

and then $\sigma:[0,\infty)\to P$3. For plateau switching,

$\sigma:[0,\infty)\to P$4

with $\sigma:[0,\infty)\to P$5 and $\sigma:[0,\infty)\to P$6 (Parry et al., 8 Jul 2026).

3. Response functions, detailed balance, and the small-gap limit

The operational temperature is defined from excitation and de-excitation probabilities. For a detector gap $\sigma:[0,\infty)\to P$7, the response functions are

$\sigma:[0,\infty)\to P$8

$\sigma:[0,\infty)\to P$9

In first-order perturbation theory,

$\nu$0

and the finite-time detailed-balance temperature is

$\nu$1

In the ASSF long-time limit,

$\nu$2

so at fixed $\nu$3,

$\nu$4

as $\nu$5 (Parry et al., 8 Jul 2026).

For uniform linear acceleration $\nu$6 in Minkowski vacuum, the Wightman spectrum obeys $\nu$7, giving

$\nu$8

the Unruh temperature $\nu$9 (Parry et al., 8 Jul 2026). The significance of the ASSF construction is that it provides a mathematically explicit route from finite-time switching to such asymptotic temperature statements.

To control the simultaneous long-time and small-gap limit, the 2026 analysis imposes two additional conditions for large $\pi_i$0: a uniform $\pi_i$1 bound,

$\pi_i$2

and localisation of the first moment,

$\pi_i$3

Under these assumptions together with the ASSF conditions, $\pi_i$4 grows at most linearly in $\pi_i$5, which is the regime in which the small-frequency suppression criterion can be analyzed (Parry et al., 8 Jul 2026).

4. Necessity of sign changes and small-frequency suppression

The central result of the 2026 paper is a no-go theorem for non-negative switchings. Let $\pi_i$6 be an ASSF satisfying, for large $\pi_i$7, the uniform bound, the localisation bound, and positivity

$\pi_i$8

Let $\pi_i$9 with $\eta_j$0 and $\eta_j$1. Then the small-frequency suppression (SFS) condition

$\eta_j$2

fails. Consequently,

$\eta_j$3

so no nonzero limiting temperature arises unless $\eta_j$4 changes sign (Parry et al., 8 Jul 2026).

The proof strategy makes the obstruction explicit. Positivity gives $\eta_j$5. The ASSF limit implies $\eta_j$6, hence $\eta_j$7. Using the uniform bound and interpolation, one obtains

$\eta_j$8

on an interval $\eta_j$9, with the derivative bound supplied by the first-moment condition. It follows that

$\rho_{ij}$0

so the SFS ratio is bounded below by a positive constant, contradicting suppression (Parry et al., 8 Jul 2026).

The physical interpretation given in the paper is that, in $\rho_{ij}$1-dimensional circular motion, the Wightman spectrum $\rho_{ij}$2 has a discontinuity at $\rho_{ij}$3, and this leads to vanishing $\rho_{ij}$4 in the small-gap, long-time limit for any non-negative switching. Sign-changing ASSFs instead achieve small-frequency suppression through destructive interference of long-time tails, so that $\rho_{ij}$5 and $\rho_{ij}$6 in a simultaneous $\rho_{ij}$7, $\rho_{ij}$8 limit (Parry et al., 8 Jul 2026).

5. Experimental implications and constructive recipes

The 2026 analysis is explicitly motivated by current work toward experimental verification of the circular motion Unruh effect in analogue spacetime experiments (Parry et al., 8 Jul 2026). Within that setting, the theorem sharpens the design criterion: low-gap observation of a nonzero limiting temperature cannot be obtained with switchings that remain non-negative throughout the interaction.

The paper identifies sign-changing couplings as the operative mechanism. In practice, such couplings arise in entanglement-harvesting protocols, including two-pulse laser schemes in condensed-matter analogues of quantum fields such as Bose–Einstein condensates and superfluid helium films. Implementing an ASSF with zero net area under $\rho_{ij}$9 is presented as a direct recipe for observing the circular Unruh effect at low detector gaps (Parry et al., 8 Jul 2026).

A plausible implication is that the mathematical condition of sign change is not merely a technical artifact of a proof but a concrete spectral-engineering requirement: the switching must actively suppress low-frequency Fourier weight rather than merely extend interaction time. That interpretation is consistent with the role of destructive interference in the SFS condition.

6. ASSFs in switched-system stability theory

In switched systems, ASSFs refer to switching laws rather than detector couplings. For the linear continuous-time model

$2+1$0

with switching signal $2+1$1, the 2013 formulation defines three asymptotic quantities: the asymptotic switching frequency

$2+1$2

the asymptotic fraction of activation

$2+1$3

and the asymptotic transition density

$2+1$4

for admissible transitions $2+1$5 (Kundu et al., 2013).

The stability analysis uses multiple Lyapunov-like functions

$2+1$6

satisfying

$2+1$7

with $2+1$8 for asymptotically stable modes, $2+1$9 for marginally stable modes, and $\{\chi_\lambda(\tau)\}_{\lambda>0}$0 for unstable modes. At a switch $\{\chi_\lambda(\tau)\}_{\lambda>0}$1,

$\{\chi_\lambda(\tau)\}_{\lambda>0}$2

Chaining the flow and jump estimates gives

$\{\chi_\lambda(\tau)\}_{\lambda>0}$3

and the associated ASSF theorem yields global asymptotic stability when the asymptotic switching averages make this exponent strictly negative (Kundu et al., 2013).

This framework was generalized in 2017 to nonlinear switched systems

$\{\chi_\lambda(\tau)\}_{\lambda>0}$4

with instantaneous switching frequency $\{\chi_\lambda(\tau)\}_{\lambda>0}$5, transition fractions $\{\chi_\lambda(\tau)\}_{\lambda>0}$6, and activation fractions $\{\chi_\lambda(\tau)\}_{\lambda>0}$7. In that setting, a switching signal belongs to the ASSF class if

$\{\chi_\lambda(\tau)\}_{\lambda>0}$8

and Theorem 5 states that this condition implies global asymptotic stability (Kundu et al., 2017).

The 2017 paper emphasizes that classical dwell-time, average-dwell-time, mode-dependent dwell-time, and unstable-mode constraints are recovered as special cases of this asymptotic inequality (Kundu et al., 2017). In that sense, ASSFs in switched-systems theory function as a unifying asymptotic criterion, replacing point-wise restrictions by long-run averages.

A related but distinct asymptotic-switching line is the fast-switching averaging theory for diffusions and jump chains. There, one studies $\{\chi_\lambda(\tau)\}_{\lambda>0}$9 and proves convergence of the switched process $C^1$0 to an averaged limit $C^1$1 under ergodicity of the modulator, using semimartingale techniques rather than ASSF terminology (Christensen et al., 2014). This suggests that “asymptotic scaling” has become a broader methodological theme across several switching problems, even when the specific ASSF definition is domain-dependent.