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arxiv: 2605.21355 · v1 · pith:AVCI6RB3new · submitted 2026-05-20 · 🧮 math-ph · math.FA· math.MP· math.SP· quant-ph

Essentially singular limits of Jacobi operators and applications to higher-order squeezing

Felix Fischer , Daniel Burgarth , Davide Lonigro This is my paper

Pith reviewed 2026-05-21 03:11 UTC · model grok-4.3

classification 🧮 math-ph math.FAmath.MPmath.SPquant-ph
keywords Jacobi operatorsself-adjoint extensionsstrong resolvent convergenceessentially singular limithigher-order squeezingquantum opticsdiscrete WKBAiry asymptotics
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The pith

Jacobi operators with vanishing coupling parameter converge in strong resolvent sense to different self-adjoint extensions of the limit operator along different sequences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies a one-parameter family of Jacobi operators whose diagonal entries are scaled by λ greater than or equal to zero. For every positive λ the operators are self-adjoint, yet the formal limit at λ equal to zero is only symmetric and possesses a one-parameter family of self-adjoint extensions. Uniform bounds on square-summable generalized eigenvectors are obtained by combining discrete WKB methods with Airy-function asymptotics; these bounds control the small-λ regime. Using the bounds, the authors prove that every sequence λ_j tending to zero admits a subsequence along which the operators converge in the strong resolvent sense to some self-adjoint extension, and that every extension arises in this manner for a suitable sequence. The same mechanism is applied to higher-order squeezing operators from quantum optics, where the limit selects a symmetry-compatible subclass of extensions rather than a unique distinguished one.

Core claim

For every sequence λ_j → 0 one can extract a subsequence along which the corresponding Jacobi operators converge to some self-adjoint extension of the limiting operator; conversely, every such extension can be obtained in this way. This behavior is called an essentially singular limit.

What carries the argument

Uniform bounds on square-summable generalized eigenvectors for small λ, derived via discrete WKB methods combined with Airy-function asymptotics, which control the strong resolvent convergence to the family of extensions.

If this is right

  • Convergence occurs in the strong resolvent sense along suitable subsequences.
  • Every self-adjoint extension of the limiting symmetric operator arises as a strong resolvent limit point.
  • In the squeezing-operator application the vanishing limit does not pick out a single physically preferred extension but only a symmetry-compatible subclass.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result indicates that physically relevant realizations of singular limits may depend on the precise manner in which the parameter is sent to its critical value.
  • Similar essentially singular behavior could appear in other parameter-dependent families of differential or difference operators whose domains change at the limit point.
  • Numerical diagonalization of the finite Jacobi matrices for successively smaller λ could test whether the predicted Airy-type decay of the eigenvectors is visible in concrete spectra.

Load-bearing premise

The uniform bounds on square-summable generalized eigenvectors in the small-λ regime hold for the family of Jacobi operators under the stated conditions for self-adjointness when λ is positive.

What would settle it

Existence of a sequence λ_j → 0 such that no subsequence of the associated Jacobi operators converges in the strong resolvent sense to any self-adjoint extension of the λ=0 operator.

Figures

Figures reproduced from arXiv: 2605.21355 by Daniel Burgarth, Davide Lonigro, Felix Fischer.

Figure 1
Figure 1. Figure 1: Numerical plot of the values of the Weyl 𝑚-function 𝑀(𝑧, 𝜆) at 𝑧 = i associated with the Jacobi operator 𝒥(𝜆) for different values of 𝜆 > 0, for the choice 𝑎𝑛 = 3−3∕2√ (3𝑛 + 1, 3), 𝑓𝑛 = 𝑛 3 , where (⋅, ⋅) denotes the Pochhammer symbol, cf. Eq. (2.16). As 𝜆 → 0, the points spiral around the limit circle of the limiting operator, shown in black. The result yields an explicit construction through which the fu… view at source ↗
Figure 2
Figure 2. Figure 2: Wigner functions of Ψ3,3 (𝐾, 𝑇) (cf. Eq. (2.21)) for different values of 𝐾 and fixed time 𝑇 = 1, together with the limiting states Φ3 (0, 𝑇) and Φ3 (∞, 𝑇)(cf. Eq. (2.22)) corresponding to the two self-adjoint extensions with parameters 𝑡 = 0 and 𝑡 = ∞, respectively. Different subsequences converge to different limiting states. Following the proof of Theorem 2.7, we numerically construct a sequence (𝐾𝑗 )𝑗∈ℕ… view at source ↗
Figure 3
Figure 3. Figure 3: Logarithmic plot of the eigenvalues 𝐸 (𝑗) (𝜆) of the Jacobi operator 𝐽(𝜆), de￾fined as in Definition 2.4, with coefficients 𝑎𝑛 = 3−3∕2√ (3𝑛 + 1, 3) and 𝑓𝑛 = 𝑛 3 , where (⋅, ⋅) denotes the Pochhammer symbol, cf. Eq. (2.16). Note that this op￾erator coincides with 𝐴 (0) 3,3 (𝜆) from Proposition 2.9. Monotonicity, continuity and lim𝜆→0 𝐸 (𝑗) (𝜆) = −∞ are clearly visible. Equipped with these results, we can fi… view at source ↗
Figure 4
Figure 4. Figure 4: Numerical evaluation of the dominant and the recessive solution of Eq. (3.5) for the family of Jacobi operators 𝐽(𝜆), defined as in Definition 2.4, with coefficients 𝑎𝑛 = 4−1∕2√ (4𝑛 + 1, 4) and 𝑓𝑛 = 𝑛 4 , where (⋅, ⋅) denotes the Pochham￾mer symbol, cf. Eq. (2.16), for two different values of 𝜆. The left plot corresponds to 𝜆 = 1∕791, while the right plot corresponds to 𝜆 = 1∕1191. The different background… view at source ↗
Figure 5
Figure 5. Figure 5: Plot of sup𝑛≥𝑛0 𝑟 𝜆𝑗 ,i 𝑛 , with 𝑟 𝜆,𝑧 𝑛 defined in Eq. (4.229), for the sequence 𝜆𝑗 = (10𝑗) −1 and different values of 𝑛0 . The underlying Jacobi operator 𝐽(𝜆) is parametrized as in Definition 2.4 with the sequences (𝑎𝑛 )𝑛∈ℕ and (𝑓𝑛 )𝑛∈ℕ from Eq. (4.232). As predicted by Theorem 3.34, the ratios remain uniformly bounded from above by a constant independent of 𝜆𝑗 . At the same time, for the displayed value… view at source ↗
Figure 6
Figure 6. Figure 6: Plot of the three regions 𝑆−1, 𝑆0 , 𝑆1 ⊂ ℂ as defined in Eq. (A.4), cf. also [156, p. 413]. The Stokes rays 𝜔 𝑗ℝ− are indicated in blue. The Airy functions Ai𝑗 from Eq. (A.3) are recessive in 𝑆𝑗 and dominant in 𝑆𝑗±1, in accordance with the Stokes phenomenon for Eq. (A.1). Of particular importance are the boundaries of the sectors 𝑆𝑗 , given by 𝜔 𝑗±1ℝ−. Along these rays, the asymptotic behavior of Ai𝑗 chang… view at source ↗
read the original abstract

We study a family of Jacobi operators in which the diagonal entries are multiplied by a coupling parameter $\lambda\geq0$. Under suitable conditions, the operator is self-adjoint for every $\lambda>0$, while the formal limit at $\lambda=0$ is a symmetric Jacobi operator admitting a one-parameter family of self-adjoint extensions. A central ingredient of our analysis is the derivation of uniform bounds for square-summable generalized eigenvectors in the small-$\lambda$ regime, which combines discrete WKB methods with Airy-function asymptotics. Using these estimates, we analyze the limiting behavior $\lambda\to0$ in the strong resolvent sense, proving that for every sequence $\lambda_j\to0$ one can extract a subsequence along which the corresponding Jacobi operators converge to some self-adjoint extension of the limiting operator; conversely, every such extension can be obtained in this way. We call this behavior an essentially singular limit, by analogy with essential singularities in complex analysis. As an application, we study higher-order squeezing operators arising in quantum optics. Using the connection with Jacobi operators, we show that when the relative strength of the free-field term tends to zero, different self-adjoint extensions of the squeezing operator are selected along different sequences. In particular, this limit does not single out a physically distinguished self-adjoint extension, but instead identifies a distinguished subclass of extensions compatible with the underlying symmetry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript studies a one-parameter family of Jacobi operators in which the diagonal entries are scaled by λ ≥ 0. For each fixed λ > 0 the operator is assumed self-adjoint under stated conditions on the off-diagonal coefficients, while the formal λ = 0 limit is a symmetric operator whose deficiency indices are (1,1) and which therefore admits a one-parameter family of self-adjoint extensions. The central technical step is the derivation of λ-uniform bounds on square-summable generalized eigenvectors for small λ, obtained by combining discrete WKB approximation away from the turning point with Airy-function asymptotics near the turning point. These bounds are then used to establish that the family exhibits an “essentially singular limit”: every sequence λ_j → 0 admits a subsequence along which the operators converge in the strong resolvent sense to some self-adjoint extension of the λ = 0 operator, and conversely every such extension arises as the strong-resolvent limit along a suitable sequence. The result is applied to higher-order squeezing operators arising in quantum optics, showing that the singular limit selects a symmetry-compatible subclass of extensions rather than a single distinguished extension.

Significance. If the uniform eigenvector bounds and the subsequent strong-resolvent convergence statements are fully rigorous, the paper supplies a precise description of how self-adjoint extensions are selected in a singular limit that is not captured by ordinary strong or norm resolvent convergence. The analogy with essential singularities is conceptually useful, and the quantum-optics application demonstrates that the phenomenon has concrete implications for the choice of domain in unbounded operators appearing in quantum mechanics. The combination of discrete WKB with Airy matching for uniform bounds is a technical contribution that may be reusable in other discrete Schrödinger problems with slowly varying coefficients.

major comments (1)
  1. The uniform bounds on ℓ²-normalized generalized eigenvectors (the key hypothesis for both the subsequence extraction and the surjectivity onto all extensions) are asserted to follow from discrete WKB plus Airy asymptotics, yet the manuscript does not display explicit λ-independent error estimates for the WKB phase or for the matching constants at the turning point. If these error terms grow with 1/λ or depend on the particular sequence λ_j, the claimed strong-resolvent convergence may fail to hold uniformly or may miss some boundary conditions. A concrete verification that the constants remain bounded independently of λ (and of the sequence) is required before the central claim can be accepted.
minor comments (3)
  1. The precise definition of the Jacobi operator family (including the precise conditions on the off-diagonal sequence that guarantee self-adjointness for λ > 0) should be stated as a numbered assumption or theorem early in the introduction rather than being scattered across the text.
  2. Notation for the limiting operator and its extensions (e.g., the parameter that labels the boundary conditions) should be introduced once and used consistently; at present the same symbol appears to be reused for different objects in the abstract and in the application section.
  3. The statement that the limit “identifies a distinguished subclass of extensions compatible with the underlying symmetry” would benefit from an explicit characterization of that subclass (for example, by a concrete condition on the boundary parameter).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We appreciate the positive assessment of the conceptual contribution and the quantum-optics application. We address the single major comment below and will strengthen the presentation of the uniform bounds in the revision.

read point-by-point responses

  1. Referee: The uniform bounds on ℓ²-normalized generalized eigenvectors (the key hypothesis for both the subsequence extraction and the surjectivity onto all extensions) are asserted to follow from discrete WKB plus Airy asymptotics, yet the manuscript does not display explicit λ-independent error estimates for the WKB phase or for the matching constants at the turning point. If these error terms grow with 1/λ or depend on the particular sequence λ_j, the claimed strong-resolvent convergence may fail to hold uniformly or may miss some boundary conditions. A concrete verification that the constants remain bounded independently of λ (and of the sequence) is required before the central claim can be accepted.

    Authors: We agree that the current manuscript states the uniform bounds without displaying fully explicit, λ-independent error estimates for the WKB phase and the Airy-matching constants. In the revised version we will insert a new subsection (approximately 3.3) that supplies these estimates. Away from the turning point we control the discrete WKB remainder by the standard Gronwall-type argument for slowly varying coefficients, yielding a phase error of O(λ) uniformly in the oscillatory region for all λ small enough. Near the turning point we match the discrete solution to the Airy function via the known asymptotic expansion of the Airy function together with a discrete variation-of-constants formula; the resulting connection coefficients differ from their λ=0 limits by at most O(λ^{1/3}), with the implied constant independent of λ and of any particular sequence λ_j→0. These bounds are uniform for all sufficiently small λ>0 and therefore guarantee that the ℓ²-normalized generalized eigenvectors remain bounded independently of λ, which in turn justifies both the subsequence extraction and the surjectivity onto every self-adjoint extension. We thank the referee for highlighting this gap; the added estimates will make the central technical step fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external standard tools

full rationale

The paper's central result establishes subsequential strong-resolvent convergence of Jacobi operators as λ→0 to self-adjoint extensions of the λ=0 limit, with the converse also holding. The load-bearing step is the derivation of uniform bounds on square-summable generalized eigenvectors for small λ, obtained by combining discrete WKB approximation with Airy-function asymptotics near turning points. These are standard external analytic techniques independent of the paper's own claims or fitted quantities. No equation or argument reduces by construction to a self-definition, a renamed fit, or a load-bearing self-citation chain; the convergence theorem follows from these bounds without circular reduction to the target statement itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard facts from operator theory plus newly derived uniform bounds; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Jacobi operator with positive λ is self-adjoint under suitable conditions on the off-diagonal entries.
    Invoked as the starting point that guarantees the family is well-defined before taking the limit.
  • ad hoc to paper Uniform bounds for square-summable generalized eigenvectors exist in the small-λ regime.
    This is the central technical ingredient whose derivation via discrete WKB and Airy asymptotics enables the resolvent convergence argument.

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