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

Abstract: We study a family of Jacobi operators in which the diagonal entries are multiplied by a coupling parameter $λ\geq0$. Under suitable conditions, the operator is self-adjoint for every $λ>0$, while the formal limit at $λ=0$ 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-$λ$ regime, which combines discrete WKB methods with Airy-function asymptotics. Using these estimates, we analyze the limiting behavior $λ\to0$ in the strong resolvent sense, proving that for every sequence $λ_j\to0$ 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.

• The paper establishes that every self-adjoint extension of the limiting Jacobi operator can be obtained through specific vanishing sequences of the coupling parameter.
• It employs discrete WKB approximations with Airy asymptotics to derive uniform bounds on square-summable solutions and characterizes the resulting spectral spiral behavior.
• The study applies these spectral results to higher-order squeezing in quantum optics, showing that regularization does not select a unique physical evolution.

Essentially Singular Limits of Jacobi Operators and Higher-Order Squeezing

Introduction and Problem Setting

The paper "Essentially singular limits of Jacobi operators and applications to higher-order squeezing" (2605.21355) provides an in-depth analysis of a family of Jacobi operators characterized by diagonal entries scaled by a coupling parameter $\lambda \geq 0$. The authors investigate the limiting behavior of this operator family as the coupling tends to zero, unveiling a nontrivial correspondence between sequences converging to zero and the spectrum of self-adjoint extensions of a limiting symmetric but non-self-adjoint Jacobi operator. This phenomenon, coined an "essentially singular limit," draws an analogy with essential singularities in complex analysis by demonstrating the capacity to approximate any self-adjoint extension via specific vanishing sequences of $\lambda$.

The methodology employs discrete WKB approximations, refined with Airy-function asymptotics, to generate uniform bounds on square-summable solutions of difference equations in the small-coupling regime. The analysis extends beyond abstract operator theory, culminating in applications to higher-order squeezing Hamiltonians in quantum optics, where the absence of a unique physically distinguished self-adjoint extension in the zero-coupling limit has substantial consequences.

Jacobi Operator Family and Limiting Behavior

Consider the family of Jacobi operators parametrized as follows: $$J(\lambda) = \begin{pmatrix} \lambda f_0 & a_0 & 0 & \cdots \ a_0 & \lambda f_1 & a_1 & \ 0 & a_1 & \lambda f_2 & \ddots \ \vdots & & \ddots & \ddots \end{pmatrix}, \quad \lambda \geq 0$$ with sequences $(a_n), (f_n)$ satisfying prescribed growth and regularity conditions: $a_n \sim n^\alpha$, $f_n \sim n^\beta$ for large $n$, with $\beta > \alpha > 4/3$.

For all $\lambda > 0$, $J(\lambda)$ is self-adjoint and bounded from below, with a purely discrete spectrum. In the singular limit $\lambda$0, the resulting Jacobi operator $\lambda$1 has deficiency indices $\lambda$2. Consequently, $\lambda$3 admits a one-parameter family of self-adjoint extensions, denoted $\lambda$4, each possessing simple, discrete spectra covering $\lambda$5 with mutually disjoint spectral points.

The central problem is to characterize the strong resolvent limits of $\lambda$6 as $\lambda$7. The main theorem establishes:

• Non-uniqueness of Limiting Extensions: For every sequence $\lambda$8, a subsequence exists such that $\lambda$9 converges in the strong resolvent sense to some $$J(\lambda) = \begin{pmatrix} \lambda f_0 & a_0 & 0 & \cdots \ a_0 & \lambda f_1 & a_1 & \ 0 & a_1 & \lambda f_2 & \ddots \ \vdots & & \ddots & \ddots \end{pmatrix}, \quad \lambda \geq 0$$0, and, reciprocally, every extension $$J(\lambda) = \begin{pmatrix} \lambda f_0 & a_0 & 0 & \cdots \ a_0 & \lambda f_1 & a_1 & \ 0 & a_1 & \lambda f_2 & \ddots \ \vdots & & \ddots & \ddots \end{pmatrix}, \quad \lambda \geq 0$$1 can be obtained in this fashion.

This phenomenon is aptly termed "essentially singular" since the limit is "maximally nonunique," mirroring essential singularity behavior in complex analytic maps, where a function can attain nearly all values along suitable vanishing sequences.

Figure 1: Numerical plot of the values of the Weyl m-function $$J(\lambda) = \begin{pmatrix} \lambda f_0 & a_0 & 0 & \cdots \ a_0 & \lambda f_1 & a_1 & \ 0 & a_1 & \lambda f_2 & \ddots \ \vdots & & \ddots & \ddots \end{pmatrix}, \quad \lambda \geq 0$$2 at $$J(\lambda) = \begin{pmatrix} \lambda f_0 & a_0 & 0 & \cdots \ a_0 & \lambda f_1 & a_1 & \ 0 & a_1 & \lambda f_2 & \ddots \ \vdots & & \ddots & \ddots \end{pmatrix}, \quad \lambda \geq 0$$3 for $$J(\lambda) = \begin{pmatrix} \lambda f_0 & a_0 & 0 & \cdots \ a_0 & \lambda f_1 & a_1 & \ 0 & a_1 & \lambda f_2 & \ddots \ \vdots & & \ddots & \ddots \end{pmatrix}, \quad \lambda \geq 0$$4 as $$J(\lambda) = \begin{pmatrix} \lambda f_0 & a_0 & 0 & \cdots \ a_0 & \lambda f_1 & a_1 & \ 0 & a_1 & \lambda f_2 & \ddots \ \vdots & & \ddots & \ddots \end{pmatrix}, \quad \lambda \geq 0$$5, illustrating that different sequences spiral to different points on the limit circle associated with self-adjoint extensions of $$J(\lambda) = \begin{pmatrix} \lambda f_0 & a_0 & 0 & \cdots \ a_0 & \lambda f_1 & a_1 & \ 0 & a_1 & \lambda f_2 & \ddots \ \vdots & & \ddots & \ddots \end{pmatrix}, \quad \lambda \geq 0$$6.

Spectral and Resolvent Analysis

The proof framework entails:

• Uniform Asymptotic Bounds: Using discrete WKB and Airy-type asymptotics, the authors derive uniform polynomial decay bounds for square-summable solutions of the recurrence relation

$$J(\lambda) = \begin{pmatrix} \lambda f_0 & a_0 & 0 & \cdots \ a_0 & \lambda f_1 & a_1 & \ 0 & a_1 & \lambda f_2 & \ddots \ \vdots & & \ddots & \ddots \end{pmatrix}, \quad \lambda \geq 0$$7

in the regime of small $$J(\lambda) = \begin{pmatrix} \lambda f_0 & a_0 & 0 & \cdots \ a_0 & \lambda f_1 & a_1 & \ 0 & a_1 & \lambda f_2 & \ddots \ \vdots & & \ddots & \ddots \end{pmatrix}, \quad \lambda \geq 0$$8. A crucial technical result is that, for any compact $$J(\lambda) = \begin{pmatrix} \lambda f_0 & a_0 & 0 & \cdots \ a_0 & \lambda f_1 & a_1 & \ 0 & a_1 & \lambda f_2 & \ddots \ \vdots & & \ddots & \ddots \end{pmatrix}, \quad \lambda \geq 0$$9 domain,

$(a_n), (f_n)$0

demonstrating uniform decay—a necessary ingredient to translate weak convergence to strong resolvent convergence.

• Spectral Curve Geometry: For fixed $(a_n), (f_n)$1, the Weyl $(a_n), (f_n)$2-functions $(a_n), (f_n)$3 for $(a_n), (f_n)$4 spiral around a circle in the complex plane as $(a_n), (f_n)$5, with each point corresponding to a distinct self-adjoint extension of $(a_n), (f_n)$6.
• Limit Parametrization: Given any $(a_n), (f_n)$7, there is a sequence $(a_n), (f_n)$8 such that $(a_n), (f_n)$9 converges to $a_n \sim n^\alpha$0 in the strong resolvent sense, as witnessed both by spectral projections and the behavior of the $a_n \sim n^\alpha$1-function—underlining the lack of selection of a "distinguished" extension.
• Spectral Divergence: All eigenvalues $a_n \sim n^\alpha$2 of $a_n \sim n^\alpha$3 diverge to $a_n \sim n^\alpha$4 as $a_n \sim n^\alpha$5, preventing spectral collapse or aggregation at any finite energy.

  Figure 2: Logarithmic plot of eigenvalues $a_n \sim n^\alpha$6 of $a_n \sim n^\alpha$7 as $a_n \sim n^\alpha$8; all diverge to $a_n \sim n^\alpha$9 demonstrating spectral dispersal in the singular limit.

Applications: Higher-Order Squeezing in Quantum Optics

An important application of these spectral-theoretic results is to the analysis of higher-order squeezing operators in quantum optics, generalizing the quadratic (conventional) squeezing Hamiltonian to $f_n \sim n^\beta$0 for $f_n \sim n^\beta$1. Physically, such Hamiltonians are relevant for generating non-Gaussian states, quantum error correction codes, and nonlinear bosonic gates.

It is well-known that for $f_n \sim n^\beta$2, $f_n \sim n^\beta$3 is not essentially self-adjoint on the natural domain, requiring the specification of a self-adjoint extension to define quantum dynamics. Previous heuristic and numeric studies speculated whether regularization—by addition of a "Kerr-type" term $f_n \sim n^\beta$4, with $f_n \sim n^\beta$5 and $f_n \sim n^\beta$6—singles out a physically preferred extension as $f_n \sim n^\beta$7.

By decomposing the regularized Hamiltonians into direct sums of Jacobi operators over symmetry-reduced subspaces, the authors show:

• Absence of Selection by Regularization: As $f_n \sim n^\beta$8, different sequences of $f_n \sim n^\beta$9 select different self-adjoint extensions of $n$0, with the set of attainable extensions precisely those compatible with the underlying sector decomposition (i.e., ones preserving $n$1-rotational symmetry).
• Physical Implication: Regularization does not resolve the "ambiguity"—distinct physical evolutions are obtained depending on the limiting procedure used for the vanishing parameter.

  Figure 3: Wigner functions of squeezed vacuum states generated by $n$2 for various $n$3 (descending) and the limiting extensions $n$4, $n$5; the different $n$6 limiting subsequences approach different self-adjoint extensions.

Technical and Numerical Illustrations

Numerical evaluation and representation further solidify the spectral and asymptotic analysis. The figures demonstrate, for explicit sequence choices, how the spectrum decomposes, the evolution of the Weyl spiral, and the practical indistinguishability of different extensions for small but nonzero coupling unless the limit is taken with infinite precision—underlining the essential singularity in physical terms.

Figure 4: Numerical evaluation of dominant and recessive eigenfunction solutions in different $n$7 regions; delineates oscillatory, turning-point, and exponential-decay regimes crucial to the uniform bounds achieved.

Figure 5: Plot of uniform normalized solution bounds $n$8, serving as evidence for sharpness of the analytic upper bounds across the three main parameter regimes.

Implications and Future Directions

The analysis has substantial theoretical implications:

• Universality of the Nonuniqueness: The phenomenon demonstrated is not restricted to discrete operators; it suggests that in broad classes of parameter-dependent operators (e.g., certain Sturm–Liouville or Dirac operators with coupling-dependent regularizations), limits as singular as the Jacobi case may arise.
• Selection Principles in Quantum Theories: The work cautions against the expectation that "physical" or "perturbative" regularizations necessarily select canonical quantum evolutions in singular Hamiltonian settings. Any such selection must involve additional, perhaps symmetry-breaking, ingredients.
• Perturbation Theory Beyond Semiboundedness: The techniques exhibit how to extend beyond the standard semibounded or Kato–Rellich paradigm, dealing directly with the nonuniqueness of the strong resolvent limit and its parameterization.
• Discretization and Simulation: The breakdown of direct numerical convergence (e.g., dependence on truncation parity, as detailed for the squeezed vacuum) foreshadows challenges for simulating singular or "borderline" quantum systems—relevant for quantum error correction and nonlinear photonics.

Regarding future developments, generalizing these results to unbounded block Jacobi matrix families and to continuous variable (differential) operators, as well as studying the dynamical consequences (quantum evolution, spectral statistics, control), could reveal new phenomena in both mathematics and quantum technology.

Conclusion

This work delivers a detailed, rigorous account of how essentially singular limits for families of Jacobi operators fundamentally defy selection of a unique self-adjoint extension, even under physically motivated regularizations. The synthesis of asymptotic analysis with operator-theoretic methods not only closes a conceptual gap in the spectral theory of unbounded discrete operators but also delivers critical insight for quantum optics and quantum information applications involving higher-order nonlinearities. The paper sets a new standard for addressing singular perturbation limits in quantum operator algebras, with implications anticipated across mathematical physics and applied quantum systems.