Papers

Quantum incommensurate systems have attracted widespread interest due to their unique physical properties. Related studies have made notable progress in recent years. To gain deeper insight, it is both significant and challenging to study a broader range of physical observables for these systems, which requires a comprehensive understanding of their spectral properties and wavefunction behavior. Based on the regularized model recently proposed, this work introduces a regularization framework, rendering physical observables for incommensurate systems mathematically well-defined and computationally accessible with theoretical guarantees. Based on this framework, taking the density of states and the electron density as representative examples, we establish their mathematical characterization, derive their planewave approximations, and provide a rigorous convergence analysis. Numerical experiments validate the effectiveness of our method, demonstrating its practical applicability for generic 1D and 2D incommensurate systems.

We consider a hierarchy of models that can be obtained from the original Nyzhnyk system by imposing conditions on parameters, introducing potentials or pseudopotentials, performing limiting processes with respect to a scaling parameter, applying differential substitutions and interpreting parts of independent and/or dependent variables as complex or real. Therefore, among the Nyzhnyk models, one can distinguish between symmetric and asymmetric, dispersive and dispersionless, standard and modified, as well as real, complex, mixed and specific models. One can also distinguish single partial differential equations or systems of such equations, as well as linear or nonlinear Lax representations in the dispersive or dispersionless cases, respectively. For each specified model, we compute the maximal Lie invariance pseudoalgebra and, in the symmetric case, find the point- and contact-symmetry pseudogroups using the megaideal-based version of the algebraic method. It is shown that relations between these pseudoalgebras and between these pseudogroups are induced by relations between the corresponding models. Defining (finite-dimensional) subalgebras are singled out in all these pseudoalgebras in the symmetric and specific cases. Based on the established correspondences between the dispersionless and dispersive models, we completely classify one- and two-dimensional subalgebras of the maximal Lie invariance pseudoalgebra of the (dispersive symmetric potential) Nyzhnyk equation and one-dimensional subalgebras of the maximal Lie invariance pseudoalgebra of its linear Lax representation.

A quantum measurement transforms a quantum statistical experiment into a classical one, but different measurements generally yield experiments with different statistical information. We introduce quantum-to-classical sufficiency: relative to a prescribed measurement class, a measurement is sufficient if its induced classical experiment Blackwell-dominates those from every other measurement in the class. This separates two components of quantum inference: reducing the measurement class through symmetry or decision-theoretic arguments, and identifying a canonical measurement in the reduced class by an experiment-level sufficiency condition. We establish a factorization criterion where the parameter enters the group-averaged experiment only through weights of an orthogonal block decomposition. The block-label measurement is then sufficient relative to all invariant measurements. For tensor-product models under unitary conjugation, this identifies weak Schur sampling as the sufficient measurement for spectral inference. Combined with risk-preserving symmetrization, it reduces invariant Bayes and minimax problems over all measurements exactly to classical decision problems based on Young diagrams. We derive sharp first-order asymptotic Bayes and minimax risks for smooth spectral functionals. We establish an analogous reduction for a noncompact group action in a continuous-variable setting. We consider displaced thermal states and functionals of the thermal parameter with an unknown displacement. Within displacement-invariant measurements, total residual photon number measurement emerges as the sufficient measurement. Combined with Hunt-Stein reduction, this yields exact finite-sample reductions of Bayes-minimax and minimax problems over all measurements to classical decision problems based on the total residual photon count, which follows a negative binomial distribution.

We establish a pointwise lower bound for the heat kernel of the elliptic operator $ \Lambda=(1+|x|^\alpha)\Delta +b|x|^{\alpha-2}x\cdot\nabla-|x|^\beta$, where $d\geq3$, $\alpha>2$, $\beta>\alpha-2$, and $b\in\mathbb R$. For every $\tau>0$, we prove that $$ \begin{aligned} p(t,x,y) &\geq C_\tau e^{\lambda_0t} \left(\frac{1+|y|^\alpha}{1+|x|^\alpha}\right)^{\frac{b}{2\alpha}} \frac{(|x||y|)^{-\frac{d-1}{2}-\frac{\beta-\alpha}{4}}}{1+|y|^\alpha}\\ &\quad\times\exp \left[ -\int_1^{|x|}\sqrt{\frac{s^\beta}{1+s^\alpha}}\,\mathrm ds -\int_1^{|y|}\sqrt{\frac{s^\beta}{1+s^\alpha}}\,\mathrm ds \right] \end{aligned} $$ for all $t\geq\tau$ and $|x|,|y|\geq1$, where $\lambda_0<0$ is the largest eigenvalue of $\Lambda$ and $C_\tau>0$ is independent of $t,x,y$. The proof applies the classical Davies - Simon argument in a weighted symmetric setting.

We study large-overtone quasi-normal modes of a massive scalar field in a four-dimensional Schwarzschild de-Sitter black hole in the presence of a wall placed at the static sphere. Dirichlet boundary condition is imposed at the wall. At leading order the quasi-normal modes with large overtones scale as $n$ for $n\gg 1$ and are inversely proportional to the bouncing time. We analyze correction terms up to $n^{-1}$ order in the asymptotic expansion.

This data article provides the datasets, symbolic derivations, and scripts used to reproduce master diagrams, stationary-phase spectra, windowed first-order coherence functions, and quantum-noise stability maps for strongly chirped dissipative solitons of the cubic-quintic complex Ginzburg-Landau equation in normal and anomalous group-delay dispersion regimes. The repository includes node-regularized normal-dispersion spectra and energies; small-parameter expansions of the branch roots; cavity-map gain-loss update relations; Airy uniformization at the normal-dispersion spectral edge; anomalous-dispersion spectra and coherence calculations; and processed tables for plotting and stability analysis. OriginLab projects are accompanied by open-format .csv/.txt numerical tables to support reuse without proprietary plotting software. Data and code repository: https://doi.org/10.5281/zenodo.22690899.

For $0<\kappa\leq4$, we construct a closed response calculus for $\mathrm{SLE}_\kappa$ weldings and a canonical Dirichlet form. The divergence covariance combines the Weil–Petersson and Velling–Kirillov forms. The integrated response gives exact changes of measure and canonical Liouville–capacity increments on conformally removable weldings.

The winding number is a simple topological invariant. In the case of chiral symmetry it characterises gapped phases of Fermions. We study statistical properties of this topological index or invariant in a chiral symplectic setting using Random Matrix Theory. We consider ensembles of Hamilton matrices in the symmetry class CII (quaternionic matrices) according to the classification in the tenfold way. We set up a parametric random matrix model and derive expressions for parametric correlations of the winding number density as well as for the discrete winding number distribution. We found a super-universality in the limit of large matrix dimensions for the one- and two-point correlators for the bulk of parameters, meaning that the results agree up to rescaling with those of the class AIII (complex matrices). In this context we employ a new method of unfolding and discover the Gaussian behaviour of the winding number distribution.

Spatial correlations can suppress scattering in disordered chains and produce perfectly transmitting resonances. The practical value of this protection, however, depends not on the resonance peak itself but on the width of the surrounding transmission window and its sensitivity to local ordering errors. We show that a resonance can remain exactly transparent at its center while arbitrarily rare adjacent-swap errors restore an inverse localization length proportional to the square of the energy detuning. For lossless single-channel chains assembled from independent blocks of fixed length and composition, this positive quadratic term is guaranteed by a local scattering invariant and holds for every arrangement and every fixed swap probability between zero and one. With exact tuning and matched contacts, the central transmission remains unity. A microscopic quantum chain exhibits both higher-order suppression of scattering in the ideal recursive arrangement and the predicted response to local exchanges. These results reveal a limitation of spatial ordering that is invisible to a measurement at the resonance alone: spatial ordering protects the resonance peak, not the transport around it. The effect can therefore be tested experimentally by measuring transmission spectra before and after exchanges, without identifying microscopic defects.

Wetting transitions on textured substrates play a central role in the design of functional surfaces, but resolving their transition mechanisms requires efficient exploration of complex energy landscapes. In particular, saddle points are essential for revealing the connectivity among metastable states and transition pathways. In this work, we develop a diffuse-domain accelerated saddle dynamics (DD-ASD) framework for mass-constrained wetting transitions on complex textured substrates. The diffuse-domain formulation embeds complex solid geometries into a Cartesian grid, avoiding body-fitted mesh construction in repeated saddle-point searches. An orthogonal projection is applied consistently to the phase-field state and unstable-direction dynamics, preserving the prescribed droplet mass at the discrete level. Momentum acceleration is incorporated into the saddle dynamics to improve the efficiency of high-index saddle searches. Under the stated local spectral and exact-eigenspace assumptions, we establish a local convergence rate of $1-\mathcal{O}(1/\sqrt{\kappa})$ on the effective mass-conserving subspace. Numerical experiments demonstrate the effectiveness of the proposed method in resolving wetting solution landscapes, transition pathways, and energy barriers on textured substrates. We further extend the framework to fully three-dimensional wetting-landscape computations.

We obtain a large number of exact hyperbolic, trigonometric, and periodic solutions in terms of Jacobi elliptic functions as well as algebraic solutions with a power law tail of the integrable local Fokas-Lennels equation and integrable nonlocal Fokas-Lennels equation. Further, we consider a one-parameter family of generalized Fokas-Lenells equations and obtain a few of their exact solutions.

A multispecies, collisionless plasma is modeled by the relativistic Vlasov-Maxwell system. Assuming the plasma is globally neutral, we show that solutions can be constructed with arbitrarily fast, polynomial rates of decay, depending upon the cancellation properties of the limiting spatial averages, as well as the masses and charges of the particles. In doing so, we establish a countably infinite number of asymptotic profiles for the charge and current density, electric and magnetic fields, and their derivatives, each of which is necessarily realized by a sufficiently smooth solution. Our approach relies upon a refined field representation that features self-similar behavior away from the boundary of the light cone. In each case we establish a linear scattering result in $L^\infty$ for every particle distribution function, namely we show that they converge as $t \to \infty$ along the transported spatial characteristics at increasingly faster rates. In the case that charge cancellation does not occur, our methods can be further extended to demonstrate polyhomogeneous expansions with logarithmic corrections.

The simulation of nonlinear ordinary differential equations on quantum computers is inherently challenging, as quantum gates are linear operators on qubit states. In this paper, we put forth a Koopman-von Neumann (KvN) operator based algorithm for solving nonlinear ordinary differential equations on a quantum computer which overcomes the innate limitations of quantum operations. In this approach, a Liouville probability density is embedded into a wavefunction, the evolution of which is governed by an operator dual to the Koopman operator. The trajectories corresponding to the nonlinear differential equations are reconstructed from the average value of Koopman observables evaluated through quantum measurements. We specifically evaluate the computational limitations of solving nonlinear equations within this Liouville embedding framework. An Ehrenfest-type estimate is derived that relates the reconstruction error to the covariance of the transported density, highlighting the competition between linear stretching and Hessian-induced folding.This leads to a key stability criterion, which depends on the local Ehrenfest-Reynolds number. We discuss the effects of measurement-induced errors including those due to Hadamard-test sampling, amplitude estimation, and bias due to covariance in probabilistic Grover-type inference on the evaluation of trajectories. The limiting bounds on the accuracy of quantum computations are numerically verified for the Lotka-Volterra system and for the quartic oscillator. We observe a rapid growth in computational errors when the Ehrenfest-Reynolds number approaches the predicted threshold. The bounds resulting from our analysis provide key guidelines for selecting the width of the initial Gaussian probability distribution associated with a system of nonlinear ordinary differential equations.

In this paper, we construct primitive KMS-symmetric quantum Markov semigroups for which the $p$-logarithmic Sobolev constant vanishes at $p=1/2$ and a correlated graph-based example for which the $p$-logarithmic Sobolev constant tends to $-\infty$ as $p\to 0^+$. These examples obstruct a positive uniform reverse-hypercontractivity rate on $(0,1)$. By contrast, we show that every primitive KMS-symmetric quantum Markov semigroup satisfies quantum reverse hypercontractivity for $1/2< p\leq q < 1$ with explicit sufficient times. For each fixed $\delta>0$, a sufficiently strong dephasing perturbation satisfies a uniform positive logarithmic Sobolev inequality on $[\delta, 1]$ and hence quantum reverse hypercontractivity for $\delta< p\leq q<1$ with fixed $0<\delta<1/2$. Furthermore, we establish reverse hypercontractivity for graph-based KMS-symmetric quantum Markov semigroups for $\delta< p\leq q< 1$ with fixed $0<\delta<1$ for connected graphs when $n \geq 3$.

We discuss in detail Neutral Measures, which are measures which believably generate any real in Cantor space. We intuitively build up the classical notions of randomness, and show that neutral measures do exist, using the language of continuous semimeasures. Finally, we construct some specific randomness tests which allow us to enforce de- sirable properties on neutral measures. As an application, we show the existence of Gibbs measures for a large class of finite range Hamilto- nians on lattice models, and establish their relationship with neutral measures.

Chimera states, characterized by the coexistence of coherent and incoherent dynamics in networks of coupled oscillators, are among the most intriguing collective phenomena in nonlinear systems and are strongly shaped by the underlying topology and initial conditions. Their analysis in large-scale networks remains computationally demanding and motivates the development of coarse-graining strategies that reduce network size while retaining the essential dynamical features of chimera behavior. In this Letter, we investigate whether equitable-partition-based quotient graphs can preserve chimera patterns under substantial reductions of network size. Using nonlocally coupled FitzHugh-Nagumo (FHN) oscillators, we exploit the fact that, for an equitable partition, the full-network dynamics restricted to the associated cluster-synchronous subspace coincide exactly with the quotient dynamics. We ask whether this structural exactness is sufficient to preserve chimera signatures after the network size reduction. Our results show that this is not necessarily the case: agreement between quotient and full-network chimera dynamics depends on the equitable partition, and exact quotient dynamics do not necessarily guarantee the transverse stability of the corresponding full-network trajectory. Importantly, when the cluster-synchronous solution becomes transversely unstable, the full network may depart from the exact quotient trajectory while remaining in a macroscopic chimera regime. Therefore, structural exactness and dynamical preservation are distinct requirements, and transverse stability provides a complementary criterion for assessing quotient-based coarse-graining of chimera states.

Equivalence groups associated with the field equations of arbitrary motions of inhomogeneous hyperelastic solids are investigated. The explicit forms of the infinitesimal generators, namely the components of the isovector field associated with the equivalence groups are derived. Additional constraints on the isovector components are established to ensure that the transformed equations are supported by a strain energy function. Possible mappings between the equations of materials with different properties are examined. Not only the transformations between homogeneous and inhomogeneous hyperelasticity equations, but also the transformations between the equations and strain energy functions of isotropic and anisotropic materials are investigated. The results are illustrated with several examples.

The random hopping Hamiltonian is a toy model for a disorder-driven topological phase transitions in one-dimensional chiral Hamiltonians. It has a vanishing Lyapunov exponent and a Dyson peak in the density of states at zero energy. This work shows how the quantum dynamics leads to a logarithmic growth of the moments of the position operator, similar as in classical Sinai diffusion. The mechanism at the origin of this phenomenon is that the eigenfunction of the smallest eigenvalue is well-approximated by the exponential of a classical random walk, leading to two essentially independent subexponential localization centers.

We prove that the regularized, stabilized sandwiched Rényi divergence of finite-dimensional quantum channels converges to their regularized relative entropy as the Rényi order tends to one. The key tool is the channel hockey-stick divergence: Gour's Stinespring approximation bound and a Schatten norm estimate amplify an asymptotic bound below one into exponential decay at higher threshold rates. For channel pairs with finite max-relative entropy, known operational connections then give exponential strong converses for parallel and adaptive discrimination, a sharp zero–one testing law, and the subchannel asymptotic equipartition property.

The ${L}_\Lambda w_{1+\infty}$ algebra is the cosmological-constant deformation of the celestial $Lw_{1+\infty}$ symmetry algebra of gravity, providing an infinite-dimensional extension of the three-dimensional conformal algebra at the AdS$_4$ boundary. We construct the Hamiltonian charges associated with these symmetries from boundary ambitwistor-space data, using LeBrun's Heaven on Earth correspondence, and translate them to spacetime via the Penrose transform. We relate the resulting expressions to light-ray operators in the boundary CFT$_3$, including ANEC operators, their conformal descendants, and their commutators.

These are extended notes based on a series of lectures given by the author at the Institut de Mathematiques de Toulouse in June 2025. The goal is to present classical problems as well as the most recent developments in spectral geometry of the Laplace operator in Euclidean domains, subject to Dirichlet, Neumann and Robin boundary conditions. Among the topics covered, there are embedded eigenvalues in unbounded domains, spectral isoperimetric inequalities, nodal-line and hot-spots conjectures, inverse problems of hearing the shape of a drum and geometrically induced eigenvalues and Hardy-type inequalities in curved tubes.

Generalized Langevin equations describe non-Markovian dynamics in which the evolution of resolved variables depends on their past. We propose a memory-conditioned diffusion method for learning stochastic flow maps of these dynamics from observed trajectories, without identifying a memory kernel or reconstructing unresolved variables. A compact, recursively updated bank of exponential filters enables the flow map to retain predictive history over multiple time scales without conditioning on long observation windows. The next-step distribution is conditioned on the current observation and this memory state, whose storage and update costs are independent of the history length for a fixed bank size. Predictive criteria guide the memory budget, with reference-assisted selection in the vector benchmark, and an optional linear projection further reduces the conditioning dimension. A kernel-based score estimator generates conditional samples without training a score network, and these samples are used to train a neural flow map for autoregressive simulation. Three numerical examples assess long-memory retention at small conditioning dimension, predictive compression in coupled vector dynamics, and non-Gaussian conditional distributions and intermittent events. The non-Gaussian example reproduces conditional asymmetry and burst statistics in a stochastic model of the plasma scrape-off layer.

The Dyson series expands the propagator of a time-dependent Hamiltonian in powers of the Hamiltonian, but its truncations are in general not unitary. The Fer expansion writes the same propagator as an infinite product of matrix exponentials, each of them unitary, and its remainder decays doubly exponentially with the number of factors. Convergence, however, is guaranteed only within a finite radius: the time integral of the norm of the Hamiltonian must be smaller than $2$. This is the best value known to date. Here we improve it by about $30\%$, raising it to about $2.6058$. The result also holds for non-Hermitian Hamiltonians that are Hermitian with respect to a fixed metric.

The axisymmetric profile \(\sin\psi(\rho)=c_0\rho\log(\rho/\rho_B)\) introduced by Naito, Okuda, and Ouyang is an exact solution of the unconstrained Helfrich shape equation for every \(\rho>0\), but its mean curvature diverges logarithmically at the dimple center. A covariant local analysis shows that in Cartesian graph coordinates the immersion is nondegenerate and belongs to \(W^{2,p}\) for every finite \(p\); it is \(C^{1,\alpha}\) for every \(\alpha<1\), but neither \(C^{1,1}\) nor \(C^2\). The conformal factor has a finite nonzero limit, so the point has branch multiplicity one and is not a branch point in the sense used in weak-immersion compactness theory. The Helfrich energy is finite, whereas the complete Euler–Lagrange operator carries the distributional residue \(\E_{\CH}=4\pi c_0\delta_p\) under the convention specified below. Hence the profile is a classical solution on the punctured surface and a finite-energy weak immersion, but it is not a free weak critical point of the unforced Canham–Helfrich functional. Stationarity is restored after adding the corresponding point-force potential or under a pinned-point constraint. Area and volume multipliers cannot cancel the Dirac residue. The apparent sign discrepancy between the shape equation used in the original papers and the modern convention is explained by the opposite definition of mean curvature. These results delimit precisely what can and cannot be inferred from modern existence and regularity theorems for Canham–Helfrich minimizers.

This is an English translation of a book originally published in Japanese by Saiensu-sha in 2015. The book is an introduction to the 3d-3d correspondence, the relation between 3d $\mathcal{N}=2$ supersymmetric gauge theories and the geometry of 3-manifolds, which arises from the compactification of the 6d $\mathcal{N}=(2,0)$ theory. The presentation is bottom-up rather than top-down: instead of starting from the 6d theory, we begin by asking what a quantum field theory is, and let the geometry emerge on its own. Along the way we discuss renormalization and low-energy effective theories, gauging as an operation which glues field theories together, the resulting $Sp(2n, \mathbb{Z})$-action on 3d theories, 3d $\mathcal{N}=2$ supersymmetric theories and their dualities, the squashed three-sphere partition function and supersymmetric localization, duality domain walls of 4d $\mathcal{N}=2$ theories, quantum Teichmuller theory, complex Chern-Simons theory and the quantization of the moduli space of flat $SL(2, \mathbb{C})$-connections on 3-manifolds, with quantum dilogarithm functions as a recurring thread. The emphasis throughout is on geometric and algebraic structures which are invisible in a single field theory, and appear only in the theory space of quantum field theories. Appendices summarize supersymmetry in various dimensions, classical and quantum dilogarithm functions, and cluster algebras. Exercises with difficulty ratings are included in each chapter.

We study the high-dimensional recovery of a signal vector $\mathbf{x}$ in the presence of sparse Wishart-like noise. We define an $N \times N$ matrix $A = J+(\theta/N)\mathbf{xx}^{\top}$, where $\mathbf{xx}^{\top}$ is the rank-one deformation of the random noise matrix $J$. We consider a Wishart-like matrix $J={X}^{\top} X$, where $X$ is a sparse $M \times N$ random matrix with entries $X_{ij} = c_{ij}W_{ij}$, with $c_{ij}$ regulating the density of non-zero elements, and $W_{ij}$ the bond weights. Using the replica method, we compute analytically the top eigenpair statistics of $A$, and their dependence on the signal strength $\theta$, the rectangularity ratio $\alpha=\sqrt{M/N}$, and the average connectivity of the noise. The spectral observables are expressed in terms of a system of Recursive Distributional Equations, which are efficiently solved via a Population Dynamics algorithm. They allow us to compute the average largest eigenvalue $\langle\lambda_1\rangle_{A}$, the average top eigenvector component density, and the average overlap between the top eigenvector of $A$ and $\mathbf{x}$. We identify a critical threshold $\theta_{\mathrm{crit}}$–depending on the average connectivity of the noise–that marks a BBP-like phase transition: below this value, $\langle\lambda_1\rangle_{A}$ is unaffected by the signal, and the overlap vanishes. Thus, the signal is not recoverable from the top eigenvector of $A$. For $\theta>\theta_{\mathrm{crit}}$, the signal-related outlier eigenvalue becomes $\langle\lambda_1\rangle_{A}$ and the overlap is nonzero, allowing for recovery of the signal. The results are in excellent agreement with numerical diagonalisation. We show that in the dense limit, the recovery threshold and eigen-statistics converge to the results predicted by the classical BBP transition for additive rank-one deformations of dense Wishart matrices.

We introduce a class of $2d$ quantum spin systems based on a unitary spherical fusion category with a chosen object. These models have frustration-free, reflection positive, but generally non-commuting local interactions, generalizing the Levin-Wen string-net models. Based on the theory developed in arXiv:2510.20662, we explicitly realize the Temperley-Lieb-Jones algebra, and more generally fusion spin chains as their boundary algebras.

We prove the Kazhdan-Lusztig correspondence for the Lie superalgebra $\mathfrak{osp}_{1|2n}$ at irrational levels, that is, we show the category $\mathrm{KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ of finite-length even ordinary modules for the affine vertex operator superalgebra of $\mathfrak{osp}_{1|2n}$ at level $k \in \mathbb{C} \setminus \mathbb{Q}$ is braided tensor equivalent to the category of finite-dimensional even weight modules for the quantum group of $\mathfrak{osp}_{1|2n}$ at parameter $q = e^{\pi i/(2k+2n+1)}$. We also prove that ${\rm KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ is braided tensor equivalent to the category ${\rm KL}_\ell^{\rm ns}(\mathfrak{so}_{2n+1})$ of finite-length ordinary modules with non-spinorial top level for the affine vertex operator algebra of $\mathfrak{so}_{2n+1}$ at level $\ell$ such that $ \frac{1}{\ell+ 2n-1} = \frac{1}{2k+2n+1} + 1 \ \ ({\rm mod}\ 2\mathbb Z).$ Consequently, by gluing vertex operator (super)algebras via tensor categories, we construct a few new families of simple conformal vertex (super)algebras, including the mixed kernel VOAs that were the missing ingredient for proving certain Feigin-Frenkel type dualities in previous work of the first-named author with Linshaw, Nakatsuka, and Sato.

Motivated by a recent article of R. Vanlaere (2026), we investigate the spectral asymptotics of a singular harmonic oscillator arising from the radial reduction of the magnetic Schrödinger operator on the unit disk in the presence of a constant magnetic field and an Aharonov-Bohm flux. The corresponding eigenvalue problem is equivalent to the study of the zeros of the Kummer confluent hypergeometric function with respect to its first parameter. Combining now standard semiclassical methods with the uniform asymptotic theory of Whittaker and confluent hypergeometric functions developed by Dunster (1989), Gabutti-Gatteschi (2001) and improved quite recently by Dunster (2026), we obtain a comprehensive description of the different spectral regimes. When the boundary lies in the classically forbidden region, we derive the exponentially small asymptotic correction to the eigenvalue caused by the tunneling effect. For energy levels above the tunneling regime, we derive a shifted Bohr-Sommerfeld asymptotics for highly excited states at a fixed magnetic field. Finally, for the transition regime where the energy level touches the boundary, we derive a three-term asymptotic expansion of the eigenvalue. In an appendix, we translate these spectral results into explicit asymptotic formulas for the zeros of the Kummer function.

We establish a sufficient condition under which overlapping PEPS boundary states satisfy the approximate factorization criterion for a spectral gap. The condition is formulated in terms of the boundary response induced by cutting bonds through the center of a rectangular region. We decompose this response into a contribution common to all four associated regions and a remainder measured in the instantaneous boundary metric. A single linear transport absorbs the common contribution and yields compatible factorization operators, with an error controlled solely by the remainder. We apply this framework to Ising PEPS and to the spin-3/2 AKLT model on the hexagonal lattice. For Ising PEPS, the required boundary-response estimate reduces to a Dobrushin-Shlosman-type condition. For the hexagonal AKLT model, a rooted expansion in paths and loops isolates the common response, while a local comparison of boundary states together with a scalar Kotecký-Preiss estimate controls the remaining terms. In both cases, the factorization error decays exponentially with the overlap width, up to a prefactor proportional to the cut length. For AKLT, we establish the corresponding physical projector estimate and obtain a uniform spectral gap. For Ising PEPS, the gap implication additionally requires compatible injective regional contractions, as specified in the paper. More generally, the method provides a systematic route from locality of PEPS boundary response to spectral gaps of two-dimensional parent Hamiltonians.