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Jeremy L. Marzuola

Publications and source records attributed to Jeremy L. Marzuola.

At least 19 recordsLinked to original sources
arXiv2026

Eigenfunction asymptotics and nodal domain estimates for dumbbell domains

We study the nodal structure of Neumann eigenfunctions on planar dumbbell domains as the width of the neck joining two fixed end domains tends to zero. Under simplicity and non-degeneracy assumptions, we show that sufficiently thin necks force an increasing number of nodal domains as the limiting eigenvalue grows. At the bottom of the spectrum this forcing can attain Courant's upper bound: in particular, the third Neumann eigenfunction has exactly three nodal domains and is Courant sharp. At the same time, nodal deficiency of an eigenfunction on an end domain can persist on the full dumbbell, giving eigenfunctions that are not Courant sharp. The main analytic difficulty is that the singular limit may vanish on one or both end domains even though these regions contribute nodal domains for every positive neck width. We overcome this by deriving refined asymptotics for the first nonzero profiles on the vanishing regions, expressed through Neumann Green's functions with poles at the neck attachment points and obtained by a two-dimensional matched asymptotic analysis with logarithmic terms.

math.AP↗
arXiv2026

On Analyticity of Solitons for the Periodic Dispersion-Managed Nonlinear Schrödinger Equation

We explore the regularity of energy maximizers for the Lagrangian of a periodic dispersion managed fiber optic at fixed intensity, on a torus of length $L$ and with vanishing average dispersion. We show that the Fourier coefficients decay at a polynomial rate, and then upgrade this to exponential decay, so that the maximizers are analytic in space for large enough $L$. In addition, by an asymptotic comparison to the optimizers on the real line, we prove that the solutions are non-trivial. We also consider a conjecture that the maximizer necessarily has an underlying symmetry inherent to both the energy functional and the resulting Euler-Lagrange equation. All of our results are supported with illustrative numerical experiments.

math.AP↗
arXiv2026

Recovering elastic subdomains with strain-gradient elastic interfaces from force measurements: the antiplane shear setting

We introduce and study a new inverse problem for antiplane shear in elastic bodies with strain-gradient interfaces. The setting is a homogeneous isotropic elastic body containing an inclusion separated by a thin interface endowed with higher-order surface energy. Using displacement-stress measurements on the exterior boundary, expressed through a certain Dirichlet-to-Neumann map, we show uniqueness in recovering both the shear and interface parameters, as well as the shape of the inclusion. To address the inverse shape problem, we adapt the factorization method to account for the complications introduced by the higher-order boundary operator and its nontrivial null space. The resulting characterization relies on pairs of sampling points rather than a single-point test used in classical factorization methods. After fixing an interior reference point, the reconstruction procedure reduces to a single-point sampling algorithm. Focusing on the latter stage, numerical experiments illustrate the feasibility of the proposed reconstruction method and suggest that the framework has potential for the nondestructive detection of interior inhomogeneities, including damaged subvolumes.

math.AP↗
arXiv2026

Probing the influence of topological and geometric disorder on the spectrum of the differential Laplacian operator on networks

Metric networks are network-shaped, one-dimensional structures on which one can solve differential equations to simulate a wide range of physical systems including conjugated molecules, photonic crystals, quantum mechanics in waveguide networks, and acoustic metamaterials. More concretely, a metric network is a network whose edges are each assigned a notion of length and a coordinate describing position. One can then define function spaces and differential operators on these objects to model the aforementioned systems. Recent software advancements have made it feasible to analyze partial differential equations on large, compact metric networks with a vast array of structures. Here, we generate compact metric network structures using the spatial tessellations of two-dimensional hyperuniform point patterns, which have suppressed large-scale density fluctuations relative to typical disordered point patterns. This choice of structure is inspired by the exotic physical properties of network materials with these structures in other contexts. Then, we characterize the eigenvalue spectrum structure of the differential Laplace operator on these networks. In particular, we find that gaps can form in the eigenvalue spectra of these networks whose widths increase when the distribution of edge lengths is narrow and as the number of triangular faces increases. Importantly, many of the structures we consider are realizable in Euclidean space, meaning they are well-suited for practical applications in, e.g., metamaterial design. This work can thus be used to inform the design of metric network-based systems with spectral gaps with tunable widths and locations.

cond-mat.dis-nn↗
arXiv2026

Transmission and Reflection coefficients for Schrödinger Operators with Truncated Periodic Potentials that support defect states

We consider scattering waves through truncated periodic potentials with perturbations that support localized gap eigenstates. In a small complex neighborhood around an assumed positive bound state of the model operator, we prove the existence of a distinct zero reflection state, or transmission resonance. We compare its location to a previously found scattering resonance and use the properties of solutions near these interesting points to analyze the behavior of transmission and reflection coefficients of scattering solutions near the assumed bound state. By example, we also discuss the truncated simple harmonic oscillator and compare the analysis to the crystalline case.

math.AP↗
arXiv2025

Cauchy Data for 1D singular Schrödinger operators

We study semiclassical 1-D Schrödinger operators of the form $Pu = -h^2 u'' \,+\,x^γW(x) u$ on a finite interval $[0,b]$ for $0 < γ\in \mathbb{R} \setminus \mathbb{Q}$. We show that that the WKB expansions of solution can be extended on $[h^{1-ε},b]$, for any $ε>0$. Using a different approximation near $0$ and a matching procedure, we obtain the Cauchy Data at $0$ of such WKB solutions. This allows us to derive singular Bohr-Sommerfeld rules. We also pay special attention to uniformity in $W$ for our expansions.

math-ph↗
arXiv2025

Discrete Bound States in a Toy Model for Weak Turbulence and Implications for the Invariant Measure

A model Hamiltonian dynamical system has been derived to study frequency cascades in the cubic defocusing nonlinear Schrödinger equation on the torus. Here, we explore the framework for exploring a canonical ensemble formulation of the dynamics through classification of energy minimizers for fixed mass and characterizing the invariant measure in a neighborhood of those minimizers.

math.CA↗
arXiv2025

On the Spectrum of Schrödinger Operators Interacting at Two Distinct Scales

Schrödinger operators of the form $Δ- W$ on $L^2_{\text{rad}}(\mathbb{R}^3)$, the space of radially symmetric square integrable functions are relevant in a variety of physical contexts. The potential $W$ is taken to be radially symmetric (i.e. $W(x) = W(|x|)$) and to decompose into two components with distinct spatial scales: $W=W_\varepsilon= V_0+V_{1,\varepsilon}$. The second component $V_{1,\varepsilon}(|x|) = \varepsilon^2V_1(\varepsilon |x|)$ represents a scaled potential that becomes increasingly delocalized as $\varepsilon \to 0$. We will assume that both potentials $V_0(r), V_1(r)$ exhibit certain decay properties as $r \to \infty$. We show how the eigenvalue count on the positive real axis is built out of the spectra associated with the two reduced eigenvalue problems on their separate scales. The result is that the total number of eigenvalues of $Δ- W$ is the sum of the number of positive eigenvalues of $Δ- V_0$ and $Δ- V_1$. Our analysis combines dynamical systems techniques with a separation of scales argument, providing a novel framework for studying spectral properties of differential operators where multiple spatial scales interact.

math-ph↗
arXiv2025

Edge states in square lattice media and their deformations

Edge states are time-harmonic solutions of conservative wave systems which are plane wave-like parallel to and localized transverse to an interface between two bulk media. We study a class of 2D edge Hamiltonians modeling a medium which slowly interpolates between periodic bulk media via a domain wall across a "rational" line defect. We consider the cases of (1) periodic bulk media having the symmetries of a square lattice, and (2) linear deformations of such media. Our bulk Hamiltonians break time-reversal symmetry due to perturbation by a magnetic term, which opens a band gap about the band structure degeneracies of the unperturbed bulk Hamiltonian. In case (1), these are quadratic band degeneracies; in case (2), they are pairs of conical degeneracies. We demonstrate that this band gap is traversed by two distinct edge state curves, consistent with the bulk-edge correspondence principle of topological physics. Blow-ups of these curves near the bulk band degeneracies are described by effective (homogenized) edge Hamiltonians derived via multiple-scale analysis which control the bifurcation of edge states. In case (1), the bifurcation is governed by a matrix Schrödinger operator; in case (2), it is governed by a pair of Dirac operators. We present analytical results and numerical simulations for both the full 2D edge Hamiltonian spectral problem and the spectra of effective edge Hamiltonians.

math-ph↗
arXiv2025

On non-local exchange and scattering operators in domain decomposition methods

We study non-local exchange and scattering operators arising in domain decomposition algorithms for solving elliptic problems on domains in $\mathbb{R}^2$. Motivated by recent formulations of the Optimized Schwarz Method introduced by Claeys, we rigorously analyze the behavior of a family of non-local exchange operators $Π_γ$, defined in terms of boundary integral operators associated to the fundamental solution for $-Δ+ γ^{-2}$, with $γ> 0$. Our first main result establishes precise estimates comparing $Π_γ$ to its local counterpart $Π_0$ as $γ\to 0$, providing a quantitative bridge between the classical and non-local formulations of the Optimized Schwarz Method. In addition, we investigate the corresponding scattering operators, proving norm estimates that relate them to their classical analogues through a detailed analysis of the associated Dirichlet-to-Neumann operators. Our results clarify the relationship between classical and non-local formulations of domain decomposition methods and yield new insights that are essential for the analysis of these algorithms, particularly in the presence of cross points and for domains with curvilinear polygonal boundaries.

math.NA↗
arXiv2024

QGLAB: A MATLAB Package for Computations on Quantum Graphs

We describe QGLAB, a new MATLAB package for analyzing partial differential equations on quantum graphs. The software is built on the existing, object-oriented MATLAB directed-graph class, inheriting its structure and adding additional easy-to-use features. The package allows one to construct a quantum graph and accurately compute the spectrum of elliptic operators, solutions to Poisson problems, the linear and nonlinear time evolution of a variety of PDEs, the continuation of branches of steady states (including locating and switching branches at bifurcations) and more. It overcomes the major challenge of discretizing quantum graphs -- the enforcement of vertex conditions -- using non-square differentiation matrices. It uses a unified framework to implement finite-difference and Chebyshev discretizations of differential operators on a quantum graph. For simplicity, the package overloads many built-in MATLAB functions to work on the class.

math.NA↗
arXiv2024

Stability of spectral partitions with corners

A spectral minimal partition of a manifold is a decomposition into disjoint open sets that minimizes a spectral energy functional. While it is known that bipartite minimal partitions correspond to nodal partitions of Courant-sharp Laplacian eigenfunctions, the non-bipartite case is much more challenging. In this paper, we unify the bipartite and non-bipartite settings by defining a modified Laplacian operator and proving that the nodal partitions of its eigenfunctions are exactly the critical points of the spectral energy functional. Moreover, we prove that the Morse index of a critical point equals the nodal deficiency of the corresponding eigenfunction. Some striking consequences of our main result are: 1) in the bipartite case, every local minimum of the energy functional is in fact a global minimum; 2) in the non-bipartite case, every local minimum of the energy functional minimizes within a certain topological class of partitions. Our results are valid for partitions with non-smooth boundaries; this introduces considerable technical challenges, which are overcome using delicate approximation arguments in the Sobolev space $H^{1/2}$.

math.AP↗
arXiv2024

Targeting influence in a harmonic opinion model

Influence propagation in social networks is a central problem in modern social network analysis, with important societal applications in politics and advertising. A large body of work has focused on cascading models, viral marketing, and finite-horizon diffusion. There is, however, a need for more developed, mathematically principled \emph{adversarial models}, in which multiple, opposed actors strategically select nodes whose influence will maximally sway the crowd to their point of view. In the present work, we develop and analyze such a model based on harmonic functions and linear diffusion. We prove that our general problem is NP-hard and that the objective function is monotone and submodular; consequently, we can greedily approximate the solution within a constant factor. Introducing and analyzing a convex relaxation, we show that the problem can be approximately solved using smooth optimization methods. We illustrate the effectiveness of our approach on a variety of example networks.

cs.SI↗
arXiv2024

Homology of spectral minimal partitions

A spectral minimal partition of a manifold is its decomposition into disjoint open sets that minimizes a spectral energy functional. It is known that bipartite spectral minimal partitions coincide with nodal partitions of Courant-sharp Laplacian eigenfunctions. However, almost all minimal partitions are non-bipartite. To study those, we define a modified Laplacian operator and prove that the nodal partitions of its Courant-sharp eigenfunctions are minimal within a certain topological class of partitions. This yields new results in the non-bipartite case and recovers the above known result in the bipartite case. Our approach is based on tools from algebraic topology, which we illustrate by a number of examples where the topological types of partitions are characterized by relative homology.

math.AP↗
arXiv2024

Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map

It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set, or, equivalently, a family of operators with delta function potentials supported on the nodal set. In this paper we explicitly describe this flow for a Schrödinger operator with separable potential on a rectangular domain, and determine a mechanism by which lower energy eigenfunctions do or do not contribute to the nodal deficiency.

math.AP↗
arXiv2023

Newton polygons and resonances of multiple delta-potentials

We prove explicit asymptotics for the location of semiclassical scattering resonances in the setting of $h$-dependent delta-function potentials on $\mathbb{R}$. In the cases of two or three delta poles, we are able to show that resonances occur along specific lines of the form $\Im z \sim -γh \log(1/h).$ More generally, we use the method of Newton polygons to show that resonances near the real axis may only occur along a finite collection of such lines, and we bound the possible number of values of the parameter $γ.$ We present numerical evidence of the existence of more and more possible values of $γ$ for larger numbers of delta poles.

math.AP↗
arXiv2023

Damping for fractional wave equations and applications to water waves

Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give the explicit decay rates for the energy, but do not address reflection/transmission of waves at the interface of the damping. Still for a subset of the models considered, this represents the first result proving the decay of the energy of the surface wave models.

math.AP↗