arXiv subjects

Gregory Berkolaiko

Publications and source records attributed to Gregory Berkolaiko.

At least 19 recordsLinked to original sources

arXiv2026

Two escape rates for negative eigenvalues of quantum graphs with a shrinking core

We study the negative spectrum of the Laplacian on a metric graph with general vertex matching conditions and with two length scales: a compact core whose edges have length of order a small parameter $ε$, together with finitely many edges of infinite length. As $ε\to0$, some negative eigenvalues may escape to $-\infty$, and we describe precisely how. There are exactly two rates of escape, $ε^{-1}$ and the fractional rate $ε^{-2/3}$. We determine the number of eigenvalues of each rate, together with their leading coefficients, explicitly from the vertex conditions. The analysis rests on the Dirichlet-to-Neumann map of the graph and on an implicit Rellich-type theorem, that identifies the power-law rates of the solution branches of a nonlinear 2-parameter matrix pencil with the leading orders of a one-parameter Hermitian family.

math.SP↗

arXiv2026

Cycle intersection form and oscillation of graph eigenfunctions

For a real symmetric matrix $H$ strictly supported on a finite simple graph, it is shown that the inertia of a weighted intersection form on the cycle space of the graph, with weights derived from a non-vanishing eigenvector of $H$, governs oscillation data on the graph. Specifically, the null space of the form controls eigenvalue multiplicity, while its Morse index determines the number of sign changes across edges. In the case of a simple eigenvalue, the Hessian at zero of the eigenvalue branch of the discrete magnetic Schrödinger operator is identified with the dual of the cycle intersection form. Applications are given to stability analysis of coupled oscillator networks, to the local behavior of dispersion relations for (decorated) strained graphene, and to nodal-domain counts.

math.SP↗

arXiv2026

The calculus of Duistermaat's triple index

In this paper we develop a systematic calculus for the Duistermaat index, a symplectic invariant defined for triples of Lagrangian subspaces. Introduced nearly half a century ago, this index has lately been the subject of renewed attention, due to its central role in eigenvalue interlacing problems on quantum graphs (and more abstractly for self-adjoint extensions of symmetric operators). Here we give an axiomatic characterization of the index that leads to elementary proofs of its fundamental properties. We also relate the index to other quantities often appearing in symplectic geometry, such as the Hörmander--Kashiwara--Wall index and the Maslov index. Among other things, this leads to a curious formula for the Morse index of a difference of Hermitian matrices.

math.SP↗

arXiv2026

Quantum graph models of quantum chaos: an introduction and some recent applications

Quantum graphs are a paradigmatic model for quantum chaos as well as for spectral theory. We give a concise didactical introduction to quantum graphs, or Schrödinger Hamiltonians on metric graphs, with a focus on results related to quantum chaos, periodic orbit theory and spectral theory. We summarise related seminal results, and give an overview over a few more recent developments.

quant-ph↗

arXiv2025

The Loring--Schulz-Baldes Spectral Localizer Revisited

The spectral localizer, introduced by Loring in 2015 and Loring and Schulz-Baldes in 2017, is a method to compute the (infinite volume) topological invariant of a quantum Hamiltonian on $\ZZ^d$, as the signature of the (finite) localizer matrix. We present a direct and elementary spectral-theoretic proof treating the $d=1$ and $d=2$ cases on an almost equal footing. Moreover, we re-interpret the localizer as a higher-dimensional topological insulator via the bulk-edge correspondence.

math-ph↗

arXiv2025

The Duistermaat index and eigenvalue interlacing for self-adjoint extensions of a symmetric operator

Eigenvalue interlacing is a useful tool in linear algebra and spectral analysis. In its simplest form, the interlacing inequality states that a rank-one positive perturbation shifts each eigenvalue up, but not further than the next unperturbed eigenvalue. For different types of perturbations, this idea is known as Weyl interlacing, Cauchy interlacing, Dirichlet--Neumann bracketing and so on. We prove a sharp version of the interlacing inequalities for ``finite-dimensional perturbations in boundary conditions'', expressed as bounds on the spectral shift between two self-adjoint extensions of a fixed symmetric operator with finite and equal defect numbers. The bounds are given in terms of the Duistermaat index, a topological invariant describing the relative position of three Lagrangian planes in a symplectic space. Two of the Lagrangian planes describe the self-adjoint extensions being compared, while the third corresponds to the Friedrichs extension, which acts as a reference point. Along the way several auxiliary results are established, including one-sided continuity properties of the Duistermaat triple index, smoothness of the (abstract) Cauchy data space without unique continuation-type assumptions, and a formula for the Morse index of an extension of a non-negative symmetric operator.

math.SP↗

arXiv2025

Eigenvalues of the discrete p-Laplacian via graph surgery

We develop a Hellmann--Feynman type perturbation theory for the discrete signed $p$-Laplacian and apply it to a parametrized perturbation by edge cuts. We show that the eigenvalues of the signed $p$-Laplacian can be characterized as citical values of the parameter-dependent eigenvalues of a simpler graph.

math.AP↗

arXiv2025

Smooth critical points of eigenvalues on the torus of magnetic perturbations of graphs

Motivated by the nodal distribution universality conjecture for discrete operators on graphs and by the spectral analysis of their maximal abelian covers, we consider a family of Hermitian matrices $h_α$ obtained by varying the complex phases of individual matrix elements. This family is parametrized by a $β$-dimensional torus, where $β$ is the first Betti number of the underlying graph. The eigenvalues of each matrix are ordered, enabling us to treat the $k$-th eigenvalue $λ_k$ as a function on the torus. We classify the smooth critical points of $λ_k$, describe their structure and Morse index in terms of the support and nodal count, that is, the number of sign changes between adjacent vertices of the corresponding eigenvector. In general, the families under consideration exhibit critical submanifolds rather than isolated critical points. These critical manifolds appear frequently and cannot be removed through perturbations. We provide an algorithmic way of determining all critical submanifolds by investigating finitely many eigenvalue problems: the $2^β$ real symmetric matrices $h_α$ in the family under consideration as well as their principal minors.

math-ph↗

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

Morse inequalities for ordered eigenvalues of generic self-adjoint families

In many applied problems one seeks to identify and count the critical points of a particular eigenvalue of a smooth parametric family of self-adjoint matrices, with the parameter space often being known and simple, such as a torus. Among particular settings where such a question arises are the Floquet--Bloch decomposition of periodic Schrödinger operators, topology of potential energy surfaces in quantum chemistry, spectral optimization problems such as minimal spectral partitions of manifolds, as well as nodal statistics of graph eigenfunctions. In contrast to the classical Morse theory dealing with smooth functions, the eigenvalues of families of self-adjoint matrices are not smooth at the points corresponding to repeated eigenvalues (called, depending on the application and on the dimension of the parameter space, the diabolical/Dirac/Weyl points or the conical intersections). This work develops a procedure for associating a Morse polynomial to a point of eigenvalue multiplicity; it utilizes the assumptions of smoothness and self-adjointness of the family to provide concrete answers. In particular, we define the notions of non-degenerate topologically critical point and generalized Morse family, establish that generalized Morse families are generic in an appropriate sense, establish a differential first-order conditions for criticality, as well as compute the local contribution of a topologically critical point to the Morse polynomial. Remarkably, the non-smooth contribution to the Morse polynomial turns out to depend only on the size of the eigenvalue multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family; it is expressed in terms of the homologies of Grassmannians.

math.SP↗

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

Quotients of graph operators by symmetry representations

A finite dimensional operator that commutes with some symmetry group admits quotient operators, which are determined by the choice of associated representation. Taking the quotient isolates the part of the spectrum supporting the chosen representation and reduces the complexity of the spectral problem. Yet, such a quotient operator is not uniquely defined. Here we present a computationally simple way of choosing a special basis for the space of intertwiners, allowing us to construct a quotient that reflects the structure of the original operator. This quotient construction generalizes previous definitions for discrete graphs, which either dealt with restricted group actions or only with the trivial representation. We also extend the method to quantum graphs, which simplifies previous constructions within this context, answers an open question regarding self-adjointness and offers alternative viewpoints in terms of a scattering approach. Applications to isospectrality are discussed, together with numerous examples and comparisons with previous results.

math-ph↗

arXiv2023

Exotic eigenvalues and analytic resolvent for a graph with a shrinking edge

We consider a metric graph consisting of two edges, one of which has length $\varepsilon$ which we send to zero. On this graph we study the resolvent and spectrum of the Laplacian subject to a general vertex condition at the connecting vertex. Despite the singular nature of the perturbation (by a short edge), we find that the resolvent depends analytically on the parameter $\varepsilon$. In contrast, the negative eigenvalues escape to minus infinity at rates that could be fractional, namely, $\varepsilon^0$, $\varepsilon^{-2/3}$ or $\varepsilon^{-1}$. These rates take place when the corresponding eigenfunction localizes, respectively, only on the long edge, on both edges, or only on the short edge.

math.SP↗

arXiv2023

Exotic eigenvalues of shrinking metric graphs

Eigenvalue spectrum of the Laplacian on a metric graph with arbitrary but fixed vertex conditions is investigated in the limit as the lengths of all edges decrease to zero at the same rate. It is proved that there are exactly four possible types of eigenvalue asymptotics. The number of eigenvalues of each type is expressed via the index and nullity of a form defined in terms of the vertex conditions.

math.SP↗

arXiv2023

Impediments to diffusion in quantum graphs: geometry-based upper bounds on the spectral gap

We derive several upper bounds on the spectral gap of the Laplacian with standard or Dirichlet vertex conditions on compact metric graphs. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total length of the graph, as well as further metric quantities introduced here for the first time, such as the avoidance diameter. Using known results about Ramanujan graphs, a class of expander graphs, we also prove that some of these metric quantities, or combinations thereof, do not to deliver any spectral bounds with the correct scaling.

math.SP↗

arXiv2022

Stability of spectral partitions and the Dirichlet-to-Neumann map

The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to compute, or even estimate. Here we compare two recently obtained formulas for the nodal deficiency, one in terms of an energy functional on the space of equipartitions of the manifold, and the other in terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. We relate these two approaches by giving an explicit formula for the Hessian of the equipartition energy in terms of the Dirichlet-to-Neumann map. This allows us to compute Hessian eigenfunctions, and hence directions of steepest descent, for the equipartition energy in terms of the corresponding Dirichlet-to-Neumann eigenfunctions. Our results do not assume bipartiteness, and hence are relevant to the study of spectral minimal partitions.

math.AP↗

arXiv2022

Universality of nodal count distribution in large metric graphs

An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph's non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph's first Betti number $β$. We study the distribution of the nodal surplus values in the countably infinite set of the graph's eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing $β$. We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus.

math-ph↗