Multiple Eigenvalues Research Articles

First‐order asymptotic perturbation theory for extensions of symmetric operators

AbstractThis work offers a new prospective on asymptotic perturbation theory for varying self‐adjoint extensions of symmetric operators. Employing symplectic formulation of self‐adjointness, we use a version of resolvent difference identity for two arbitrary self‐adjoint extensions that facilitates asymptotic analysis of resolvent operators via first‐order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati‐type differential equation and the first‐order asymptotic expansion for resolvents of self‐adjoint extensions determined by smooth one‐parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard–Rellich‐type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self‐adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig–Penney model, elliptic second‐order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity.

Quantum Multiple Eigenvalue Gaussian filtered Search: an efficient and versatile quantum phase estimation method

Quantum phase estimation is one of the most powerful quantum primitives. This work proposes a new approach for the problem of multiple eigenvalue estimation: Quantum Multiple Eigenvalue Gaussian filtered Search (QMEGS). QMEGS leverages the Hadamard test circuit structure and only requires simple classical postprocessing. QMEGS is the first algorithm to simultaneously satisfy the following two properties: (1) It can achieve the Heisenberg-limited scaling without relying on any spectral gap assumption. (2) With a positive energy gap and additional assumptions on the initial state, QMEGS can estimate all dominant eigenvalues to ϵ accuracy utilizing a significantly reduced circuit depth compared to the standard quantum phase estimation algorithm. In the most favorable scenario, the maximal runtime can be reduced to as low as log⁡(1/ϵ). This implies that QMEGS serves as an efficient and versatile approach, achieving the best-known results for both gapped and gapless systems. Numerical results validate the efficiency of our proposed algorithm in various regimes.

Some new results on the property (R) under compact perturbations

The property (R) for a bounded linear operator T on a Hilbert space means that the points λ in the approximate point spectrum of T for which T − λI is upper semi-Browder are exactly the isolated points of the spectrum of T which are eigenvalues of finite multiplicity. For an operator T, if T + K satisfies the property (R) for all compact operators K, then T is said to have the stability of the property (R). In this paper, we give new criteria for the fulfillment of the stability of the property (R) for an operator. Moreover, the property (R) for complex symmetric operators is discussed.

Backward Stability of the Schur Decomposition under Small Perturbations

In the present paper, we show the backward stability of the Schur decomposition for a given matrix under small perturbations. We also present the forward stability problem for matrices with multiple eigenvalues and show conditions under which the forward stability fails.

The Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian

The Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian

Spectrum and resolvent of multi-channel systems with internal energies and common boundary conditions

In the article the spectrum and resolvent of the so-called multichannel systems with nonzero internal energies were investigated. The spectrum and resolvent of multichannel Sturm-Liouville systems with non-zero internal energies mi2 and general boundary conditions were investigated. These systems describe the propagation of partial waves in the theory of quantum physics. The importance of studying the spectral characteristics of these systems is presented in the well-known books of the theory of quantum physics. The finiteness of the number of eigenvalues was proved, the multiplicity of positive eigenvalues was investigated, and as well as the resolvent kernel of the system was found.

On separability of semigroups: application to models of physics and discrete optimization

Nilpotent semigroups are used in various fields of physics to model processes like system decay in quantum mechanics, state transitions in statistical mechanics, and simplifying dynamical systems. They also assist in optimizing control processes. The set of nilpotent matrices forms a semigroup and plays a key role in Jordan decomposition, which has many physical applications. In quantum mechanics, the Jordan form helps analyze linear operators on Hilbert spaces, especially with systems involving eigenvalue multiplicities. In vibration analysis, it aids in studying normal modes of mechanical systems with degenerate frequencies. Additionally, Jordan decomposition simplifies control theory for linear time-invariant systems, stability analysis in dynamical systems, and the behavior of coupled oscillators, as well as solving systems of linear differential equations. Certainly, the semigroup theory has even more applications in theoretical computer science: suffice it to say, for example, that some authors identify semigroups and finite automata. In this paper, we address the problem of P-separability of semigroups with respect to three key predicates: equality-separability, divisibility-separability, and subsemigroup-separability, achieved through homomorphisms into a nilpotent semigroup. We present some significant results that advance the understanding of these concepts.

On spectral simplicity of the Hodge Laplacian and curl operator along paths of metrics

We prove that the curl operator on closed oriented 3 3 -manifolds, i.e., the square root of the Hodge Laplacian on its coexact spectrum, generically has 1 1 -dimensional eigenspaces, even along 1 1 -parameter families of C k \mathcal {C}^k Riemannian metrics, where k ≥ 2 k\geq 2 . We show further that the Hodge Laplacian in dimension 3 3 has two possible sources for nonsimple eigenspaces along generic 1 1 -parameter families of Riemannian metrics: either eigenvalues coming from positive and from negative eigenvalues of the curl operator cross, or an exact and a coexact eigenvalue cross. We provide examples for both of these phenomena. In order to prove our results, we generalize a method of Teytel [Comm. Pure Appl. Math. 52 (1999), pp. 917–934], allowing us to compute the meagre codimension of the set of Riemannian metrics for which the curl operator and the Hodge Laplacian have certain eigenvalue multiplicities. A consequence of our results is that while the simplicity of the spectrum of the Hodge Laplacian in dimension 3 3 is a meagre codimension 1 1 property with respect to the C k \mathcal {C}^k topology as proven by Enciso and Peralta-Salas in [Trans. Amer. Math. Soc. 364 (2012), pp. 4207–4224], it is not a meagre codimension 2 2 property.

Threshold approximations for the exponential of a factorized operator family with correctors taken into account

In a Hilbert space H \mathfrak H , consider a family of selfadjoint operators (a quadratic operator pencil) A ( t ) A(t) , t ∈ R t\in \mathbb {R} , of the form A ( t ) = X ( t ) ∗ X ( t ) A(t) = X(t)^* X(t) , where X ( t ) = X 0 + t X 1 X(t) = X_0 + t X_1 . It is assumed that the point λ 0 = 0 \lambda _0=0 is an isolated eigenvalue of finite multiplicity for the operator A ( 0 ) A(0) . Let F ( t ) F(t) be the spectral projection of the operator A ( t ) A(t) for the interval [ 0 , δ ] [0,\delta ] . Approximations for F ( t ) F(t) and A ( t ) F ( t ) A(t)F(t) for | t | ≤ t 0 |t| \leq t_0 (the so-called threshold approximations) are used to obtain approximations in the operator norm on H \mathfrak H for the operator exponential exp ⁡ ( − i τ A ( t ) ) \exp (-i \tau A(t)) , τ ∈ R \tau \in \mathbb {R} . The numbers δ \delta and t 0 t_0 are controlled explicitly. Next, the behavior for small ε > 0 \varepsilon >0 of the operator exp ⁡ ( − i ε − 2 τ A ( t ) ) \exp (-i \varepsilon ^{-2} \tau A(t)) multiplied by the “smoothing factor” ε s ( t 2 + ε 2 ) − s / 2 \varepsilon ^s (t^2 + \varepsilon ^2)^{-s/2} with a suitable s > 0 s>0 is studied. The obtained approximations are given in terms of the spectral characteristics of the operator A ( t ) A(t) near the lower edge of the spectrum. The results are aimed at application to homogenization of the Schrödinger-type equations with periodic rapidly oscillating coefficients.

Linear operator theory of phase mixing

ABSTRACT We study solutions of the collisionless Boltzmann equation (CBE) in a functional Koopman representation. This facilitates the use of linear spectral techniques characteristic of the analysis of Schrödinger-type equations. For illustrative purposes, we consider the classical phase mixing of a non-interacting distribution function in a quartic potential. Solutions are determined perturbatively relative to a harmonic oscillator. We impose a form of coarse-graining by choosing a finite-dimensional basis to represent the distribution function and time evolution operators, which sets a minimum length-scale on phase space structure. We observe a relationship between the dimension of the representation and the multiplicity of the harmonic oscillator eigenvalues. System dynamics are understood in terms of degenerate subspaces of the linear operator spectra. Each subspace is associated with a mode of the harmonic oscillator, the first two being bending and breathing structures. The quartic potential splits the degenerate eigenvalues within each subspace. This facilitates the formation of spiral structure as deformations from the harmonic oscillator modes. We ultimately argue that this construction provides a promising avenue for study of self-interacting systems experiencing phase mixing, which is an outstanding problem in the context of the Gaia DR2 vertical phase space spirals.

An inverse eigenvalue problem for structured matrices determined by graph pairs

Given a pair of real symmetric matrices A,B∈Rn×n with nonzero patterns determined by the edges of any pair of chosen graphs on n vertices, we consider an inverse eigenvalue problem for the structured matrix C=[ABIO]∈R2n×2n. We conjecture that C can attain any spectrum that is closed under conjugation. We use a structured Jacobian method to prove this conjecture for A and B of orders at most 4 or when the graph of A has a Hamilton path, and prove a weaker version of this conjecture for any pair of graphs with a restriction on the multiplicities of eigenvalues of C.

Minimal graphs with eigenvalue multiplicity of n − d

For a connected graph G with order n, let e(G) be the number of its distinct eigenvalues and d be the diameter. We denote by mG(μ) the eigenvalue multiplicity of μ in G. It is well known that e(G)≥d+1, which shows mG(μ)≤n−d for any real number μ. A graph is called minimal if e(G)=d+1. In 2013, Wong et al. characterize all minimal graphs with mG(0)=n−d. In this paper, by applying the star complement theory, we prove that if G is not a path and mG(μ)=n−d, then μ∈{0,−1}. Furthermore, we completely characterize all minimal graphs with mG(−1)=n−d.

Derivation of Bose's Entropy Spectral Density from the Multiplicity of Energy Eigenvalues.

The modern textbook analysis of the thermal state of photons inside a three-dimensional reflective cavity is based on the three quantum numbers that characterize photon's energy eigenvalues coming out when the boundary conditions are imposed. The crucial passage from the quantum numbers to the continuous frequency is operated by introducing a three-dimensional continuous version of the three discrete quantum numbers, which leads to the energy spectral density and to the entropy spectral density. This standard analysis obscures the role of the multiplicity of energy eigenvalues associated to the same eigenfrequency. In this paper we review the past derivations of Bose's entropy spectral density and present a new analysis of energy spectral density and entropy spectral density based on the multiplicity of energy eigenvalues. Our analysis explicitly defines the eigenfrequency distribution of energy and entropy and uses it as a starting point for the passage from the discrete eigenfrequencies to the continuous frequency.

Seidel matrices, Dilworth number and an eigenvalue-free interval for cographs

A graph G=(VG,EG) is said to be a cograph if the path P4 does not appear among its induced subgraphs. The vicinal preorder ≺ on the vertex set VG is defined in terms of inclusions between neighborhoods. The minimum number ∇(G) of ≺-chains required to cover G is called the Dilworth number of G. In this paper it is proved that for a cograph G, the multiplicity of every Seidel eigenvalue λ≠±1 does not exceed ∇(G). This bound turns out to be tight and can be further improved for threshold graphs. Moreover, it is shown that cographs with at least two vertices have no Seidel eigenvalues in the interval (−1,1).

Upper Bounds for the Eigenvalue Multiplicities of a Fourth-Order Differential Operator on a Graph

Upper Bounds for the Eigenvalue Multiplicities of a Fourth-Order Differential Operator on a Graph

Index of embedded networks in the sphere

In this paper, we will compute the Morse index and nullity of some embedded stationary geodesic networks in the sphere. The key theorem in the computation is that the index (and nullity) for the whole network is related to the index (and nullity) of small networks and the Dirichlet-to-Neumann map defined in this paper. Finally, we will show that for all stationary triple junction networks in S2, there is only one eigenvalue (without multiplicity) −1, which is less than 0, and the corresponding eigenfunctions are locally constant. Besides, the multiplicity of eigenvalues 0 is 3 for these networks, and their eigenfunctions are generated by the rotations on the sphere.

A note on the persistence of multiplicity of eigenvalues of fractional Laplacian under perturbations

We consider the eigenvalue problem for the fractional Laplacian (−Δ)s, s∈(0,1), in a bounded domain Ω with Dirichlet boundary condition. A recent result (see Fall et al., 2023) states that, under generic small perturbations of the coefficient of the equation or of the domain Ω, all the eigenvalues are simple. In this paper we give a condition for which a perturbation of the coefficient or of the domain preserves the multiplicity of a given eigenvalue. Also, in the case of an eigenvalue of multiplicity ν=2 we prove that the set of perturbations of the coefficients which preserve the multiplicity is a smooth manifold of codimension 2 in C1(Rn).

Adjusting exceptional points using saturable nonlinearities

We study the impact of saturable nonlinearity on the presence and location of exceptional points in a non-Hermitian dimer system. The inclusion of the saturable nonlinearity leads to the emergence of multiple eigenvalues, exceeding the typical two found in the linear counterpart. To identify the exceptional points, we calculate the nonlinear eigenvalues both from a polynomial equation for the defined population imbalance and through a fully numerical method. Our results reveal that exceptional points can be precisely located by adjusting the non-equal saturable nonlinearities in the detuning space.

Graphs with large multiplicity of −2 in the spectrum of the eccentricity matrix

The eccentricity matrix of a simple connected graph is obtained from the distance matrix by keeping only the entries that are largest in at least one of their row or column. This matrix can be seen as a counterpart to the standard adjacency matrix, since the latter one is also obtained from the distance matrix but this time by keeping only the entries equal to 1. It is known that, for λ∉{−1,0} and a fixed i∈N, when n passes N there is only a finite number of graphs with n vertices having λ as an eigenvalue of multiplicity n−i in the spectrum of the adjacency matrix. This phenomenon motivates us to consider graphs with large multiplicity of an eigenvalue of the eccentricity matrix. In this context, we determine all connected graphs with n vertices for which −2 has the multiplicity n−i, where i≤5. Infinite families of graphs with this spectral property are encountered. Results of this paper can be compared to results concerning graphs with large multiplicity of zero in the spectrum of the adjacency matrix; at present the graphs for which this multiplicity is n−i, where i≤4, are known. Our results also become meaningful in the framework of the median eigenvalue problem since, for sufficiently large n, the median eigenvalue of the obtained graphs is always −2.

Spectral stability of periodic traveling wave solutions for a double dispersion equation

ABSTRACT In this article, we investigate the spectral stability of periodic traveling waves for a cubic-quintic and double dispersion equation. Using the quadrature method, we find explict periodic waves and we also present a characterization for all positive and periodic solutions for the model using the monotonicity of the period map in terms of the energy levels. The monotonicity of the period map is also useful to obtain the quantity and multiplicity of non-positive eigenvalues for the associated linearized operator and to do so, we use tools of the Floquet theory. Finally, we prove the spectral stability by analyzing the difference between the number of negative eigenvalues of a convenient linear operator restricted to the space constituted by zero-mean periodic functions and the number of negative eigenvalues of the matrix formed by the tangent space associated to the low order conserved quantities of the evolution model.