Multiple Eigenvalues Research Articles

Heisenberg-limited quantum phase estimation of multiple eigenvalues with few control qubits

Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentially-large sparse matrices.The maximum rate at which these eigenvalues may be learned, –known as the Heisenberg limit–, is constrained by bounds on the circuit complexity required to simulate an arbitrary Hamiltonian. Single-control qubit variants of quantum phase estimation that do not require coherence between experiments have garnered interest in recent years due to lower circuit depth and minimal qubit overhead. In this work we show that these methods can achieve the Heisenberg limit, also when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a `phase function' g(k)=∑jAjeiϕjk with unknown eigenphases ϕj and overlaps Aj at quantum cost O(k), we show how to estimate the phases {ϕj} with (root-mean-square) error δ for total quantum cost T=O(δ−1). Our scheme combines the idea of Heisenberg-limited multi-order quantum phase estimation for a single eigenvalue phase [Higgins et al (2009) and Kimmel et al (2015)] with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series analysis for the QEEP problem [Somma (2019)] or the matrix pencil method. For our algorithm which adaptively fixes the choice for k in g(k) we prove Heisenberg-limited scaling when we use the time-series/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well.

Approximating Matrix Eigenvalues by Subspace Iteration with Repeated Random Sparsification

Traditional numerical methods for calculating matrix eigenvalues are prohibitively expensive for high-dimensional problems. Iterative random sparsification methods allow for the estimation of a single dominant eigenvalue at reduced cost by leveraging repeated random sampling and averaging. We present a general approach to extending such methods for the estimation of multiple eigenvalues and demonstrate its performance for several benchmark problems in quantum chemistry.

The Second Variation for Null-Torsion Holomorphic Curves in the 6-Sphere

In the round 6-sphere, null-torsion holomorphic curves are fundamental examples of minimal surfaces. This class of minimal surfaces is quite rich: By a theorem of Bryant, extended by Rowland, every closed Riemann surface may be conformally embedded in the round 6-sphere as a null-torsion holomorphic curve. In this work, we study the second variation of area for compact null-torsion holomorphic curves \(\Sigma \) of genus g and area \(4\pi d\), focusing on the spectrum of the Jacobi operator. We show that if \(g \le 6\), then the multiplicity of the lowest eigenvalue \(\lambda _1 = -2\) is equal to 4d. Moreover, for any genus, we show that the nullity is at least \(2d + 2 - 2g\). These results are likely to have implications for the deformation theory of asymptotically conical associative 3-folds in \({\mathbb {R}}^7\), as studied by Lotay.

Ramification of multiple eigenvalues for the Dirichlet-Laplacian in perforated domains

Taking advantage from the so-called Lemma on small eigenvalues by Colin de Verdière, we study ramification for multiple eigenvalues of the Dirichlet Laplacian in bounded perforated domains. The asymptotic behavior of multiple eigenvalues turns out to depend on the asymptotic expansion of suitable associated eigenfunctions.We treat the case of planar domains in details, thanks to the asymptotic expansion of a generalization of the so-called u-capacity which we compute in dimension 2. In this case multiple eigenvalues are proved to split essentially by different rates of convergence of the perturbed eigenvalues or by different coefficients in front of their expansion if the rate of two eigenbranches turns out to be the same.

On the Rayleigh–Taylor instability for the two-phase Navier–Stokes equations in cylindrical domains

We are considering the two-phase Navier–Stokes equations with surface tension, modelling the dynamic behaviour of two immiscible and incompressible fluids in a cylindrical domain, which are separated by a sharp interface forming a contact angle with the fixed boundary. In the case that the heavy fluid is situated on top of the light fluid, one expects the effect which is known as Rayleigh–Taylor instability . Our main result implies the existence of a critical surface tension with the following property: In the case that the surface tension of the interface separating the two fluids is smaller than the critical surface tension, Rayleigh–Taylor instability occurs. On the contrary, if the surface tension of the interface is larger than the critical value, one has exponential stability of the flat interfaces. The last part of this article is concerned with the bifurcation of nontrivial equilibria in multiple eigenvalues. The invariance of the corresponding bifurcation equation with respect to rotations and reflections yields the existence of bifurcating subcritical equilibria.

A uniform bound on the multiplicity of Dirichlet eigenvalues in planar annuli

We study the multiplicity of Dirichlet eigenvalues in the planar annulus Ωr={x∈R2:r<|x|<1} when r is an algebraic number. Using the algebraic independence results of the Bessel functions developed by Siegel and Shildlovskii, we prove that all Dirichlet eigenvalues here have multiplicity at most 6.

Adaptive trust-region-based algorithm for the discrete eigenvalue evaluation of the direct nonlinear Fourier transform

We propose an adaptive method for numerical computation of discrete eigenvalues of the direct nonlinear Fourier transform (NFT). The approach is based on trust region algorithm and modified objective function, which can alleviate the problem of sensitivity to initial values of the Newton–Raphson method, and enhance the robustness as well as reduce the computational complexity. The reliability and performance of the novel, to the best of our knowledge, approach have been demonstrated to a single eigenvalue and multiple eigenvalues of the NFT. Meanwhile, the proposed method can be used not only to solve the complex pulses with a large number of discrete eigenvalues, but also to solve those with eigenvalues having extremely small difference. The results show that the proposed approach represents a significant improvement in comparison with previous reports in the computation accuracy and complexity.

Direct and inverse problems for nonlocal heat equation with boundary conditions of periodic type

A mathematical model of the process of heat diffusion in a closed metal wire is considered. This wire is wrapped around a thin sheet of insulation material. We assume that the insulation is slightly permeable. Because of this, the temperature at the point of the wire on one side of the insulation influences the diffusion process in the wire on the other side of the insulation. Thus, the standard heat equation will change and an extra term with involution will be added. When modeling of this process there arises an initial-boundary value problem for a one-dimensional heat equation with involution and with a boundary condition of periodic type with respect to a spatial variable. We prove the well-posedness of the formulated problem in the class of strong generalized solutions. The use of the method of separation of variables leads to a spectral problem for an ordinary differential operator with involution at the highest derivative. All eigenfunctions of the problem are constructed. In the case when all eigenvalues of the problem are simple, the system of eigenfunctions does not form an unconditional basis. A criterion when this spectral problem can have an infinite number of multiple eigenvalues is proved. Corresponding root subspaces consist of one eigenfunction and one associated function. We prove that the system of root functions forms an unconditional basis and can be used for constructing a solution of the heat conduction problem by the method of separation of variables. We also consider an inverse problem. This is the problem on restoring (simultaneously with solving) of an unknown stationary source of external influence with respect to an additionally known final state. The existence of a unique solution of this inverse problem and its stability with respect to initial and final data are proved.

A note on the multiplicities of the eigenvalues of a tree

Let T be a tree of order n with μ as an eigenvalue of multiplicity mT(μ). Wong et al. showed that mT(μ)≤n−43 when n>6 and μ2 is an integer at least 2. In this paper, we generalize this result by first showing that if T is a tree of order n and t is a positve integer, then (i) if n≤2t+2, then mT(t)≤1; (ii) If n≥2t+3, then mT(t)≤n−(t+2)t+1, with equality if and only if T has a vertex v such that T−v=kK1,t for some integer k≥2.

On the multiplicity of positive eigenvalues of a graph

Let T be a tree on n(≥7) vertices with λ as a positive eigenvalue of multiplicity k. If λ2≥2 is an integer, then we prove that k≤⌊n−43⌋ and all extremal graphs attaining the upper bound are characterized. This result revises and improves the main conclusion of Wong, Zhou and Tian (2020). Moreover, applying this result we investigate the eigenvalue multiplicity of unicyclic graphs. Let G be a unicyclic graph of order n(≥11), which contains λ (λ2≥2 is an integer) as a positive eigenvalue of multiplicity m. Then it is proved that m≤⌊n−23⌋, and all extremal graphs attaining the upper bound are determined. These two upper bounds improve the conclusions of Rowlinson (2010, 2011), respectively.

Fully computable a posteriori error bounds for eigenfunctions

For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple eigenvalues, under the settings of target eigenvalue problems. Algorithm I is based on the Rayleigh quotient and the min-max principle that characterizes the eigenvalue problems. The formula for the error estimate provided by Algorithm I is easy to compute and applies to problems with limited information of Rayleigh quotients. Algorithm II, as an extension of the Davis–Kahan method, takes advantage of the dual formulation of differential operators along with the Prager–Synge technique and provides greatly improved accuracy of the estimate, especially for the finite element approximations of eigenfunctions. Numerical examples of eigenvalue problems of matrices and the Laplace operators over convex and non-convex domains illustrate the efficiency of the proposed algorithms.

Linear Instability of Breathers for the Focusing Nonlinear Schrödinger Equation

Relying upon tools from the theory of integrable systems, we discuss the linear instability of the Kuznetsov–Ma breathers and the Akhmediev breathers of the focusing nonlinear Schrödinger equation. We use the Darboux transformation to construct simultaneously the breathers and the exact solutions of the Lax system associated with the breathers. We obtain a full description of the Lax spectra for the two breathers, including multiplicities of eigenvalues. Solutions of the linearized NLS equations are then obtained from the eigenfunctions and generalized eigenfunctions of the Lax system. While we do not attempt to prove completeness of eigenfunctions, we aim to determine the entire set of solutions of the linearized NLS equations generated by the Lax system in appropriate function spaces.

Sampling expansions associated with quaternion difference equations

Starting with a quaternion difference equation with boundary conditions, a parameterized sequence that is complete in the finite-dimensional quaternion Hilbert space is derived. By employing the parameterized sequence as the kernel of the discrete transform, we construct a quaternion function space whose elements have sampling expansions. It is shown that the sample points of the sampling expansions are the eigenvalues of an associated quaternion difference operator. For the operator that has multiple standard eigenvalues, we provide a method to find the ‘missing sample points’. Through formulating the boundary-value problems, we make a connection between a class of tridiagonal quaternion matrices and the polynomials with quaternion coefficients. We prove that for a tridiagonal symmetric quaternion matrix, one can always associate a quaternion characteristic polynomial whose roots are the eigenvalues of the matrix. By representing colour image pixels as quaternions, an application of the proposed sampling theorem in colour image encryption is discussed. Throughout the paper, examples are given to illustrate the results.

A few results on permittivity variations in electromagnetic cavities

We study the eigenvalues of time-harmonic Maxwell's equations in a cavity upon changes in the electric permittivity ε of the medium. We prove that all the eigenvalues, both simple and multiple, are locally Lipschitz continuous with respect to ε. Next, we show that simple eigenvalues and the symmetric functions of multiple eigenvalues depend real analytically upon ε and we provide an explicit formula for their derivative in ε. As an application of these results, we show that for a generic permittivity all the Maxwell eigenvalues are simple.

Reductional Method in Perturbation Theory of Generalized Spectral E. Schmidt Problem

In this a paper perturbations of multiple eigenvalues of Schmidt spectral problems is considered. At the usage of the reductional method suggested in the articles [11, 12] the investigation of the multiple Schmidt perturbation eigenvalues is reduced to the investigation of perturbation of simple ones. At the end, as application of the obtained results the problem about the boundary perturbation for the system of two Sturm–Liouville problems with Schmidt spectral parameter is considered.

Full and Partial Eigenvalue Placement for Minimum Norm Static Output Feedback Control

The controller design for linear time-invariant state space systems seems to be straightforward and well established. This is not true for static output feedback control, which is still a challenging task. This paper deals with controller design based on eigenvalue assignment. We consider the placement of distinct as well as multiple real eigenvalues or complex conjugate pairs. The desired eigenvalue configurations are characterised in terms of algebraic divisibility of the characteristic polynomial of the closed-loop system. We also consider the problem of partial eigenvalue placement, where not all eigenvalues are fixed by feedback. Degrees of freedom in the controller design are used for the minimization of various matrix norms of the feedback gain matrix.

Convergence and optimality of adaptive multigrid method for multiple eigenvalue problems

A new type of adaptive multigrid method is presented for multiple eigenvalue problems based on multilevel correction scheme and adaptive multigrid method. Different from the classical adaptive finite element method which requires to solve eigenvalue problems on the adaptively refined triangulations, with our approach we just need to solve several linear boundary value problems in the current refined space and an eigenvalue problem in a very low dimensional space. Further, the involved boundary value problems are solved by an adaptive multigrid iteration. Since there is no eigenvalue problem to be solved on the refined triangulations, which is quite time-consuming, the proposed method can achieve the same efficiency as that of the adaptive multigrid method for the associated linear boundary value problems. Besides, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically.

S1 attractor bifurcation analysis for a three-species cooperating model

In this paper, the dynamic bifurcation of the three-species cooperating model is considered. It worth noting that the main theory of this paper is the Center manifold reduction and attractor bifurcation theory, which is developed by Ma [1,2]. The main work of this paper shows that if the algebraic multiplicity of the first eigenvalue is 2, there exists an S1 attractor bifurcation, and the number of its singular points can only be eight. Besides, we show that the simplified governing equations bifurcate to an S1 attractor, when the Control parameter λ crosses a critical value λ 0.

On the singular two-parameter eigenvalue problem II

In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems, which is a key relation for many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems. In 2009, Muhič and Plestenjak extended the above relation to a class of singular two-parameter eigenvalue problems with coprime characteristic polynomials and such that all finite eigenvalues are algebraically simple. They introduced a way to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems. Using new tools, in particular the stratification theory, we extend this connection to singular two-parameter eigenvalue problems with possibly multiple eigenvalues and such that characteristic polynomials can have a nontrivial common factor.

Асимптотика решений системы сингулярно возмущенных дифференциальных уравнений

Solutions of nonlinear singularly perturbed differential equations are studied in this work. The matrix functions have multiple eigen values, which change the stability conditions the region under consideration several times. For these eigen values, the stability conditions are satisfied, but the stability change points, initial and critical, coincide. This is not typical for ongoing work in this direction. It is not clear what the estimate is at this point. If we consider this point as the starting point, then to the right of it will be, to the left more, at the point itself . If this point is taken as the point of change of stability, then we obtain a similar estimate. If this point is a critical point, then we will not approach this point. Therefore, we choose the starting point so that it remains the area under consideration. To obtain the asymptotics of solutions, the method of successive approximations is used. The solution of the stated problem is considered in the complex plane. Eigen values are analytic functions, therefore, using level lines, both parts of analytic functions can be covered with lines of the region under consideration. When investigating the solution, integration paths were chosen and the corresponding estimate was obtained. As a result, the uniqueness and uniform convergence of solutions are proved.