Multiple Eigenvalues Research Articles

Исследование решений системы сингулярно возмущенных дифференциальных уравнений

Solutions of linear systems of singularly perturbed differential equations are investigated in the work, in the case when the matrix function had multiple eigenvalues. And also in the study of solutions to a system of singularly perturbed differential equations, we apply the level line method. We define a stable and unstable interval. We take the starting point in stable intervals. Passing to the complex domain, we define the domain that we study for solutions of the problem under consideration. We divide the defined areas near the singular point into several areas. In each area, we estimate the solutions of the problem. To do this, we choose the integration path and prove the lemma and theorem. As a result, we will prove the asymptotic proximity of the solutions of the perturbed and unperturbed problems.

Construction of self‐adjoint differential operators with prescribed spectral properties

AbstractIn this expository article some spectral properties of self‐adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or potentials such that the essential or discrete spectrum of a Schrödinger operator of a certain type (e.g. the Neumann Laplacian) coincides with a predefined subset of the real line. Another aim is to emphasize that the spectrum of a differential operator on a bounded domain or bounded interval is not necessarily discrete, that is, eigenvalues of infinite multiplicity, continuous spectrum, and eigenvalues embedded in the continuous spectrum may be present. This unusual spectral effect is, very roughly speaking, caused by (at least) one of the following three reasons: The bounded domain has a rough boundary, the potential is singular, or the boundary condition is nonstandard. In three separate explicit constructions we demonstrate how each of these possibilities leads to a Schrödinger operator with prescribed essential spectrum.

Generalized derivatives of eigenvalues of a symmetric matrix

Sensitivity theory is established for eigenvalues of a symmetric matrix depending on an arbitrary number of parameters; a sequence of symmetric eigenvalue problems is derived whose solution yields first-order generalized derivative information in the form of Clarke subgradients. This is accomplished using lexicographic directional differentiation, a recently developed tool in nonsmooth analysis. The findings here include sensitivities of multiple eigenvalues of symmetric matrices depending nonsmoothly on parameters and sensitivities of nonsmooth functions of symmetric matrix eigenvalues. The approach here is computationally relevant (with an algorithm given), and classical theory is recovered along the way (such as if an eigenvalue is simple). Thanks to the flexibility of this approach, the present theory is amenable to compositions of symmetric matrix eigenvalue problems with other problems, such as optimization or dynamic problems.

Rigid local systems and the multiplicative eigenvalue problem

We give a construction that produces irreducible complex rigid local systems on $\mathbb{P}_C^1-\{p_1,\ldots ,p_s\}$ via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups $\mathrm{SU}(n)$ (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices of these polytopes give all possible unitary irreducible rigid local systems. As a consequence we obtain that the ranks of unitary irreducible rigid local systems, including those with finite global monodromy, on $\mathbb{P}^1-S$ are bounded above if we fix the cardinality of the set $S=\{p_1,\ldots ,p_s\}$ and require that the local monodromies have orders that divide $n$ for a fixed $n$. Answering a question of N.~Katz, we show that there are no irreducible rigid local systems of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing $n$, when $n$ is a prime number. We also show that all unitary irreducible rigid local systems on $\mathbb{P}_C^1-S$ with finite local monodromies arise as solutions to the Knizhnik-Zamalodchikov equations on conformal blocks for the special linear group. Along the way, generalizing previous works of the author and J.~Kiers, we give an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for $\mathrm{SU}(n)$.

On the multiplicity of the least signless Laplacian eigenvalue of a graph

Let G be a simple and connected graph of order n(≥6). The eigenvalues of the signless Laplacian matrix Q(G) of G are called the Q-eigenvalues of G. Let qn(G) be the least Q-eigenvalue of G. In this paper, we investigate the multiplicity of qn(G) and characterize all n-vertex connected graphs with qn(G) of multiplicity n−3. Moreover, it is shown that all these graphs are determined by their signless Laplacian spectra.

Quasi-periodic solutions for a Schrödinger equation under periodic boundary conditions with given potential

This paper is concerned with the cubic nonlinear Schrödinger equationiut−uxx+V(x)u+|u|2u=0,x∈T=R/2πZ subject to periodic boundary conditions, where the potential V(x) is real analytic in |Imx|<r and i=−1. It is proved that there exist many quasi-periodic solutions of the nonlinear Schrödinger equation with given potential V(x) by a infinite dimensional KAM theorem dealing with multiple eigenvalues.

Dynamical transition and bifurcation of hydromagnetic convection in a rotating fluid layer

We study the stability and dynamic transition of a rotating electrically conducting fluid layer in the presence of an external magnetic field based on the Boussinesq approximation. By analyzing the spectrum of the linear part of the model and verifying the validity of the principle of exchange of stability, we take a hybrid approach combining theoretical analysis with numerical computation to study the transition from a simple real eigenvalue, a pair of complex conjugate eigenvalues and a real eigenvalue of multiplicity two, respectively. The center manifold reduction theory is applied to reduce the infinite dimensional system to the corresponding finite dimensional one together with several non-dimensional transition numbers that determine the dynamic transition types. Careful numerical computations are performed to determine these transition numbers as well as related flow patterns. Our results indicate that both continuous and jump transitions can occur at certain parameter region.

Spectral properties of a class of treelike networks generated by motifs with single root node

In this paper, we introduce a class of treelike networks generated by motifs with single root node. We study the spectral properties of treelike networks by studying the behaviors of the corresponding adjacency matrix and Laplacian matrix. First, we get the adjacency matrix and Laplacian matrix of this treelike network. Then, we analyze symmetry of eigenvalues, multiplicity of eigenvalue [Formula: see text] and prove that its largest eigenvalue is strictly monotonically increasing and convergent. Finally, we introduce a new method to prove that the largest eigenvalue of the Laplacian matrix is strictly monotonically increasing and convergent. Also, we obtain that its smallest nonzero eigenvalue is strictly monotonically decreasing and convergent.

Graphs with high second eigenvalue multiplicity

Jiang, Tidor, Yao, Zhang, and Zhao recently showed that connected bounded degree graphs have sublinear second eigenvalue multiplicity (always referring to the adjacency matrix). This result was a key step in the solution to the problem of equiangular lines with fixed angles. It led to the natural question: what is the maximum second eigenvalue multiplicity of a connected bounded degree $n$-vertex graph? The best known upper bound is $O(n/\log\log n)$. The previously known best known lower bound is on the order of $n^{1/3}$ (for infinitely many $n$), coming from Cayley graphs on $\text{PSL}(2,q)$. Here we give constructions showing a lower bound on the order of $\sqrt{n/\log n}$. We also construct Cayley graphs with second eigenvalue multiplicity at least $n^{2/5}-1$. Earlier techniques show that there are at most $O(n/\log\log n)$ eigenvalues (counting multiplicities) within $O(1/\log n)$ of the second eigenvalue. We give a construction showing this upper bound on approximate second eigenvalue multiplicity is tight up to a constant factor. This demonstrates a barrier to earlier techniques for upper bounding eigenvalue multiplicities.

Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part II: Limiting absorption, limiting amplitude principles and interface resonance

This paper is concerned with the time-dependent Maxwell’s equations for a plane interface between a negative material described by the Drude model and the vacuum, which fill, respectively, two complementary half-spaces. In a first paper, we have constructed a generalized Fourier transform which diagonalizes the Hamiltonian that represents the propagation of transverse electric waves. In this second paper, we use this transform to prove the limiting absorption and limiting amplitude principles, which concern, respectively, the behavior of the resolvent near the continuous spectrum and the long time response of the medium to a time-harmonic source of prescribed frequency. This paper also underlines the existence of an interface resonance which occurs when there exists a particular frequency characterized by a ratio of permittivities and permeabilities equal to −1 across the interface. At this frequency, the response of the system to a harmonic forcing term blows up linearly in time. Such a resonance is unusual for wave problem in unbounded domains and corresponds to a non-zero embedded eigenvalue of infinite multiplicity of the underlying operator. This is the time counterpart of the ill-posdness of the corresponding harmonic problem.

The proportionality principle for Osserman manifolds

We give the necessary and sufficient conditions for Jacobi operators that determine an algebraic curvature tensor. This motivates us to introduce the new concept of Jacobi-proportional Riemannian tensors, whose special case is the Rakić duality principle. We prove that all known Osserman tensors (both two-root Osserman and Clifford) are Jacobi-proportional. After the results given by Nikolayevsky, it is known that possible counterexamples of the Osserman conjecture can occur in dimension 16 only, while the reduced Jacobi operator has an eigenvalue of multiplicity 7 or 8. We prove that Jacobi-proportional Osserman tensors that do not satisfy the Osserman conjecture are 2-root with multiplicities 8 and 7, or 3-root with multiplicities 7, 7, and 1.

Graphs with eigenvalue −1 of multiplicity 2θ(G)+ρ(G)−1

Let G be a graph, m(G,λ) be the multiplicity of eigenvalue λ of A(G), θ(G) be the cyclomatic number of G, ρ(G) be the number of pendent vertices of G. Wang et al. proved that m(G,λ)≤2θ(G)+ρ(G)−1 for an arbitrary eigenvalue λ if G is not a cycle, and left an open problem to characterize the extremal graphs attaining the upper bound. We solve the open problem when λ=−1, that is a graph G satisfies m(G,−1)=2θ(G)+ρ(G)−1 if and only if it is obtained from a tree T with m(T,−1)=2θ(T)+ρ(T)−1 by attaching θ(G) cycles of order a multiple of 3 to θ(G) quasi-pendent vertices of T and delete related θ(G) pendent vertices, where ρ(T)≥θ(G).

The minimum number of multiplicity 1 eigenvalues among real symmetric matrices whose graph is a 2-linear tree

For a tree T, denotes the minimum number of eigenvalues of multiplicity 1 among all real symmetric matrices, whose graph is T. It is known that . A tree is linear if all its vertices of degree at least 3 lie on a single induced path, and k-linear if there are k of these high degree vertices. is known for generalized stars (1-linear trees), and is determined here for 2-linear trees. Three (and higher) linear trees offer considerable additional complications.

Nonlinear Fourier transform assisted high-order soliton characterization

Nonlinear Fourier transform (NFT), based on the nonlinear Schrödinger equation, is implemented for the description of soliton propagation, and in particular focused on propagation of high-order solitons. In nonlinear frequency domain, a high-order soliton has multiple eigenvalues depending on the soliton amplitude and pulse-width. During the propagation along the standard single mode fiber (SSMF), their eigenvalues remain constant, while the corresponding discrete spectrum rotates along with the SSMF transmission. Consequently, we can distinguish the soliton order based on its eigenvalues. Meanwhile, the discrete spectrum rotation period is consistent with the temporal evolution period of the high-order solitons. The discrete spectrum contains nearly 99.99% energy of a soliton pulse. After inverse-NFT on discrete spectrum, soliton pulse can be reconstructed, illustrating that the eigenvalues can be used to characterize soliton pulse with good accuracy. This work shows that soliton characteristics can be well described in the nonlinear frequency domain. Moreover, as a significant supplement to the existing means of characterizing soliton pulses, NFT is expected to be another fundamental optical processing method besides an oscilloscope (measuring pulse time domain information) and a spectrometer (measuring pulse frequency domain information).

Periodic Indefinite Sturm-Liouville Problems With One Turning Point

Multiplicity of eigenvalues of the regular indefinite Sturm- Liouville problem ?y"" + qy = ?wy on [a, b] subject to periodic boundary conditions is discussed. A necessary and sufficient condition for the existence of non-simple real eigenvalues is proved. Eigenfunctions corresponding to non-simple real eigenvalues are obtained. In this article, we discuss the interlacing property in one turning point case with periodic boundary conditions.

The multiplicities of eigenvalues of a graph

For a connected graph G, let e(G) be the number of its distinct eigenvalues and d be the diameter. It is well known that e(G)≥d+1. This shows η≤n−d, where η and n denote the nullity and the order of G, respectively. A graph is called minimal if e(G)=d+1. In this paper, we characterize all trees satisfying η(T)=n−d or n−d−1. Applying this result, we prove that a caterpillar is minimal if and only if it is a path or an even caterpillar, which extends a result by Aouchiche and Hansen. Furthermore, we completely characterize all connected graphs satisfying η=n−d. For any non-zero eigenvalue of a tree, a sharp upper bound of its multiplicity involving the matching number and the diameter is provided.

Tachyons with any spin

In earlier work, we showed how to handle the Group Theoretical issue of the Little Group for spin 1/2 tachyons by introducing a special metric in the vector space of one-particle states. Here that technique is extended to tachyons of any spin. Examining the bi-linear algebra of the generating matrices for spin 5/2, we find a complete basis for the Gell-Mann matrices that form the Lie algebra for SU(3). A Dirac-like equation is developed for tachyons of any integer-plus-one-half spin; and it shows multiple distinct mass eigenvalues. The primary model shows a mass spectrum (in the case of [Formula: see text]) that roughly mimics the known data on masses of the three neutrinos; the model can be tweaked to fit that experimental data precisely.

Extended flexibility of Lyapunov exponents for Anosov diffeomorphisms

Bochi-Katok-Rodriguez Hertz proposed recently a program on the flexibility of Lyapunov exponents for conservative Anosov diffeomorphisms, and obtained partial results in this direction. For conservative Anosov diffeomorphisms with strong hyperbolic properties we establish extended flexibility results for their Lyapunov exponents. We give examples of Anosov diffeomorphisms with the strong unstable exponent larger than the strong unstable exponent of the linear part. We also give examples of derived from Anosov diffeomorphisms with the metric entropy larger than the entropy of the linear part. These results rely on a new type of deformation which goes beyond the previous Shub-Wilkinson and Baraviera-Bonatti techniques for conservative systems having some invariant directions. In order to estimate the Lyapunov exponents even after breaking the invariant bundles, we obtain an abstract result which gives bounds on exponents of some specific cocycles and which can be applied in various other settings. We also include various interesting comments in the appendices: our examples are C 2 \mathcal {C}^2 robust, the Lyapunov exponents are continuous (with respect to the map) even after breaking the invariant bundles, a similar construction can be obtained for the case of multiple eigenvalues.

Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term

We study the following class of double-phase nonlinear eigenvalue problems − div ⁡ [ ϕ ( x , | ∇ u | ) ∇ u + ψ ( x , | ∇ u | ) ∇ u ] = λ f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain from R N and the potential functions ϕ and ψ have ( p 1 ( x ) ; p 2 ( x ) ) variable growth. The primitive of the reaction term of the problem (the right-hand side) has indefinite sign in the variable u and allows us to study functions with slower growth near + ∞ , that is, it does not satisfy the Ambrosetti–Rabinowitz condition. Under these hypotheses we prove that for every parameter λ ∈ R + ∗ , the problem has an unbounded sequence of weak solutions. The proofs rely on variational arguments based on energy estimates and the use of Fountain Theorem.

On multiplicities of eigenvalues of a spectral problem on a prolate tree

On multiplicities of eigenvalues of a spectral problem on a prolate tree