Unique Eigenvalue Research Articles

Asymptotic profiles of a nonlocal dispersal SIS epidemic model with saturated incidence

Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (J. Differ. Equ. 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.

Reconstruction of averaging indicators for highly heterogeneous media

We propose a new imaging algorithm capable of producing quantitative indicator functions for unknown and possibly highly oscillating media from multi-static far field measurements of scattered fields at a fixed frequency. The algorithm exploits the notion of modified transmission eigenvalues and their determination from measurements. We propose in particular the use of a new modified background obtained as the limit of a metamaterial background. It has the advantage of having a unique non zero eigenvalue, which is particularly suited for the proposed imaging procedure. We show the efficiency of this new algorithm on some two dimensional experiments and emphasise its superiority with respect to some classical approaches such as the linear sampling method.

Operator Space Manifold Theory: Modeling Quantum Operators with a Riemannian Manifold

The Half-Transform Ansatz (HTA) is a proposed method to solve hyper-geometric equations in Quantum Phase Space by transforming a differential operator to an algebraic variable and including a specific exponential factor in the wave function, but the mechanism which provides this solution scheme is not known. Analysis of the HTA’s application to the Hydrogen atom suggests an underlying mechanism which the HTA is a part of. Observations of exponential factors that act on the wave function naturally suggest modeling quantum operator definitions as a point on a Riemannian manifold in the 4D Operator Space, a novel idea we call the Operator Space Manifold Theory. On this manifold, we explore the concepts of superposition, regions of unique energy eigenvalues, and translation operators. We also find the theoretical backing to derive the HTA and how Operator Space Manifold Theory can be used to describe and solve quantum systems by manipulating how a quantum state perceives position and momentum.

Bound states of weakly deformed soft waveguides

In this paper we consider the two-dimensional Schrödinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the function R ∋ x ↦ d + ε f ( x ), where d > 0 is a constant, ε > 0 is a small parameter, and f is a compactly supported continuous function. We prove that if ∫ R f d x > 0, then the respective Schrödinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small ε > 0 and we obtain the asymptotic expansion of this eigenvalue in the regime ε → 0. An asymptotic expansion of the respective eigenfunction as ε → 0 is also obtained. In the case that ∫ R f d x < 0 we prove that the discrete spectrum is empty for all sufficiently small ε > 0. In the critical case ∫ R f d x = 0, we derive a sufficient condition for the existence of a unique bound state for all sufficiently small ε > 0.

Spectrum of the Dirichlet Laplacian in a thin cubic lattice

We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of widthε ≪ 1) which have a square cross section. This spectrum coincides with the union of segments which all go to +∞ asεtends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of lengthO(e−δ/ε),δ> 0, while the length of the next spectral segments isO(ε). To establish these results, we need to study in detail the properties of the Dirichlet LaplacianAΩin the geometry Ω obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max–min arguments as well as a well-chosen Poincaré–Friedrichs inequality, we prove thatAΩhas a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance forAΩ, that is no non trivial bounded solution at the threshold frequency forAΩ. This implies that the correct 1D model of the lattice for the next spectral segments is a system of ordinary differential equations set on the limit graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.

Existence condition of an eigenvalue of the three particle Schr¨odinger operator on a lattice

We consider the three-particle discrete Schrodinger operator Hµ,γ(К), К ϵТ3 associated to a system of three particles (two particle are fermions with mass 1 and third one is an another particle with mass m = 1/y < 1) interacting through zero range pairwise potential µ> 0 on the three-dimensional lattice Z3. It is proved that for γϵ(1, γ0) (γ0≈4,7655) the operator Hµ,γ(π), π=(π,π,π), has no eigenvalue and has only unique eigenvalue with multiplicity three for γ>γ0 lying right of the essential spectrum for sufficiently large µ.

An Eigenvalue Problem for Prescribed Curvature Equations

Abstract We study an eigenvalue problem for prescribed $\sigma _{k}$-curvature equations of star-shaped, $k$-convex, closed hypersurfaces. We establish the existence of a unique eigenvalue and its associated hypersurface, which is also unique, provided that the given data is even. Moreover, we show that the hypersurface must be strictly convex. A crucial aspect of our proof involves deriving uniform estimates in $p$ for $L_{p}$-type prescribed curvature equations.

From breather solutions to lump solutions: A construction method for the Zakharov equation

Periodic solutions of the Zakharov equation are investigated. By performing the limit operation λ 2l–1 → λ 1 on the eigenvalues of the Lax pair obtained from the n-fold Darboux transformation, an order-n breather-positon solution is first obtained from a plane wave seed. It is then proven that an order-n lump solution can be further constructed by taking the limit λ 1 → λ 0 on the breather-positon solution, because the unique eigenvalue λ 0 associated with the Lax pair eigenfunction Ψ(λ 0) = 0 corresponds to the limit of the infinite-periodic solutions. A convenient procedure of generating higher-order lump solutions of the Zakharov equation is also investigated based on the idea of the degeneration of double eigenvalues in multi-breather solutions.

Computational exploration of Cu(II)-en chelate-catalyzed hydrolysis of O-isopropyl methylphosphonofluoridate

In contrast to un-catalyzed hydrolysis of organophosphorus (OP) compounds, metal ions or/and their complexes with chelating ligands show catalytic effects in several ways depending upon the nature of the metal, ligand, substrate, and medium. It is known that Cu(II)-en chelate containing copper complexes accelerate the hydrolysis of OP compounds. However, the mechanism for this rate enhancement in the Cu(II)-en chelate catalytic hydrolysis reaction of sarin remains unexplored. We have examined possible mechanisms involving a Cu(II)-en with hydroxide nucleophile for the reaction pathway of the hydrolysis of O-isopropyl methylphosphonofluoridate (sarin) computationally. The density functional (B3LYP) employed in this study has reproduced the experimental Gibb’s free energy of activation value 15.5 kcal/mol for alkaline hydrolysis of sarin. Earlier proposal of push–pull mechanism for metal ion chelate-catalyzed hydrolysis of OP compounds has been found to be unfavorable in the present study. The role of water molecules in catalyzing the hydrolysis of sarin with Cu(II)-en chelate is crucial. The catalytic process involving Cu(II)-en chelate with one water molecule is the more plausible pathway to achieve the hydrolysis of sarin with Cu(II)-en chelate complexes. The most popular B3LYP method was used for optimization of given geometries. Except LANL2DZ for Cu atom, all the atoms are described using the 6–31 + G(d) basis set. The stability test has been performed for the wave functions as we are dealing with the open-shell molecules in order to ensure stable electronic configuration form, and the stable wavefunction is used as the initial configuration for the subsequent optimization. Harmonic frequency calculations and thermodynamic corrections were performed at the same level of theory. PCM method has been used for solvation effects. In order to ensure that each saddle point is linked to a minimum, IRC calculations were performed in forward and reverse directions to ensure the eigenvectors associated with the unique negative eigenvalues of the Hessian matrix. All energies discussed are solvated Gibbs free energies corrected to 298.15 K for the relative stability of the chemical structure. All calculations were performed using the Gaussian 09 code.

Semiclassical Measures for Higher-Dimensional Quantum Cat Maps

Consider a quantum cat map M associated with a matrix A∈Sp(2n,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\in {{\\,\ extrm{Sp}\\,}}(2n,{\\mathbb {Z}})$$\\end{document}, which is a common toy model in quantum chaos. We show that the mass of eigenfunctions of M on any nonempty open set in the position–frequency space satisfies a lower bound which is uniform in the semiclassical limit, under two assumptions: (1) there is a unique simple eigenvalue of A of largest absolute value and (2) the characteristic polynomial of A is irreducible over the rationals. This is similar to previous work (Dyatlov and Jin in Acta Math 220(2):297–339, 2018; Dyatlov et al. in J Am Math Soc 35(2):361–465, 2022) on negatively curved surfaces and (Schwartz in The full delocalization of eigenstates for the quantized cat map, 2021) on quantum cat maps with n=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=1$$\\end{document}, but this paper gives the first results of this type which apply in any dimension. When condition (2) fails we provide a weaker version of the result and discuss relations to existing counterexamples. We also obtain corresponding statements regarding semiclassical measures and damped quantum cat maps.

On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction

We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schrödinger equation with a point interaction and a focusing power nonlinearity. The Schrödinger operator with a point interaction (−Δα)α∈R describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The operator −Δα always has a unique simple negative eigenvalue eα. We prove that if the frequency of the standing wave is close to −eα, it is stable. Moreover, if the frequency is sufficiently large, we have the stability in the L2-subcritical or critical case, while the instability in the L2-supercritical case.

Projected CAP-EOM-CCSD method for electronic resonances.

The complex absorbing potential equation-of-motion coupled-cluster (CAP-EOM-CC) method is routinely used to investigate metastable electronic states in small molecules. However, the requirement of evaluating eigenvalue trajectories presents a barrier to larger simulations, as each point corresponding to a different value of the CAP strength parameter requires a unique eigenvalue calculation. Here, we present a new implementation of CAP-EOM-CCSD that uses a subspace projection scheme to evaluate resonance positions and widths at the overall cost of a single electronic structure calculation. We analyze the performance of the projected CAP-EOM-CC scheme against the conventional scheme, where the CAP is incorporated starting from the Hartree-Fock level, for various small and medium sized molecules, and investigate its sensitivity to various parameters. Finally, we report resonance parameters for a set of molecules commonly used for benchmarking CAP-based methods, and we report estimates of resonance energies and widths for 1- and 2-cyanonaphtalene, molecules that were recently detected in the interstellar medium.

Multiconfiguration Dirac-Hartree-Fock theory for copper Kα and Kβ diagram lines, satellite spectra, and ab initio determination of single and double shake probabilities

The x-ray spectra of copper $K\ensuremath{\alpha}$ and $K\ensuremath{\beta}$ remain a topic of great interest due to the complex open-shell processes involved, with many discrepancies among theories and experiments. We present high-accuracy theory of copper $K\ensuremath{\alpha}$ and $K\ensuremath{\beta}$ diagram and satellite spectra in the multiconfiguration Dirac-Hartree-Fock method. Diagram spectra are expanded to $5s$ with simultaneous convergence of 28 000 configuration state functions (CSFs) $(K\ensuremath{\alpha})$ and to $6g$ with simultaneous convergence of 91 000 CSFs $(K\ensuremath{\beta})$, achieving eigenvalue convergence to $\ifmmode\pm\else\textpm\fi{}0.03$ eV or 0.000 25%, approximately a factor of 10 improvement over past work. It is necessary to invoke biorthogonalization, developments of the active space approach, analysis of markers for theoretical convergence of eigenvalues, and the question of self-consistency for both $K\ensuremath{\alpha}$ and $K\ensuremath{\beta}$ spectra. We make use of gauge convergence, eigenvalue convergence, and $A$-coefficient convergence. The satellite spectra are a major outcome---without these, it is not possible to make use of the increased accuracy of the diagram computations. The Cu $K\ensuremath{\alpha}\phantom{\rule{4pt}{0ex}}3{d}^{8}$ double shake satellite spectrum alone contains 1506 unique eigenvalues (transitions) and required simultaneous convergence of 593 000 CSFs. Ab initio shakeoff probabilities for $1s,\phantom{\rule{0.16em}{0ex}}2s,\phantom{\rule{0.16em}{0ex}}2p,\phantom{\rule{0.16em}{0ex}}3s,\phantom{\rule{0.16em}{0ex}}3p,\phantom{\rule{0.16em}{0ex}}3d$, and $4s$ subshells as a result of the $K$-shell photoionization process are presented. Portable spectral representations are provided in the supplemental material, discussed in the text. We investigate the meaning behind shake, shakeoff, and shakeup processes and the computational potential to investigate these at the current time. We separate the notions of total shake, single shake, and double shake, explain how these are observable in high-quality experimental spectra, and how these calculations can be used experimentally.

ASYMPTOTICS FOR THE EIGENVALUE OF THE ONE-PARTICLE DISCRETE SCHRÖDINGER OPERATOR ON THE TWO-DIMENSIONAL LATTICE

We investigate the one-particle discrete Schrödinger operator ̂ on the two-dimensional lattice with extended discrete Laplacian and the Kronecker‘s delta function concentrated on . We show that the operator ̂ has a unique eigenvalue and we find the asymptotics for this eigenvalue as .

Beamforming Performance Analysis of Millimeter-Wave 5G Wireless Networks

With the rapid growth in the number of mobile devices and user connectivity, the demand for higher system capacity and improved quality-of-service is required. As the demand for high-speed wireless communication grows, numerous modulation techniques in the frequency, temporal, and spatial domains, such as orthogonal frequency division multiplexing (OFDM), time division multiple access (TDMA), space division multiple access (SDMA), and multiple-input multiple-output (MIMO), are being developed. Along with those approaches, electromagnetic waves’ orbital angular momentum (OAM) is attracting attention because it has the potential to boost the wireless communication capacity. Antenna electromagnetic radiation can be described by a sum of Eigen functions with unique eigenvalues, as is well known. In order to address such issues, the millimeter-wave (mmWave) communication is proposed which is considered as one of the potential technology for 5G wireless networks. The intrinsic feature of all electromagnetic waves is OAM. The OAM beams’ unique qualities have led to a slew of new uses. Broadband OAM generators, on the other hand, have gotten very little attention, especially in the mmWave frequency band. The use of OAM in conjunction with mmWave can reduce the beam power loss, enhance the received signal quality, and hence increase the system capacity. The transmitter and receiver antennas must be coaxial and parallel to achieve precise mode detection. The proposed mmWave integrated with OAM system model is discussed in this study. The channel model is created using the channel transition characteristics. The simulation results demonstrate that the proposed system model is a good way to boost the system capacity.

ON THE SPECTRUM OF THE TWO-PARTICLE SCHRÖDINGER OPERATOR WITH POINT POTENTIAL: ONE DIMENTIONAL CASE

In the paper a one-dimensional two-particle quantum system interacted by two identical point interactions is considered. The corresponding Schr\"{o}\-dinger operator (energy operator) $h_\varepsilon$ depending on $\varepsilon,$ is constructed as a self-adjoint extension of the symmetric Laplace operator. The main results of the work are based to the study of the operator $h_\varepsilon.$ First the essential spectrum is described. The existence of unique negative eigenvalue of the Schr\"{o}dinger operator is proved. Further, this eigenvalue and corresponding eigenfunction are found.

Generalized quantum circuit differentiation rules

Variational quantum algorithms that are used for quantum machine learning rely on the ability to automatically differentiate parametrized quantum circuits with respect to underlying parameters. Here, we propose the rules for differentiating quantum circuits (unitaries) with arbitrary generators. Unlike the standard parameter shift rule valid for unitaries generated by operators with spectra limited to at most two unique eigenvalues (represented by involutory and idempotent operators), our approach also works for generators with a generic non-degenerate spectrum. Based on a spectral decomposition, we derive a simple recipe that allows explicit derivative evaluation. The derivative corresponds to the weighted sum of measured expectations for circuits with shifted parameters. The number of function evaluations is equal to the number of unique positive non-zero spectral gaps (eigenvalue differences) for the generator. We apply the approach to relevant examples of two-qubit gates, among others showing that the fSim gate can be differentiated using four measurements. Additionally, we present generalized differentiation rules for the case of Pauli string generators, based on distinct shifts (here named as the triangulation approach), and analyse the variance for derivative measurements in different scenarios. Our work offers a toolbox for the efficient hardware-oriented differentiation needed for circuit optimization and operator-based derivative representation.

Convergent Expansions of Eigenvalues of the Generalized Friedrichs Model with a Rank-One Perturbation

We study analytic behavior of eigenvalues of the generalized Friedrichs model \(H_\mu (p)\), with a rank-one perturbation, depending on parameters \(\mu >0\) and \(p\in {\mathbb {T}}^2\). Under certain conditions, the existence of a unique eigenvalue lying below the essential spectrum has been shown in Lakaev (Abstract Appl Anal 2012, 2012). Here, we obtain an absolutely convergent expansion for that eigenvalue at \(\mu (p)\), the coupling constant threshold. The expansion is dependent to a large extent on whether the lower bound of the essential spectrum is a threshold resonance, a threshold eigenvalue or neither of them.

A framework for optimal quantum spatial search using alternating phase-walks

We present a novel methodological framework for quantum spatial search, generalising the Childs & Goldstone () algorithm via alternating applications of marked-vertex phase shifts and continuous-time quantum walks. We determine closed form expressions for the optimal walk time and phase shift parameters for periodic graphs. These parameters are chosen to rotate the system about subsets of the graph Laplacian eigenstates, amplifying the probability of measuring the marked vertex. The state evolution is asymptotically optimal for any class of periodic graphs having a fixed number of unique eigenvalues. We demonstrate the effectiveness of the algorithm by applying it to obtain search on a variety of graphs. One important class is the n × n 3 rook graph, which has N = n 4 vertices. On this class of graphs the algorithm performs suboptimally, achieving only overlap after time . Using the new alternating phase-walk framework, we show that overlap is obtained in phase-walk iterations.

Graphs with [formula omitted] main eigenvalues

An eigenvalue μ of a graph G of order n is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector e. Characterizing graphs with s(2≤s≤n) number of main eigenvalues is a long-standing problem in algebraic and graph theory. For graphs with n−1 main eigenvalues, we show that the unique non-main eigenvalue and the corresponding eigenvector can be obtained from its walk matrix W without solving the characteristic equation. We construct infinitely many graphs with n−1 main eigenvalues by the union and join, and the corona operations. We also determine the automorphism group of such graphs. We enumerate all connected graphs with n≤8 vertices and n−1 main eigenvalues, up to isomorphism. For the case n=7 we state the spectrum, the complement and the structure of the Galois group of the graph.