Point Spectrum Research Articles

On Numerical Approximations of the Koopman Operator

We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode Decomposition-type methods in the context of Finite Section theory of infinite dimensional operators, and provide an example of a mixing map for which the finite section method fails. Under assumptions on the underlying dynamics, we provide the first result on the convergence rate under sample size increase in the finite-section approximation. We study the error in the Krylov subspace version of the finite section method and prove convergence in pseudospectral sense for operators with pure point spectrum. Since Krylov sequence-based approximations can mitigate the curse of dimensionality, this result indicates that they may also have low spectral error without an exponential-in-dimension increase in the number of functions needed.

About essential spectra of unbounded Jacobi matrices

We study spectral properties of unbounded Jacobi matrices with periodically modulated or blended entries. Our approach is based on uniform asymptotic analysis of generalized eigenvectors. We determine when the studied operators are self-adjoint. We identify regions where the point spectrum has no accumulation points. This allows us to completely describe the essential spectrum of these operators.

Distribution pattern of fluctuating wind pressures on cooling towers in grouped rectangular arrangement

The cooling towers in this study are in a typical grouped arrangement. Traditional research on grouped-tower interference effects usually emphasizes equivalent static wind loads and ignores quantitative definition of interference effects under aerodynamic load, especially those for fluctuating wind pressure distribution patterns. An eight-tower rectangular arrangement with various aerodynamic interference characteristics was selected as the research object on which a rigid-model synchronous-pressure-measurement wind-tunnel test was performed. Fluctuating time-variant pressures on the tower surfaces for different wind directions was obtained. In three typical incoming flow modes: “windward flow”, “crosswind flow” and “leeward flow”, the fluctuating wind pressure distribution around circumferential directions can be summarized as “V” type, “M” type, and half “V” + half “M” type. The interference patterns can be defined as “weak interference”, “asymmetric strong interference” or “symmetric strong interference”, according to which a 9-item fitting formula of fluctuating wind pressure is proposed and a recommended distribution is given. Interference involving aerodynamic correlation and spectra characters of distributed wind loads are further discussed, and a fitting expression of a specific point spectrum is given. The research strategies and conclusions on fluctuating wind pressure interference provide a loading mode reference for evaluating wind resistance design of similar groups of structures.

Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension one

In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let $P\\colon L^2(\\mathbb{R})\\to L^2(\\mathbb{R})$ have the form $$ P:=-\\frac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $\\mathbf{1}{(-\\infty,\\lambda^2]}(P)$ has a full asymptotic expansion in powers of $\\lambda$. In particular, our class of potentials $W$ is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, the class of potentials includes certain potentials with \_dense pure point spectrum. The proof combines the gauge transform methods of Parnovski–Shterenberg and Sobolev with Melrose's scattering calculus.

Local Ergodic Theorems for $$C_0$$-Semigroups

Let \(\{T(t)\}_{t\ge 0}\) be a \(C_0\)-semigroup of bounded linear operators on the Banach space X into itself and let A be their infinitesimal generator. In this paper, we show that if T(t) is uniformly ergodic, then A does not have the single valued extension property, which implies that A must have a non-empty interior of the point spectrum. Furthermore, we introduce the local mean ergodic for \(C_0\)-semigroup T(t) at a vector \(x\in X\) and we establish some conditions implying that T(t) is a local mean ergodic at x.

The Spectrum of Second Order Quantum Difference Operator

In this study, we give another generalization of second order backward difference operator ∇2 by introducing its quantum analog ∇q2. The operator ∇q2 represents the third band infinite matrix. We construct its domains c0(∇q2) and c(∇q2) in the spaces c0 and c of null and convergent sequences, respectively, and establish that the domains c0(∇q2) and c(∇q2) are Banach spaces linearly isomorphic to c0 and c, respectively, and obtain their Schauder bases and α-, β- and γ-duals. We devote the last section to determine the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the operator ∇q2 over the Banach space c0 of null sequences.

A Spectral Theorem for the Semigroup Generated by a Class of Infinitely Many Master Equations

In this article we investigate the spectral properties of the infinitesimal generator of an infinite system of master equations arising in the analysis of the approach to equilibrium in statistical mechanics. The system under consideration thus consists of infinitely many first-order differential equations governing the time evolution of probabilities susceptible of describing jumps between the eigenstates of a differential operator with a discrete point spectrum. The transition rates between eigenstates are chosen in such a way that the so-called detailed balance conditions are satisfied, so that for a large class of initial conditions the given system possesses a global solution which converges exponentially rapidly toward a time independent probability of Gibbs type. A particular feature and a challenge of the problem under consideration is that in the infinite-dimensional functional space where the initial-value problem is well posed, the infinitesimal generator is realized as a non normal and non dissipative compact operator, whose spectrum therefore does not exhibit a spectral gap around the zero eigenvalue.

Scattering analysis of a quantum impulsive boundary value problem with spectral parameter

We are interested in scattering and spectral analysis of an impulsive boundary value problem (IBVP) generated with a $q$-difference equation with eigenparameter in boundary condition in addition to impulsive conditions. We work on the Jost solution and scattering function of this problem, and by using the scattering solutions, we establish the resolvent operator, continuous spectrum and point spectrum of this problem. Furthermore, we discuss asymptotic behavior of the Jost solution and properties of eigenvalues.Also, we illustrate our results by a detailed example which is the special case of main problem.

Singular rank one perturbations

In this paper, A = B + V represents a self-adjoint operator acting on a Hilbert space H. We set a general theoretical framework and obtain several results for singular perturbations of A of the type Aβ = A + βτ*τ for τ being a functional defined in a subspace of H. In particular, we apply these results to Hβ = −Δ + V + β|δ⟩⟨δ|, where δ is the singular perturbation given by δ(φ) = ∫Sφ dσ, where S is a suitable hypersurface in Rn. Using the fact that the singular perturbation τ*τ is a sort of rank one perturbation of the operator A, it is possible to prove the invariance of the essential spectrum of A under these singular perturbations. The main idea is to apply an adequate Krein’s formula in this singular framework. As an additional result, we found the corresponding relationship between the Green’s functions associated with the operators H0 = Δ + V and Hβ, and we give a result about the existence of a pure point spectrum (eigenvalues) of Hβ. We also study the case β goes to infinity.

Bayesian surface regression versus spatial spectral nonparametric curve regression

COVID-19 incidence is analyzed at the provinces of the Spanish Communities in the Iberian Peninsula during the period February–October, 2020. Two infinite-dimensional regression approaches, surface regression and spatial curve regression, are proposed. In the first one, Bayesian maximum a posteriori (MAP) estimation is adopted in the approximation of the pure point spectrum of the temporal regression residual autocorrelation operator. Thus, an alternative to the moment-based estimation methodology developed in Ruiz-Medina, Miranda and Espejo (2019) is derived. Additionally, spatial curve regression is considered. A nonparametric estimator of the spectral density operator, based on the spatial periodogram operator, is computed to approximate the spatial correlation between curves. Dimension reduction is achieved by projection onto the empirical eigenvectors of the long-run spatial covariance operator. Cross-validation procedures are implemented to test the performance of the two functional regression approaches.

Spectral Properties of Stochastic Processes Possessing Finite Propagation Velocity.

This article investigates the spectral structure of the evolution operators associated with the statistical description of stochastic processes possessing finite propagation velocity. Generalized Poisson–Kac processes and Lévy walks are explicitly considered as paradigmatic examples of regular and anomalous dynamics. A generic spectral feature of these processes is the lower boundedness of the real part of the eigenvalue spectrum that corresponds to an upper limit of the spectral dispersion curve, physically expressing the relaxation rate of a disturbance as a function of the wave vector. We also analyze Generalized Poisson–Kac processes possessing a continuum of stochastic states parametrized with respect to the velocity. In this case, there is a critical value for the wave vector, above which the point spectrum ceases to exist, and the relaxation dynamics becomes controlled by the essential part of the spectrum. This model can be extended to the quantum case, and in fact, it represents a simple and clear example of a sub-quantum dynamics with hidden variables.

On left and right west-stampfli decomposition

In this paper we define and investigate the decomposition of a Hilbert space operator T in the form T = K+Q where K is a compact and the approximate points spectrum (or the surjectivity spectrum) of Q is identical to the set of all accumulation point of the approximate point spectrum ( or the surjectivity spectrum) of T. Also, we provide the relation between operators having these decomposition and left (or right) Stampfli operators.

Spectral inclusions between C0-quasi-semigroups and their generators

In this paper, we show a spectral inclusion of a dierent spectra of a C0-quasi-semigroup and its generator and precisely for ordinary, point, approximate point, residual, essential and regular spectra.

New approach to property (ω) for functions of operators

Let be an infinite dimensional complex Hilbert space and be the algebra of all bounded linear operators on . For , we say T satisfies property (ω) if σa (T) \\ σea (T) = π 00(T), where π 00(T) = {λ ∈ isoσ(T) : 0 < n(T − λI) < ∞}. In this paper, we research on the property (ω) for functions of operators by using the new spectrum σvea (T) which is a variant of the Weyl essential approximate point spectrum. At the same time, the stability of SVEP as well as the relationships between the two parts is also given.

Subdivisions of the spectra for D(r,0,s,0,t) operator on certain sequence spaces

In this paper we have examined the approximate point spectrum, defect spectrum and compression spectrum of the operator D(r,0,s,0,t) on the sequence spaces c0, c, and $bv_p (1&lt;p&lt;\infty)$.

CR embeddability of quotients of the Rossi sphere via spectral theory

We look at the action of finite subgroups of [Formula: see text] on [Formula: see text], viewed as a CR manifold, both with the standard CR structure as the unit sphere in [Formula: see text] and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of [Formula: see text] to the asymptotic distribution of the Kohn Laplacian’s eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum.

Green synthesis and antimicrobial activity of substituted diethyl (((5-(ethylthio)-1,3,4-thiadiazol-2-yl)amino)(phenyl)methyl)phosphonates

The one-pot, three-component, and facile synthesis of substituted diethyl (((5-(ethylthio)-1,3,4-thiadiazol-2-yl)amino)(phenyl)methyl)phosphonates (4a–j) has been achieved from the reaction of 2-amino-5-ethylthio-1,3,4-thiadiazole (1), various aromatic aldehydes (2a–j) and diethylphosphite (3) using Kabachnik–Fields reaction in the presence SnO2 nanoparticles as an efficient heterogeneous catalyst under solvent-free MWI condition. Furthermore, these series of new compounds were characterized by melting point, IR, NMR, and mass spectra. The newly synthesized compounds were evaluated for their antimicrobial activity which showed good activity compared to standard.

Denjoy–Wolff theory and spectral properties of weighted composition operators on Hol(D)

We study the spectrum σ(T) of a weighted composition operator T induced by a weight m∈Hol(D) and a holomorphic self-map φ on the unit disc, which is not an elliptic automorphism. If φ has a unique fixed point in D, we show that σ(T) is a bounded discrete set such that σ(T)∖{0} is a set of eigenvalues with multiplicity one. If φ has a Denjoy–Wolff point α on the unit circle, we first prove that the point spectrum is C∖{0} whenever m≠0 is constant. Moreover, the multiplicity of each eigenvalue is infinite. Then we describe classes of m for which the point spectrum of T is either empty or equal to C∖{0}.

Quasi-Fredholm spectrum for operator matrices

For A ? L(X), B ? L(Y) and C ? L(Y,X) we denote by MC the operator matrix defined on X ? Y by MC = (A C 0 B). In this paper, we prove that ?qF(A) ? ?qF(B) ? [ C?L(Y,X) ?qF(MC) ? ?p(B) ? ?p(A?), where ?qF(.) (resp. ?p(.)) denotes the quasi-Fredholm spectrum (resp. the point spectrum). Furthermore, we consider some sufficient conditions for MC to be quasi-Fredholm and sufficient conditions to have ?qF(A) ? ?qF(B) = ? C?L(Y,X) ?qF(MC).

APPLICATION OF TARTARIC ACID DERIVATIVES IN ENANTIOSEPARATION OF RS-IBUPROFEN

A novel Tartaric acid derivative-based chiral resolving agent has been discovered for the enantiosepration of racemic Ibuprofen [(RS)-2-(4-(2-methyl propyl) phenyl) propanoic acid]. The processinvolves optimization of diastereomeric salt formed between (+) Ibuprofen and optically pure tartaric acid derivatives, followed by separating the individual enantiomer by acid salt preparation. Characterization of the synthesized compound has been carried out by elemental analysis, melting point, FT-IR, UV, HPLC, and 1H NMR spectra. The developed separation method is simple, cost-effective, and industrially viable.