Time-dependent Operator Research Articles

Direct and inverse scattering at fixed energy for massless charged Dirac fields by Kerr-Newman-de Sitter black holes

In this paper, we study the direct and inverse scattering theory at fixed energy for massless charged Dirac fields evolving in the exterior region of a Kerr-Newman-de Sitter black hole. In the first part, we establish the existence and asymptotic completeness of time-dependent wave operators associated to our Dirac fields. This leads to the definition of the time-dependent scattering operator that encodes the far-field behavior (with respect to a stationary observer) in the asymptotic regions of the black hole: the event and cosmological horizons. We also use the miraculous property (quoting Chandrasekhar) - that the Dirac equation can be separated into radial and angular ordinary differential equations - to make the link between the time-dependent scattering operator and its stationary counterpart. This leads to a nice expression of the scattering matrix at fixed energy in terms of stationary solutions of the system of separated equations. In a second part, we use this expression of the scattering matrix to study the uniqueness property in the associated inverse scattering problem at fixed energy. Using essentially the particular form of the angular equation (that can be solved explicitely by Frobenius method) and the Complex Angular Momentum technique on the radial equation, we are finally able to determine uniquely the metric of the black hole from the knowledge of the scattering matrix at a fixed energy.

Acceleration of Relativistic Electron Dynamics by Means of X2C Transformation: Application to the Calculation of Nonlinear Optical Properties.

The Liouville-von Neumann equation based on the four-component matrix Dirac-Kohn-Sham Hamiltonian is transformed to a quasirelativistic exact two-component (X2C) form and then used to solve the time evolution of the electronic states only. By this means, a significant acceleration by a factor of 7 or more has been achieved. The transformation of the original four-component equation of motion is formulated entirely in matrix algebra, following closely the X2C decoupling procedure of Ilias and Saue [ J. Chem. Phys. 2007 , 126 , 064102 ] proposed earlier for a static (time-independent) case. In a dynamic (time-dependent) regime, however, an adiabatic approximation must in addition be introduced in order to preserve the block-diagonal form of the time-dependent Dirac-Fock operator during the time evolution. The resulting X2C Liouville-von Neumann electron dynamics (X2C-LvNED) is easy to implement as it does not require an explicit form of the picture-change transformed operators responsible for the (higher-order) relativistic corrections and/or interactions with external fields. To illustrate the accuracy and performance of the method, numerical results and computational timings for nonlinear optical properties are presented. All of the time domain X2C-LvNED results show excellent agreement with the reference four-component calculations as well as with the results obtained from frequency domain response theory.

Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time

In this paper we study maximal Lp-regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the Lp-boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of m-th order elliptic operators A with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an Lp(Lq)-theory for such equations for p,qin (1, infty ). In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.

Global wellposedness for a class of reaction–advection–anisotropic-diffusion systems

We prove existence and uniqueness of global solutions for a class of reaction-advection-anisotropic-diffusion systems whose reaction terms have a triangular structure. We thus extend previous results to the case of time-space dependent anisotropic diffusions and with time-space dependent advection terms. The corresponding models are in particular relevant for transport processes inside porous media and in situations in which additional migration occurs. The proofs are based on optimal $L^p$-maximal regularity results for the general time-dependent linear operator dual to the one involved in the considered systems. As an application, we prove global well-posedness for a prototypical class of chemically reacting systems with mass-action kinetics, involving networks of reactions of the type $C_1+\ldots+C_{P-1} \rightleftharpoons C_{P} $. Finally, we analyze how a classical a priori $L^2$-estimate of the solutions, which holds with this kind of nonlinear reactive terms, extends to our general anisotropic-advection framework. It does extend with the same assumptions for isotropic diffusions and is replaced by an $L^{(N+1)/N}$-estimate in the general situation.

Quantum optical electromagnetic field and Rabi oscillation in time-varying medium

In this paper, we consider the quantum optical electromagnetic wave in time-varying media, where the electric permittivity ϵ(t) and the magnetic permeability μ(t) depend on time explicitly as ϵ(t) = ϵ0(1+qt) and μ(t) = μ0(1+qt). For these time-varying parameters, we solve Maxwell’s equations. To construct time-dependent coherent states for this light, we adopt the invariant method proposed by Lewis and Riesenfeld and express the fields in terms of the time-dependent step operators for the invariant. We also discuss the deformed Rabi oscillation for both on-resonance and off-resonance.

Quantum mechanical expansion of variance of a particle in a weakly non-uniform electric and magnetic field

We have solved the Heisenberg equation of motion for the time evolution of the position and momentum operators for a non-relativistic spinless charged particle in the presence of a weakly non-uniform electric and magnetic field. It is shown that the drift velocity operator obtained in this study agrees with the classical counterpart, and that, using the time dependent operators, the variances in position and momentum grow with time. The expansion rate of variance in position and momentum are dependent on the magnetic gradient scale length, however, independent of the electric gradient scale length. In the presence of a weakly non-uniform electric and magnetic field, the theoretical expansion rates of variance expansion are in good agreement with the numerical analysis. It is analytically shown that the variance in position reaches the square of the interparticle separation, which is the characteristic time much shorter than the proton collision time of plasma fusion. After this time, the wavefunctions of the neighboring particles would overlap, as a result, the conventional classical analysis may lose its validity. The broad distribution of individual particle in space means that their Coulomb interactions with other particles become weaker than that expected in classical mechanics.

Classical-like behavior in quantum walks with inhomogeneous, time-dependent coin operators

Although quantum walks exhibit peculiar properties that distinguish them from random walks, classical behavior can be recovered in the asymptotic limit by destroying the coherence of the pure state associated to the quantum system. Here I show that this is not the only way: I introduce a quantum walk driven by an inhomogeneous, time-dependent coin operator, which mimics the statistical properties of a random walk at all time scales. The quantum particle undergoes unitary evolution and, in fact, the high correlation evidenced by the components of the wave function can be used to revert the outcome of an accidental measurement of its chirality.

Dynamic response of a viscoelastic beam impacted by a viscoelastic sphere

In the present paper, we consider the problem on a transverse impact of a viscoelastic sphere upon a viscoelastic Bernoulli–Euler beam, the viscoelastic features of which are defined via the fractional derivative standard linear solid models. As this takes place, only Young’s time-dependent operators are preassigned, while the bulk moduli are considered to be constant values, since the bulk relaxation for the majority of materials is far less than the shear relaxation. Beam’s displacement subjected to the concentrated contact force is found by the method of expansion in terms of eigen (beam) functions. The contact force driven displacement of the impactor, which is the sum of the beam’s displacement at the place of contact and the indentation of the impactor into the target, is defined from the equation of motion of the material point with the mass equal to sphere’s mass. Within the contact domain, the contact force is defined by the modified Hertzian contact law with the time-dependent rigidity function. For decoding the viscoelastic operators involving in the problem under consideration, the algebra of Rabotnov’s fractional operators is employed. A nonlinear integro-differential equation is obtained either in terms of the contact force or in the local bearing of the target and impactor materials. Using the duration of contact as a small parameter, approximate analytical solutions have been found, which allow one to define the maximal contact force and the duration of contact for colliding bodies.

Posttreatment FDG PET/CT in predicting survival of patients with ovarian carcinoma.

BackgroundThe purposes of this study are to evaluate the prognostic value of posttreatment 18F-FDG PET/CT in predicting the survival of patients with ovarian carcinoma and to determine incremental value of combining posttreatment PET/CT with traditional prognostic factors in a multivariate model.MethodsThis was an IRB-approved retrospective study. From July 2001 to July 2011, 48 patients who completed initial therapy for ovarian carcinoma with concurrent 3- to 9-month initial posttreatment 18F-FDG PET/CT and serum CA-125 were identified from the radiology database. Prognostic value of posttreatment PET/CT, CA-125, age, race, and tumor stage were determined from Cox proportional hazard model using univariate and multivariate analyses. Time-dependent receiver operator curves were also calculated at various follow-up intervals.ResultsIn a univariate model, overall survival (OS) was associated with PET/CT (hazard ratio = 4.18; 95 % CI 1.49–11.70) and CA-125 (hazard ratio = 11.09; 95 % CI 4.27–28.79). When the effects of posttreatment PET/CT and CA-125 were combined in the multivariate analysis, hazard ratio for PET/CT increased to 4.84 (95 % CI 1.59–14.73, p = 0.005) and hazard ratio for CA-125 increased to 14.43 (95 % CI 4.65–44.84, p < 0.001). In the subset of patients with negative CA-125, posttreatment PET/CT had a hazard ratio of 2.98 (95 % CI 0.86–10.37), supporting the role of posttreatment PET/CT in risk stratification of patients with negative CA-125. Time-dependent receiver operator curves showed that the combination of PET/CT and CA-125 improved prognostic accuracy compared to PET/CT or CA-125 alone at 12-, 24-, 30-, and 36-month follow-up.ConclusionsPosttreatment PET/CT can predict the survival of patients with ovarian carcinoma. The addition of posttreatment PET/CT to the CA-125 serum biomarker has an incremental value in improving prognostic accuracy, particularly in the subset of patients with negative CA-125.

Unitary quantum evolution for time-dependent quasi-Hermitian systems with nonobservable Hamiltonians

It has been argued that it is incompatible to maintain unitary time-evolution for time-dependent non-Hermitian Hamiltonians when the metric operator is explicitly time-dependent. We demonstrate here that the time-dependent Dyson equation and the time-dependent quasi-Hermiticity relation can be solved consistently in such a scenario for a time-dependent Dyson map and time-dependent metric operator, respectively. These solutions are obtained at the cost of rendering the non-Hermitian Hamiltonian to be a non-observable operator as it ceases to be quasi-Hermitian when the metric becomes time-dependent.

A hierarchy of local coupled cluster singles and doubles response methods for ionization potentials.

We present a hierarchy of local coupled cluster (CC) linear response (LR) methods to calculate ionization potentials (IPs), i.e., excited states with one electron annihilated relative to a ground state reference. The time-dependent perturbation operator V(t), as well as the operators related to the first-order (with respect to V(t)) amplitudes and multipliers, thus are not number conserving and have half-integer particle rank m. Apart from calculating IPs of neutral molecules, the method offers also the possibility to study ground and excited states of neutral radicals as ionized states of closed-shell anions. It turns out that for comparable accuracy IPs require a higher-order treatment than excitation energies; an IP-CC LR method corresponding to CC2 LR or the algebraic diagrammatic construction scheme through second order performs rather poorly. We therefore systematically extended the order with respect to the fluctuation potential of the IP-CC2 LR Jacobian up to IP-CCSD LR, keeping the excitation space of the first-order (with respect to V(t)) cluster operator restricted to the m=½⊕3/2 subspace and the accuracy of the zero-order (ground-state) amplitudes at the level of CC2 or MP2. For the more expensive diagrams beyond the IP-CC2 LR Jacobian, we employ local approximations. The implemented methods are capable of treating large molecular systems with hundred atoms or more.

Optimal Control Problems Governed by Time Dependent Subdifferential Operators with Delay

ABSTRACTThis article is devoted to the study, in the infinite dimensional setting, of a Bolza-type problem governed by a class of functional evolution inclusion which involves a time-dependent subdifferential operator with a time-delay perturbation. We present a relaxation result associated with such equations where the controls are Young measures in order to show the existence of an optimal solution under a suitable convexity assumption.

Quantum mechanical grad-B drift velocity operator in a weakly non-uniform magnetic field

This paper presents the analytical solution for quantum mechanical grad-B drift velocity operator by solving the Heisenberg equation of motion. Using the time dependent operators, it is shown the analytical solution of the position operators in x̂(t) and ŷ(t) of the particle in the presence of a weakly non-uniform magnetic field. It is also shown numerically that the grad-B drift velocity operator agrees with the classical counterpart.

Nonclassical Properties for Two Coupled N-Two-Level Atom and a Single Two-Level Atom Under an External Magnetic Field

We study the interaction between a single two-level atom and N two-level atoms under the effect of a uniform magnetic field. The exact solution is obtained and the expectation value of the time-dependent quantum operators calculated using the Block state (the generalized coherent state). We discuss numerically the atomic inversion where the phenomenon of collapse and revival is observed. The change in the value of the atomic angle plays a role in variance squeezing, where it is pronounced for 𝜃 = π/3. Entropy squeezing is discussed and occurred in the first quadrature. The degree of entanglement through linear entropy is examined where the system shows partial entanglement and at a certain value of parameters displays nearly maximum entanglement.

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$ C n

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.

An Operatorial Description of Desertification

We propose a simple theoretical model for desertification processes based on three actors (soil, seeds, and plants) on a two-dimensional lattice. Each actor is described by a time dependent fermionic operator, and the dynamics is ruled by a self-adjoint Hamilton-like operator. We show that even taking into account only a few parameters, accounting for external actions on the ecosystem or the response to positive feedbacks, the model provides a plausible description of the desertification process, and can be adapted to different ecological landscapes. We first describe the simplified model in one cell. Then, we define the full model on a two-dimensional region, taking into account additional factors such as nonhomogeneities, the competition for resources between plants, and the spread of seeds due to the action of wind or animals. This allows us to explore the effects of positive feedback on slowing down, stopping, or reversing the desertification process.

Convergence results for families of univalent mappings on the unit ball in C^n

Let e S t (B n ) be the family of normalized univalent mappings on the Euclidean unit ball B n in C n , which have generalized parametric representation with respect to time-dependent operators A ∈ e A, where e A is a family of measurable mappings from (0,∞) into L(C n ) with some particular properties. Also, let e RT(idBn,(NA(t))t∈(T0,T)) be the time-T -reachable family of the control system C((T0,T),(NA(t))t∈(T0,T)), where A ∈ e A and T0 ≥ 0. In this paper we obtain certain convergence results for the families e

Parabolic Littlewood–Paley inequality for a class of time-dependent pseudo-differential operators of arbitrary order, and applications to high-order stochastic PDE

In this paper we prove a parabolic version of the Littlewood–Paley inequality for a class of time-dependent local and non-local operators of arbitrary order, and as an application we show that this inequality gives a fundamental estimate for an Lp-theory of high-order stochastic partial differential equations.

Influence of extracapsular nodal spread extent on prognosis of oral squamous cell carcinoma.

An objective definition of clinically relevant extracapsular nodal spread (ECS) in head and neck squamous cell carcinoma (SCC) is unavailable. Pathologic review of 245 pathologically positive oral cavity SCC neck dissection specimens was performed. The presence/absence of ECS, its extent (in millimeters), and multiple nodal and primary tumor risk factors were related to disease-specific survival (DSS) at a follow-up of 73 months. ECS was detected in 109 patients (44%). DSS was significantly better for patients without ECS than patients with ECS. Time-dependent receiver operator curve (ROC) analysis identified a prognostic cutoff for ECS extent at 1.7 mm. In multivariate analyses, DSS was significantly lower for patients with major ECS compared with patients with minor ECS, but not significantly different between patients with minor ECS and patients without ECS. ECS is clinically relevant in oral cavity SCC when it has extended more than 1.7 mm beyond the nodal capsule. © 2015 Wiley Periodicals, Inc. Head Neck 38: E1192-E1199, 2016.

Set-valued perturbation for time dependent subdifferential operator

In a separable Hilbert space, we consider an evolution inclusion involving time-dependent subdifferential of a proper convex lower semicontinuous function with a set-valued perturbation depending on both time and state variable. We prove, under a compactness condition on the perturbation, that there exists at least one absolutely continuous solution.