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We analyze in detail the discrete--time quantum walk on the line by separating the quantum evolution equation into Markovian and interference terms. As a result of this separation, it is possible to show analytically that the quadratic…
Before we proposed an algebraic technics for the Hamiltonian approach to the evolution systems of partial differential equations, including systems with constraints. Here we further develop this approach and present the defining system of…
Discussions of quantum mechanics often loosely claim that time evolution logically must be unitary, in order for the probabilistic interpretation of the amplitudes of the state vector to make sense at all times. We discuss from first…
Under broad conditions, evolutions due to two different Hamiltonians are shown to lead at some moment to orthogonal states. For two spin-1/2 systems subject to precession by different magnetic fields the achievement of orthogonalization is…
The connection of the 'time' evolution of the eigenstates of the reflectionless potentials of the Lax hierarchy to the more general case of the 'time' evolution of the eigenstates of the Schroedinger equation for potentials with…
Higher-order nonlinear time-evolution equations have widespread applications in science and engineering, such as in solid mechanics, materials science, and fluid mechanics. This paper mainly studies a direct time-parallel algorithm for…
The short-time and global behaviour are studied for autonomous linear evolution equations defined by generators of uniformly bounded holomorphic semigroups in a Hilbert space. A general criterion for log-convexity in time of the norm of the…
Time dependence is a universal phenomenon in nature, and a variety of mathematical models in terms of dynamical systems have been developed to understand the time-dependent behavior of real-world problems. Originally constructed to analyze…
There is a sub-class of the solutions to Quantum Tetrahedron Equation related to the algebraical Pentagon Equation. The Quantum Tetrahedron Equation defines an evolution operator in wholly discrete three dimensional space-time. In this…
We are concerned with periodic problems for nonlinear evolution equations at resonance of the form $\dot u(t) = - A u(t) + F (t,u(t))$, where a densely defined linear operator $A\colon D(A)\to X$ on a Banach space $X$ is such that $-A$…
In this article, we show that a technique for showing well-posedness results for evolutionary equations in the sense of [13] established in [16] applies to a broader class of non-autonomous integro-differential-algebraic equations. Using…
We study the obstacle problem for the Evolutionary p-Laplace Equation when the obstacle is discontinuous and without regularity in the time variable. Two quite different procedures yield the same solution.
In studying the time evolution of isolated many-body quantum systems, a key focus is determining whether the system undergoes relaxation and reaches a steady state at a given point in time. Traditional approaches often rely on specific…
We study a class of nonlinear eigenvalue problems which involves a convolution operator as well as a superlinear nonlinearity. Our variational existence proof is based on constrained optimization and provides a one-parameter family of…
In this paper, we discuss time evolution and adiabatic approximation in $PT$-symmetric quantum mechanics. we give the time evolving equation for a class of $PT$-symmetric Hamiltonians and some conditions of the adiabatic approximation for…
We study abstract linear and nonlinear evolutionary systems with single or multiple delay feedbacks, illustrated by several concrete examples. In particular, we assume that the operator associated with the undelayed part of the system…
A new method of determining B\"acklund transformations for nonlinear partial differential equations of the evolution type is introduced. Using the Hilbert space approach the problem of finding B\"acklund transformations is brought down to…
The cyclic evolutions and associated geometric phases induced by time-independent Hamiltonians are studied for the case when the evolution operator becomes the identity (those processes are called {\it evolution loops}). We make a detailed…
The classical and quantum dynamics of simple time-reparametrization- invariant models containing two degrees of freedom are studied in detail. Elimination of one ``clock'' variable through the Hamiltonian constraint leads to a description…
Linear evolution equations are considered usually for the time variable being defined on an interval where typically initial conditions or time-periodicity of solutions are required to single out certain solutions. Here we would like to…