Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Abstract

Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.

Coupled Multiphysics Modelling of Sensors for Chemical, Biomedical, and Environmental Applications with Focus on Smart Materials and Low-Dimensional Nanostructures

“…At the same time, it is often more convenient to use a generalized form of the quantum Liouville equation where the Hamiltonian operator: is split into two Hamiltonians describing the nuclear motion for the lower and upper electronic states, as: where and are the effective potentials of the nuclei in the lower and upper electronic states, respectively, and T is the kinetic energy. Among other applications, this generalized approach is important for calculating molecular (vibrational and electronic) spectra (e.g., [ 9 ]), and more generally, for a diverse range of problems involving systems at finite temperatures, open quantum systems, and dissipative environments [ 10 , 11 ]. Taking into account (2)–(4), the generalized quantum Liouville equation then becomes: …”

“…Subsequently, among the most common semi-empirical numerical approaches, the method has been demonstrated to be robust, reliable, and computationally efficient, in particular for low-dimensional semiconductor nanostructures [ 30 , 40 , 41 ]. The construction of multiband Hamiltonians for this method in solving coupled multiscale problems is closely interwoven with some of the fundamental questions of quantum mechanics and mathematical physics [ 11 ] which are becoming increasingly important for the development of data-driven models.…”

Nonlocal Models in the Analysis of Brain Neurodegenerative Protein Dynamics with Application to Alzheimer's Disease

“…where the first two equations correspond to the healthy and toxic variants of the protein Aβ and the last two equations play the same role for τP. Model (1) is an example of nonlocal models which become increasingly important in diverse areas of applications 32 . Here, a 0 and b 0 are the mean production rates of healthy proteins, a 1 , b 1 , a 1 and b 1 are the mean clearance rates of healthy and toxic proteins, and a 2 and b 2 represent the mean conversion rates of healthy proteins to toxic proteins.…”

Quantum mechanics with spatial non-local effects and position-dependent mass

“…Therefore, it is reasonable to investigate and develop a new methodology taking into account non-local effects without using fractional calculus. We stress that non-local mathematical models are not limited to quantum mechanics but, as stated previously, to various fields of research including biomedical, fractures and damages of materials, cosmological and astrophysical sciences, among many other fields where irregular, non-differentiable, singular and discontinuous solutions arise, and where spatial non-locality emerges naturally (see [62–65] and the references therein). Let us recall that non-locality in quantum mechanics may arise in the form of convolution integrals with kernels that generally put across the interaction of spatially distant particles with other particles.…”