Boundary conditions for quantum mechanics on cones and fields around cosmic strings

In recent years, there has been an increase in the application of quantum mechanics (QM) methods to describe properties related to the ADMET profile of small molecules. The application of these methods allows calculating useful descriptors and physiochemical properties contributing to ADMET prediction. Considering that QM methods are the only one that describe the electronic state of a molecules, such methods are particularly useful for studying the metabolism of drugs; furthermore, the introduction of mixed QM and molecular mechanics (QM/MM) is also increasing the understanding of drug interaction with cytochromes from a mechanistic point of view. Finally, combining the increase number of experimental data with machine learning algorithms and QM-derived descriptors allowed the creation of an end-user software capable of affecting the drug discovery process.

CONCEPTUAL FOUNDATIONS OF QUANTUM MECHANICS: THE ROLE OF EVIDENCE THEORY, QUANTUM SETS, AND MODAL LOGIC

In this chapter, we turn our attention to applications of the Weierstrass elliptic and related functions in Quantum Mechanics. It is well-known that Bloch’s theorem implies that periodic potentials have in general a non-trivial band structure. However, it is not simple to find analytically solvable periodic potentials that demonstrate this behaviour. In this chapter, we focus on the n \(=\) 1 Lamé potential, which is a one-dimensional periodic potential that can be expressed in terms of the Weierstrass elliptic function. Its band structure can be analytically specified, using the properties of the Weierstrass functions.

In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to τ-Wigner representations in [11].In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices A∈Sp(2d,R). We find conditions on A for which the related symplectic time-frequency representation WA can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices A, i.e., their corresponding WA are members of the Cohen class.Finally, we show that symplectic time-frequency representations WA can be efficiently employed in the study of Schrödinger equations. In fact, modulation spaces and WA representations are the frame for a new definition of wave front set, providing a sharp result for propagation of micro-singularities in the case of the quadratic Hamiltonians. This new approach may have further applications in quantum mechanics and PDE's.

In certain problems in model building and Bayesian analysis, the results end up in forms connected with generalized zeta functions. This necessitates the introduction of an extended form of the generalized zeta function. Such an extended form of the zeta function is introduced in this paper. In model building situations and in various types of applications in physical, biological and social sciences and engineering, a basic model taken is the Gaussian model in the univariate, multivariate and matrix-variate situations. A real scalar variable logistic model behaves like a Gaussian model but with a thicker tail. Hence, for many of industrial applications, a logistic model is preferred to a Gaussian model. When we study the properties of a logistic model in the multivariate and matrix-variate cases, in the real and complex domains, invariably the problem ends up in the extended zeta function defined in this paper. Several such extended logistic models are considered. It is also found that certain Bayesian considerations also end up in the extended zeta function introduced in this paper. Several such Bayesian models in the multivariate and matrix-variate cases in the real and complex domains are discussed. It is stated in a recent paper that “Quantum Mechanics is just the Bayesian theory generalized to the complex Hilbert space”. Hence, the models developed in this paper are expected to have applications in quantum mechanics, communication theory, physics, statistics and related areas.

Quantum mechanics is usually defined in terms of some loosely connected axioms and rules. Such a foundation is far from the beauty of, e.g., the `principles' underlying classical mechanics. Motivated, in addition, by notorious interpretation problems, there have been numerous attempts to modify or `complete' quantum mechanics. A first attempt was based on so-called hidden variables; its proponents essentially tried to expel the non-classical nature of quantum mechanics. More recent proposals intend to complete quantum mechanics not within mechanics proper but on a `higher (synthetic) level'; by means of a combination with gravitation theory (R Penrose), with quantum information theory (C M Caves, C A Fuchs) or with psychology and brain science (H P Stapp). I think it is fair to say that in each case the combination is with a subject that, per se, suffers from a very limited understanding that is even more severe than that of quantum mechanics. This was acceptable, though, if it could convincingly be argued that scientific progress desperately needs to join forces.

In this article we consider partial abelianization of associative algebra with respect to some fixed subalgebra. This notion is a generalization of usual abelianization of an associative algebra and has some applications in Quantum Mechanics and Quantum Information Theory. Using various methods, we study partial abelianization of free product of finite-dimensional algebras.

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.

Monte-Carlo methods is the generic term given to numerical methods that use sampling of random numbers. This text is aimed at graduate students in mathematics, physics, engineering, economics, finance, and the biosciences that are interested in using Monte-Carlo methods for the resolution of partial differential equations, transport equations, the Boltzmann equation and the parabolic equations of diffusion. It includes applied examples, particularly in mathematical finance, along with discussion of the limits of the methods and description of specific techniques used in practice for each example. This is the sixth volume in the Oxford Texts in Applied and Engineering Mathematics series, which includes texts based on taught courses that explain the mathematical or computational techniques required for the resolution of fundamental applied problems, from the undergraduate through to the graduate level. Other books in the series include: Jordan & Smith: Nonlinear Ordinary Differential Equations: An introduction to Dynamical Systems; Sobey: Introduction to Interactive Boundary Layer Theory; Scott: Nonlinear Science: Emergence and Dynamics of Coherent Structures; Tayler: Mathematical Models in Applied Mechanics; Ram-Mohan: Finite Element and Boundary Element Applications in Quantum Mechanics; Elishakoff and Ren: Finite Element Methods for Structures with Large Stochastic Variations.

The continuum calculus, proposed earlier [J. Math. Phys. 17, 1988 (1976)], is here applied to the development of a functional version of Laplace transform and a method for the evaluation of path integrals containing a Gaussian-like measure. Two methods in functional integration are proposed. First, the Gaussian integral for polynomial functionals and consequently functionals that can be expanded in Taylor series are examined. Formula (2.12) is derived. Next, we define the Laplace transform in the function space through the weak distribution formulation of Skorohod. Comparison of both approaches enables us to determine the expression of the functional integeral through a series of Laplace transformations. The second formula is given in Eq. (4.5). The latter formula is applicable to all Laplace transformable functionals. As an illustration of the utility of the formulas derived, we evaluate the integral of a cosine functional by methods 1 and 2, and obtain consistent results. Further applications to quantum mechanics are also presented. We examine the cases of a free particle,the quantum harmonic oscillator, the forced oscillator, and charged particle in a magnetic field. In all cases, we obtain correct results in comparison with known expressions. A numerical procedure is employed in the calculation of infinite products. The usefulness of the p-integral method is stressed.

This study examines the intricate interplay between Hamiltonian dynamics and spectral analysis within the context of algebraic structures and symmetry groups. By integrating perspectives from symplectic geometry, Lie theory, and nonlinear dynamics, we introduce a framework that aims to unify classical conservation laws with advanced spectral decomposition techniques. Our approach synthesizes geometric mechanics with adaptive numerical strategies, revealing novel relationships between phase-space invariants and multi-scale Fourier analysis. Applications in quantum mechanics, signal processing, and machine learning illustrate the potential of Hamiltonian-preserving algorithms to maintain spectral integrity while ensuring long-term stability. These findings assist in detecting chaotic transitions and demonstrate compatibility with established FFT methodologies, thereby offering new insights into the synergy between geometric structures and spectral methods.

The intermolecular field of hydrogen and deuterium and the quantum mechanical theory of the equation of state

The nonlinear conformable fractional Schrödinger–Hirota (NCFSH) equation, describing the propagation of optical solitons in a dispersive optical fibre, is considered and its exact solutions are investigated. Abundant solutions including dark, bright and singular solutions have been obtained by two distinct methods, namely direct method and trial equation method. These methods provide a direct and efficient strategy for the NCFSH equation. The solutions have been simulated by graphs to check the types of solitons and their dynamical behaviours. With back substitution using Mathematica, all the solutions have been verified. Obtaining optical solutions to the proposed equation will play a fundamental role due to the applications in quantum mechanics, fluid dynamics, nonlinear optics and various other branches of mathematical physics.

Nonlinear systems, including atom–field interaction, are investigated due to their fundamental applications in quantum mechanics and rapidly growing fields of quantum communication, especially secure communication with chaotic dynamics. In this paper, we study the chaotic dynamics of a system consisting of an atomic Bose–Einstein condensate interacting with a quantized radiation field in a high-quality cavity with a periodically modulated length. The frequency modulation is adapted by a periodic time-dependent atom–field coupling strength. We use a semiclassical approach to decouple the atoms and field variables and then numerically solve the corresponding nonlinear dynamical equations of the system. Generally, the dynamics of the system sensitively depends on its initial conditions, thereby long-term prediction is impossible. We show that the system demonstrates the emergence of classical dynamical chaos from quantum electrodynamics. The chaotic behavior of energy transfer in the system can be enhanced by increasing the depth of frequency modulation. The strange attractor clearly illustrates that the system presents extremely exotic dynamics over a wide range of parameters. This implies that the dynamical quantities oscillate irregularly, never exactly repeating but always remaining in a bounded region of the phase space.

In this chapter we present an introduction to the theory of inner product spaces over the field of quaternions. As is known, the discovery of quaternions is attributed to the great Irish mathematician R. Hamilton*. In recent years the interest in spaces over the quaternions has increased due in part to their possible application in quantum mechanics. We shall present some facts concerning linear spaces over Q and linear operators on such spaces (Q will be used to denote the set of quaternions). Other topics are very briefly mentioned.