Search Results (9)

This article reviews several numerical methods for the time-dependent Schrödinger Equation (TDSE). We consider both the most commonly used approach—short-time propagation, which solves the TDSE by assuming that the Hamiltonian is time-independent over sufficiently small (time) intervals—as well as a number of higher-order alternatives. Our goal is to dispel the notion that the latter are too computationally demanding for practical use. To that end, we cover methods whose numerical building blocks are shared by short-time propagators or can be handled by standard libraries. Moreover, we make the case that these methods are best positioned to take advantage of parallel computing environments. One of the alternatives considered is a “double DVR” solver, which applies an expansion in a product basis of functions in space and time to obtain a solution (over all space and at multiple time points simultaneously) with a single linear system solve. To our knowledge, and despite its simplicity, this approach has not previously been applied to the TDSE. Full article

Background: Injuries of the proximal attachment of the hamstring muscles are common. The present study aimed to investigate the relationship of the proximal attachment of the hamstring muscles with neighboring structures comprehensively. Methods: A total of 97 hemipelvis from 66 cryopreserved specimens were evaluated via ultrasound, anatomical and histological samples. Results: The proximal attachment of the hamstring muscles presents a hyperechogenic line surrounding the origin of the semimembranosus and the long head of the biceps femoris muscles, as well as another hyperechogenic line covering the sciatic nerve. The anatomical and histological study confirms the ultrasound results and shows different layers forming the sacrotuberous ligament. Furthermore, it shows that the proximal attachment of the semimembranosus muscle has a more proximal origin than the rest of the hamstring muscles. Moreover, this muscle shares fibers with the long head of the biceps femoris muscle and expands to the adductor magnus muscle. The histological analysis also shows the dense connective tissue of the retinaculum covering the long head of the biceps femoris and semimembranosus muscles, as well as the expansion covering the sciatic nerve. Conclusions: These anatomical relationships could explain injuries at the origin of the hamstring muscles. Full article

A Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. This superposition rule can be obtained using the geometric features of the Lie system, its symmetries, and the symmetric properties of certain morphisms involved. Even if a superposition rule for a Lie system is known, the explicit analytic expression of its solutions frequently is not. This is why this article focuses on a novel geometric attempt to integrate Lie systems analytically and numerically. We focus on two families of methods based on Magnus expansions and on Runge–Kutta–Munthe–Kaas methods, which are here adapted, in a geometric manner, to Lie systems. To illustrate the accuracy of our techniques we analyze Lie systems related to Lie groups of the form SL$(n,\mathbb{R})$, which play a very relevant role in mechanics. In particular, we depict an optimal control problem for a vehicle with quadratic cost function. Particular numerical solutions of the studied examples are given. Full article

Quantum state processing is one of the main tools of quantum technologies. While real systems are complicated and/or may be driven by non-ideal control, they may nevertheless exhibit simple dynamics approximately confined to a low-energy Hilbert subspace. Adiabatic elimination is the simplest approximation scheme allowing us to derive in certain cases an effective Hamiltonian operating in a low-dimensional Hilbert subspace. However, these approximations may present ambiguities and difficulties, hindering a systematic improvement of their accuracy in larger and larger systems. Here, we use the Magnus expansion as a systematic tool to derive ambiguity-free effective Hamiltonians. We show that the validity of the approximations ultimately leverages only on a proper coarse-graining in time of the exact dynamics. We validate the accuracy of the obtained effective Hamiltonians with suitably tailored fidelities of quantum operations. Full article

This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm–Liouville problems. Next, a concrete implementation to the inverse Sturm–Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm–Liouville problems of higher order (for $n=2,4$) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm–Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions. Full article

We propose a unified approach for different exponential perturbation techniques used in the treatment of time-dependent quantum mechanical problems, namely the Magnus expansion, the Floquet–Magnus expansion for periodic systems, the quantum averaging technique, and the Lie–Deprit perturbative algorithms. Even the standard perturbation theory fits in this framework. The approach is based on carrying out an appropriate change of coordinates (or picture) in each case, and it can be formulated for any time-dependent linear system of ordinary differential equations. All of the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation. Full article

In this paper, based on the iterative technique, a new explicit Magnus expansion is proposed for the nonlinear stochastic equation $dy=A(t,y)ydt+B(t,y)y\circ dW.$ One of the most important features of the explicit Magnus method is that it can preserve the positivity of the solution for the above stochastic differential equation. We study the explicit Magnus method in which the drift term only satisfies the one-sided Lipschitz condition, and discuss the numerical truncated algorithms. Numerical simulation results are also given to support the theoretical predictions. Full article

In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively. Full article

The aim of the study was to evaluate the activity and the validity of the expansion of the Clinic of Ophthalmology of the University of Lithuania (since 1930 – Vytautas Magnus University) operating in Kaunas city between 1922 and 1938. The evaluation was based on the analysis of changes in inpatient and outpatient flow, the structure of cases of inpatient treatment, and the usage of beds. In the analysis, we used annual reports of the Clinic of Ophthalmology as well as data presented in statistical publications of the Department of Health for the studied period. The changes in the indices of the activity of the Clinic were evaluated using the logarithmic regression coefficient. A more rapid increase in the number of patients discharged from the Clinic of Ophthalmology was observed during 1922–1930 (on the average, by 9% per year). During 1931–1938, only the number of discharged men was increasing. During the studied period, the majority of the cases of inpatient treatment were lenticular diseases (19%), trachoma (16%), and corneal diseases (16%). During 1922–1930, the sharpest increase was observed in the number of inpatients with eyeball diseases and eye traumas (on the average, by 12.3% per year) and during 1931–1938, in the number of patients with trachoma (on the average, by 6.7% per year). The analysis of the indices of the activity of the inpatient unit confirmed the need for the expansion of the Clinic during 1922–1930, but revealed that the expansion of the material basis of the Clinic up to 50 beds during 1931–1938 was not efficient. In the outpatient unit of the Clinic of Ophthalmology, the number of visits per year and the number of admitted patients per year during the studied period increased by 2.5 and 3.5 times, respectively. Full article