arXiv subjects

Karolina Kropielnicka

Publications and source records attributed to Karolina Kropielnicka.

At least 19 recordsLinked to original sources

arXiv2026

Time-Dependent Logarithmic Perturbation Theory for Quantum Dynamics: Formulation and Applications

We present a time-dependent extension of logarithmic perturbation theory for nonrelativistic quantum dynamics governed by the Schrödinger equation, in which the logarithm of the wave function is expanded in powers of a coupling constant. The resulting hierarchy of equations defining the perturbative corrections is governed by a gauge-rotated Hamiltonian of the unperturbed system and leads to closed-integral expressions for the time-dependent corrections based on Duhamel's formula. This closed-integral structure of corrections is a hallmark of time-independent logarithmic perturbation theory and is preserved in the present extension. This structure, in particular, provides a computable expression for the instantaneous energy shift. Furthermore, dynamic energy shifts arise naturally within this framework in the form of time-averaged expectation values of pseudopotentials and can be related, for example, to AC Stark shifts and electric polarizabilities. As an illustration, we apply the method to the harmonic oscillator and the hydrogen atom, both driven by a time-dependent laser field. The harmonic oscillator provides a proof of principle for which the exact solution is recovered, while the hydrogen atom illustrates the method applied to atomic systems. Supported by numerical simulations, we demonstrate the applicability to obtain relevant physical observables with high accuracy. The present approach offers a promising alternative for analytical studies of time-dependent multi-photon processes in the perturbative regime.

quant-ph↗

arXiv2025

Fourth-order compact exponential splittings for unbounded operators

We present a derivation and error bound for the family of fourth order splittings, originally introduced by Chin and Chen, where one of the operators is unbounded and the second one bounded but time dependent, and which are dependent on a parameter. We first express the error by an iterated application of the Duhamel principle, followed by quadratures of Birkhoff-Hermite type of the underlying multivariate integrals. This leads to error estimates and bounds, derived using Peano/Sard kernels and direct estimates of the leading error term. Our analysis demonstrates that, although no single value of the parameter can minimise simultaneously all error components, an excellent compromise is the cubic Gauss--Legendre point $1/2-\sqrt{15}/10$.

math.NA↗

arXiv2024

Correlations and signaling in the Schrödinger-Newton model

The Schrödinger--Newton model is a semi-classical theory in which, in addition to mutual attraction, massive quantum particles interact with their own gravitational fields. While there are many studies on the phenomenology of single particles, correlation dynamics in multipartite systems is largely unexplored. Here, we show that the Schrödinger--Newton interactions preserve the product form of the initial state of a many-body system, yet on average agreeing with classical mechanics of continuous mass distributions. This leads to a simple test of the model, based on verifying bipartite gravitational evolution towards non-product states. We show using standard quantum mechanics that, with currently accessible single-particle parameters, two masses released from harmonic traps get correlated well before any observable entanglement is accumulated. Therefore, the Schrödinger--Newton model can be tested with setups aimed at observation of gravitational entanglement with significantly relaxed requirements on coherence time. We also present a mixed-state extension of the model that avoids superluminal signaling.

quant-ph↗

arXiv2024

Computation of some dispersive equations through their iterated linearisation

It is often the case that, while the numerical solution of the non-linear dispersive equation $\mathrm{i}\partial_t u(t)=\mathcal{H}(u(t),t)u(t)$ represents a formidable challenge, it is fairly easy and cheap to solve closely related linear equations of the form $\mathrm{i}\partial_t u(t)=\mathcal{H}_1(t)u(t)+\widetilde{\mathcal H}_2(t)u(t)$, where $\mathcal{H}_1(t)+\mathcal{H}_2(v,t)=\mathcal{H}(v,t)$. In that case we advocate an iterative linearisation procedure that involves fixed-point iteration of the latter equation to solve the former. A typical case is when the original problem is a nonlinear Schrödinger or Gross--Pitaevskii equation, while the `easy' equation is linear Schrödinger with time-dependent potential. We analyse in detail the iterative scheme and its practical implementation, prove that each iteration increases the order, derive upper bounds on the speed of convergence and discuss in the case of nonlinear Schrödinger equation with cubic potential the preservation of structural features of the underlying equation: the $\mathrm{L}_2$ norm, momentum and Hamiltonian energy. A key ingredient in our approach is the use of the Magnus expansion in conjunction with Hermite quadratures, which allows effective solutions of the linearised but non-autonomous equations in an iterative fashion. The resulting Magnus--Hermite methods can be combined with a wide range of numerical approximations to the matrix exponential. The paper concludes with a number of numerical experiments, demonstrating the power of the proposed approach.

math.NA↗

arXiv2024

Second-order exponential splittings in the presence of unbounded and time-dependent operators: construction and convergence

For linear differential equations of the form $u'(t)=[A + B(t)] u(t)$, $t\geq0$, with a possibly unbounded operator $A$, we construct and deduce error bounds for two families of second-order exponential splittings. The role of quadratures when integrating the twice-iterated Duhamel's formula is reformulated: we show that their choice defines the structure of the splitting. Furthermore, the reformulation allows us to consider quadratures based on the Birkhoff interpolation to obtain not only pure-stages splittings but also those containing derivatives of $B(t)$ and commutators of $A$ and $B(t)$. In this approach, the construction and error analysis of the splittings are carried out simultaneously. We discuss the accuracy of the members of the families. Numerical experiments are presented to complement the theoretical consideration.

math.NA↗

arXiv2024

Splitting methods for unbounded operators

This paper considers computational methods that split a vector field into three components in the case when both the vector field and the split components might be unbounded. We first employ classical Taylor expansion which, after some algebra, results in an expression for a second-order splitting which, strictly speaking, makes sense only for bounded operators. Next, using an alternative approach, we derive an error expression and an error bound in the same setting which are however valid in the presence of unbounded operators. While the paper itself is concerned with second-order splittings using three components, the method of proof in the presence of unboundedness remains valid (although significantly more complicated) in a more general scenario, which will be the subject of a forthcoming paper.

math.NA↗

arXiv2024

An elementary approach to splittings of unbounded operators

Using elementary means, we derive the three most popular splittings of $e^{(A+B)}$ and their error bounds in the case when $A$ and $B$ are (possibly unbounded) operators in a Hilbert space, generating strongly continuous semigroups, $e^{tA}$, $e^{tB}$ and $e^{t(A+B)}$. The error of these splittings is bounded in terms of the norm of the commutators $[A, B]$, $[A, [A, B]]$ and $[B, [A, B]]$.

math.FA↗

arXiv2023

Asymptotic expansions for the linear PDEs with oscillatory input terms; Analytical form and error analysis

Partial differential equations with highly oscillatory input terms are hardly ever solvable analytically and their numerical treatment is difficult. Modulated Fourier expansion used as an {\it ansatz} is a well known and extensively investigated tool in asymptotic numerical approach for this kind of problems. Although the efficiency of this approach has been recognised, its error analysis has not been investigated rigorously for general forms of linear PDEs. In this paper, we start such kind of investigations for a general form of linear PDEs with an input term characterised by a single high frequency. More precisely we derive an analytical form of such an expansion and provide a formula for the error of its truncation. Theoretical investigations are illustrated by computational simulations.

math.NA↗

arXiv2023

Third order, uniform in low to high oscillatory coefficients, exponential integrators for Klein-Gordon equations

Allowing for space- and time-dependence of mass in Klein--Gordon equations resolves the problem of negative probability density and of violation of Lorenz covariance of interaction in quantum mechanics. Moreover it extends their applicability to the domain of quantum cosmology, where the variation in mass may be accompanied by high oscillations. In this paper we propose a third-order exponential integrator, where the main idea lies in embedding the oscillations triggered by the possibly highly oscillatory component intrinsically into the numerical discretisation. While typically high oscillation requires appropriately small time steps, an application of Filon methods allows implementation with large time steps even in the presence of very high oscillation. This greatly improves the efficiency of the time-stepping algorithm. Proof of the convergence and its rate are nontrivial and require alternative representation of the equation under consideration. We derive careful bounds on the growth of global error in time discretisation and prove that, contrary to standard intuition, the error of time integration does not grow once the frequency of oscillations increases. Several numerical simulations are presented to confirm the theoretical investigations and the robustness of the method in all oscillatory regimes.

math.NA↗

arXiv2023

Effective highly accurate time integrators for linear Klein-Gordon equations across the scales

We propose an efficient approach for time integration of Klein-Gordon equations with highly oscillatory in time input terms. The new methods are highly accurate in the entire range, from slowly varying up to highly oscillatory regimes. Our approach is based on splitting methods tailored to the structure of the input term which allows us to resolve the oscillations in the system uniformly in all frequencies, while the error constant does not grow as the oscillations increase. Numerical experiments highlight our theoretical findings and demonstrate the efficiency of the new schemes.

math.NA↗

arXiv2022

Solving the linear semiclassical Schrödinger equation on the real line

The numerical solution of a linear Schrödinger equation in the semiclassical regime is very well understood in a torus $\mathbb{T}^d$. A raft of modern computational methods are precise and affordable, while conserving energy and resolving high oscillations very well. This, however, is far from the case with regard to its solution in $\mathbb{R}^d$, a setting more suitable for many applications. In this paper we extend the theory of splitting methods to this end. The main idea is to derive the solution using a spectral method from a combination of solutions of the free Schrödinger equation and of linear scalar ordinary differential equations, in a symmetric Zassenhaus splitting method. This necessitates detailed analysis of certain orthonormal spectral bases on the real line and their evolution under the free Schrödinger operator.

math.NA↗

arXiv2021

Efficient Magnus-type integrators for solar energy conversion in Hubbard models

Strongly interacting electrons in solids are generically described by Hubbardtype models, and the impact of solar light can be modeled by an additional time-dependence. This yields a finite dimensional system of ordinary differential equations (ODE)s of Schrödinger type, which can be solved numerically by exponential time integrators of Magnus type. The efficiency may be enhanced by combining these with operator splittings. We will discuss several different approaches of employing exponential-based methods in conjunction with an adaptive Lanczos method for the evaluation of matrix exponentials and compare their accuracy and efficiency. For each integrator, we use defect-based local error estimators to enable adaptive time-stepping. This serves to reliably control the approximation error and reduce the computational effort

math.NA↗

arXiv2020

Time adaptive Zassenhaus splittings for the Schrödinger equation in the semiclassical regime

Time dependent Schrödinger equations with conservative force field U commonly constitute a major challenge in the numerical approximation, especially when they are analysed in the semiclassical regime. Extremely high oscillations originate from the semiclassical parameter, and call for appropriate methods. We propose to employ a combination of asymptotic Zassenhaus splitting with time adaptivity. While the former turns the disadvantage of the semiclassical parameter into an advantage, leading to highly efficient methods with low error constants, the latter enables to choose an optimal time step and to speed up the calculations when the oscillations subside. We support the results with numerical examples.

math.NA↗

arXiv2018

The Escalator Boxcar Train Method for a System of Aged-structured Equations in the Space of Measures

The Escalator Boxcar Train (EBT) method is a well known and widely used numerical method for one-dimensional structured population models of McKendrick-von Foerster type. Recently the method, in its full generality, has been applied to aged-structured two-sex population model (Fredrickson-Hoppensteadt model), which consists of three coupled hyperbolic partial differential equations with nonlocal boundary conditions. We derive the simplified EBT method and prove its convergence to the solution of Fredrickson-Hoppensteadt model. The convergence can be proven, however only if we analyse the whole problem in the space of nonnegative Radon measures equipped with bounded Lipschitz distance (flat metric). Numerical simulations are presented to illustrate the results.

math.NA↗

arXiv2018

Magnus-Zassenhaus methods for the semiclassical Schrödinger equation with oscillatory time-dependent potentials

Schrödinger equations with time-dependent potentials are of central importance in quantum physics and theoretical chemistry, where they aid in the simulation and design of systems and processes at atomic scales. Numerical approximation of these equations is particularly difficult in the semiclassical regime because of the highly oscillatory nature of solution. Highly oscillatory potentials such as lasers compound these difficulties even further. Altogether, these effects render a large number of standard numerical methods less effective in this setting. In this paper we will develop a class of high-order exponential splitting schemes that are able to overcome these challenges by combining the advantages of integral-preserving simplified-commutator Magnus expansions with those of symmetric Zassenhaus splittings. This allows us to use large time steps in our schemes even in the presence of highly oscillatory potentials and solutions.

math.NA↗

arXiv2018

Compact schemes for laser-matter interaction in Schrödinger equation

Numerical solutions for laser-matter interaction in Schrödinger equation has many applications in theoretical chemistry, quantum physics and condensed matter physics. In this paper we introduce a methodology which allows, with a small cost, to extend any fourth-order scheme for Schrödinger equation with time-indepedent potential to a fourth-order method for Schrödinger equation with laser potential. These fourth-order methods improve upon many leading schemes of order six due to their low costs and small error constants.

math.NA↗

arXiv2018

Magnus-Lanczos methods with simplified commutators for the Schrödinger equation with a time-dependent potential

The computation of the Schrödinger equation featuring time-dependent potentials is of great importance in quantum control of atomic and molecular processes. These applications often involve highly oscillatory potentials and require inexpensive but accurate solutions over large spatio-temporal windows. In this work we develop Magnus expansions where commutators have been simplified. Consequently, the exponentiation of these Magnus expansions via Lanczos iterations is significantly cheaper than that for traditional Magnus expansions. At the same time, and unlike most competing methods, we simplify integrals instead of discretising them via quadrature at the outset -- this gives us the flexibility to handle a variety of potentials, being particularly effective in the case of highly oscillatory potentials, where this strategy allows us to consider significantly larger time steps.

math.NA↗

arXiv2016

Efficient methods for time-dependence in semiclassical Schrödinger equations

We build an efficient and unitary (hence stable) method for the solution of the semi-classical Schrödinger equation subject with explicitly time-dependent potentials. The method is based on a combination of the Zassenhaus decomposition (Bader, Iserles, Kropielnicka & Singh 2014) with the Magnus expansion of the time-dependent Hamiltonian. We conclude with numerical experiments.

math.NA↗