Piecewise Constant Potentials Research Articles

Metastable states in a 1D quantum system

One-dimensional (1D) systems in which metastable states exist are physically important, but they are usually not discussed quantitatively in textbooks. This paper presents a relatively simple 1D system involving a piecewise-constant potential for which metastable states can be easily calculated using a computer algebraic system. The metastable-state wave functions are computed and plotted for various particle energies. The Breit–Wigner approximation is used to fit the resulting resonant structure in the transmission coefficient. Connections are made between the initial analysis and poles of the scattering matrix that corresponds to the potential.

Analysing Pine Disease Spread Using Random Point Process by Remote Sensing of a Forest Stand

The application of a process model to investigate pine tree infestation caused by bark beetles is discussed. The analysis of this disease was carried out using spatial and spatio−temporal models of random point patterns. Spatial point patterns were constructed for remote sensing images of pine trees damaged by the apical bark beetle. The method of random point processes was used for their analysis. A number of known models of point pattern processes with pairwise interaction were fitted to actual data. The best model to describe the real data was chosen using the Akaike information index. The residual K−function was used to check the fit of the model to the real data. According to values of the Akaike information criterion and the residual K−function, two models were found to correspond best to the investigated data. These are the generalized Geyer model of the point process of saturation and the pair interaction process with the piecewise constant potential of a pair of points. For the first time, a spatio−temporal model of the contagious process was used for analysis of tree damage.

Numerical solution of an extra-wide angle parabolic equation through diagonalization of a 1-D indefinite Schrödinger operator with a piecewise constant potential

We present a numerical method for computing the solution of a partial differential equation (PDE) for modeling acoustic pressure, known as an extra-wide angle parabolic equation, that features the square root of a differential operator. The differential operator is the negative of an indefinite Schrödinger operator with a piecewise constant potential. This work primarily deals with the 3-piece case; however, a generalization is made the case of an arbitrary number of pieces. Through restriction to a judiciously chosen lower-dimensional subspace, approximate eigenfunctions are used to obtain estimates for the eigenvalues of the operator. Then, the estimated eigenvalues are used as initial guesses for the Secant Method to find the exact eigenvalues, up to roundoff error. An eigenfunction expansion of the solution is then constructed. The computational expense of obtaining each eigenpair is independent of the grid size. The accuracy, efficiency, and scalability of this method is shown through numerical experiments and comparisons with other methods.

Inverse problems on low-dimensional manifolds

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finite-dimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and Hölder and Lipschitz stability results in this general setting, also in the case when only a finite discretization of the measurements is available. Then, a Landweber-type reconstruction algorithm from a finite number of measurements is proposed, for which we prove global convergence, thanks to a new criterion for finding a suitable initial guess.These general results are then applied to several examples, including two classical nonlinear ill-posed inverse boundary value problems. The first is Calderón’s inverse conductivity problem, for which we prove a Lipschitz stability estimate from a finite number of measurements for piece-wise constant conductivities with discontinuities on an unknown triangle. A similar stability result is then obtained for Gel’fand-Calderón’s problem for the Schrödinger equation, in the case of piece-wise constant potentials with discontinuities on a finite number of non-intersecting balls.

A trace formula for metric graphs with piecewise constant potentials and multi-mode graphs

We generalize the scattering approach to quantum graphs to quantum graphs with piecewise constant potentials and multiple excitation modes. The free single-mode case is well-known and leads to the trace formulas of Roth (1983 C. R. Acad. Sci., Paris I 296 793–5), Kottos and Smilansky (1997 Phys. Rev. Lett. 79 4794). By introducing an effective reduced scattering picture we are able to propose new exact trace formulas in the more general settings. The latter are derived and discussed in details with some numerical examples for illustration. Our generalization is motivated by both experimental applications and fundamental theoretical considerations. The free single-mode quantum graphs are an extreme idealization of reality that, due to the simplicity of the model allows to understand a large number of generic or universal phenomena. We lift some of this idealization by considering the influence of evanescent modes that only open above threshold energies. How to do this theoretically in a closed model in general is a challenging question of fundamental theoretical interest and we achieve this here for quantum graphs.

A Space-Time Trefftz Discontinuous Galerkin Method for the Linear Schrödinger Equation

A space-time Trefftz discontinuous Galerkin method for the Schr\"odinger equation with piecewise-constant potential is proposed and analyzed. Following the spirit of Trefftz methods, trial and test spaces are spanned by non-polynomial complex wave functions that satisfy the Schro\"odinger equation locally on each element of the space-time mesh. This allows for a significant reduction in the number of degrees of freedom in comparison with full polynomial spaces. We prove well-posedness and stability of the method, and, for the one- and two- dimensional cases, optimal, high-order, h-convergence error estimates in a skeleton norm. Some numerical experiments validate the theoretical results presented.

Conditions for realizing one-point interactions from a multi-layer structure model

A heterostructure composed of N parallel homogeneous layers is studied in the limit as their widths l 1, …, l N shrink to zero. The problem is investigated in one dimension and the piecewise constant potential in the Schrödinger equation is given by the strengths V 1, …, V N as functions of l 1, …, l N , respectively. The key point is the derivation of the conditions on the functions V 1(l 1), …, V N (l N ) for realizing a family of one-point interactions as l 1, …, l N tend to zero along available paths in the N-dimensional space. The existence of equations for a squeezed structure, the solution of which determines the system parameter values, under which the non-zero tunneling of quantum particles through a multi-layer structure occurs, is shown to exist and depend on the paths. This tunneling appears as a result of an appropriate cancellation of divergences.

Anomalous thermal relaxation of Langevin particles in a piecewise-constant potential

We consider the thermal relaxation of a particle in a piecewise-constant potential landscape, subject to thermal fluctuations in the overdamped limit. We study the connection between the occurrence of the Mpemba effect, the presence of metastable states, and phase transitions as a function of the potential. We find that the Mpemba effect exists even in cases without metastable states. In the considered physical system, the borders of the areas where the effect happens correspond to either eigenvector changes of direction or to phase transitions. Finally, we discuss the topological aspects of the strong Mpemba effect and propose using topology to search for the Mpemba effect in a physical system.

“Striped” rectangular rigid box with Hermitian and non-Hermitian PT symmetric potentials

Eigenspectra of a spinless particle inside a rigid rectangular box subject to diverse inner potential distributions are investigated under both Hermitian and non-Hermitian antiunitary PT (composite parity and time-reversal) symmetric regimes. Four “stripes” conjoined widthwise, spanning the box bearing piecewise constant potentials, are studied whose exact energy eigenspectra are obtained employing matrix mechanics. Diverse real-Hermitian and complex non-Hermitian PT symmetric potential compositions are considered separately and in conjunction, unraveling peculiar retention and breakdown scenarios engendered by PT symmetry. Some states exhibit remarkable crossovers of symmetry “making” and “breaking”: a broken PT gets reinstated, while higher levels “collude” to continue with symmetry breaking. Furthermore, a charged particle in a PT symmetric electric field imposed on the striped potential backdrop reveals peculiar symmetry retention and breakdown scenarios. Depictions of prominent probability-density redistributions under the norm conserving unitary regime, as well as non-conserving post-PT-breakdown, are presented.

Solving Klein’s paradox

We figure out the famous Klein’s paradox arising from the reflection problem when a Dirac particle encounters a step potential with infinite width. The key is to piecewise solve Dirac equation in such a way that in the region where the particle’s energy E is greater (less) than the potential V, the solution of the positive (negative) energy branch is adopted. In the case of Klein–Gordon equation with a piecewise constant potential, the equation is decoupled to positive and negative energy equations, and reflection problem is solved in the same way. Both infinitely and finitely wide potentials are considered. The reflection coefficient never exceeds 1. The results are applied to discuss the transmissions of particles with no mass or with very small mass.

Quasi-relativistic description of a quantum particle moving through one-dimensional piecewise constant potentials

Using a novel quasi-relativistic wave equation, which can give precise results up to energies ~mc2, exact quantum mechanical solutions are found which corresponds to a particle with mass moving through one-dimensional piecewise constant potentials. As expected, at low particle’s speeds, the found solutions coincide with the solutions of the same problems calculated using the Schrödinger equation; however, as it should be, both solutions have a significative difference at quasi-relativistic speeds. Then, it is argued that the quasi-relativistic wave equation provides a simpler description than a fully relativistic theory or the perturbation approach for a quantum particle moving at quasi-relativistic energies through piecewise constant potentials.

The Dirac Propagator for One-Dimensional Finite Square Well

The solution of Dirac particles confined in a one-dimensional finite square well potential is solved by using the path-integral formalism for Dirac equation. The propagator of the Dirac equation in case of the bounded Dirac particles is obtained by evaluating an appropriate path integral, directly constructed from the Dirac equation. The limit of integration techniques for evaluating path integral is only valid for the piecewise constant potential. Finally, the Dirac propagator is expressed in terms of standard special functions.

Efficient simulation of the Schrödinger equation with a piecewise constant positive potential

We introduce a new method for the Monte Carlo simulation of a weak solution of the Schrödinger-type equation where the potential is piecewise constant and positive. The method, called the killing walk-on-spheres algorithm, combines the classical walk-on-spheres algorithm with killing that can be determined by using panharmonic measures. This paper continues our earlier work in which simulation of the solutions of the Yukawa and the Helmholtz partial differential equations were developed.

CALDERÓN’S INVERSE PROBLEM WITH A FINITE NUMBER OF MEASUREMENTS

We prove that an$L^{\infty }$potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace${\mathcal{W}}$. As a corollary, we obtain a similar result for Calderón’s inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces${\mathcal{W}}$, including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of$\dim {\mathcal{W}}$.

The time-dependent Schrödinger equation with piecewise constant potentials

The linear Schrödinger equation with piecewise constant potential in one spatial dimension is a well-studied textbook problem. It is one of only a few solvable models in quantum mechanics and shares many qualitative features with physically important models. In examples such as ‘particle in a box’ and tunnelling, attention is restricted to the time-independent Schrödinger equation. This paper combines the unified transform method and recent insights for interface problems to present fully explicit solutions for the time-dependent problem.

Two-body loss rates for reactive collisions of cold atoms

We present an effective two-channel model for reactive collisions of cold atoms. It augments elastic molecular channels with an irreversible, inelastic loss channel. Scattering is studied with the distorted-wave Born approximation and yields general expressions for angular momentum resolved cross sections as well as two-body loss rates. Explicit expressions are obtained for piece-wise constant potentials. A pole expansion reveals simple universal shape functions for cross sections and two-body loss rates in agreement with the Wigner threshold laws. This is applied to collisions of metastable ${^{20}}$Ne -- ${^{21}}$Ne atoms, which decay primarily through exothermic Penning-, or Associative- ionization processes. From a numerical solution of the multichannel Schr\"odinger equation using the best currently available molecular potentials, we have obtained synthetic scattering data. Using the two-body loss shape functions derived in this paper, we can match these scattering data very well.

Algorithm for Numerical Solution of Diffraction Problem on the Joint of Two Open Three-Layer Waveguides

This paper describes the algorithm for the numerical solution of the diffraction problem of waveguide modes at the joint of two open planar waveguides. For planar structures under consideration, we can formulate a scalar diffraction problem, which is a boundary value problem for the Helmholtz equation with a variable coefficient in two-dimensional space. The eigenmode problem for an open three-layer waveguide is the Sturm-Liouville problem for a second-order operator with piecewise constant potential on the axis, where the potential is proportional to the refractive index. The described problem is singular and has a mixed spectrum and therefore the Galerkin method can not be used in this definition. One way to adapt the Galerkin method for the problem solution is to artificially limit the area, which is equivalent to placing the open waveguide in question in a hollow closed waveguide whose boundaries are remote from the real boundaries of the waveguide layer of the open waveguide. Thus, we obtain a diffraction problem on a finite interval and with a discrete spectrum, which can be solved by the projection method, as done in this paper.

Helicity-dependent scattering in layered Weyl semimetals

Weyl fermions are characterized by definite helicity, a relation between orientation of the spin and momentum. We investigate controlling helicity utilizing helicity-dependent dynamics due to separation between the Weyl points. We study transport properties of a multilayer structure formed by piece-wise constant scalar and vector potentials and separations between the Weyl points. We show that the transmission spectrum is determined by a relation between the electron momentum and momenta corresponding to Klein tunneling, when the barrier is fully transparent regardless of its width. The helicity dependence of the “Klein momenta” is shown to yield a helicity filtering effect.

<em>l</em><sup>1</sup>-Error Estimates on the Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials: A Simple Proof

<em>l</em><sup>1</sup>-Error Estimates on the Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials: A Simple Proof

Solving the time-dependent Schrödinger equation via Laplace transform

We show how the Laplace transform can be used to give a solution of the time-dependent Schrodinger equation for an arbitrary initial wave packet if the solution of the stationary equation is known. The solution is obtained without summing up eigenstates nor do we need the path integral. We solve the initial value problem for three characteristic piecewise constant potentials. The results give an intuitive picture of the wave packet dynamics, reproducing explicitly all possible reflection and transmission processes. We investigate classical and quantum properties of the evolution and determine the reflection and transmission probabilities.