Search results for "Mathematica"
showing 10 items of 7971 documents
Quantum Toda Lattice: a Challenge for Representation Theory
2021
Quantum Toda lattice may solved by means of the Representation Theory of semisimple Lie groups, or alternatively by using the technique of the Quantum Inverse Scattering Method. A comparison of the two approaches, which is the purpose of the present review article, sheds a new light on Representation Theory and leads to a number of challenging questions.
Free boundary methods and non-scattering phenomena
2021
We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from t…
Perfectly matched layers for the stationary Schrodinger equation in a periodic structure
2008
We construct a perfectly matched absorbing layer for stationary Schrodinger equation with analytic slowly decaying potential in a periodic structure. We prove the unique solvability of the problem with perfectly matched layer of finite length and show that solution to this problem approximates a solution to the original problem with an error that exponentially tends to zero as the length of perfectly matched layer tends to infinity.
Coherent Quantum Tomography
2016
We discuss a quantum mechanical indirect measurement method to recover a position dependent Hamilton matrix from time evolution of coherent quantum mechanical states through an object. A mathematical formulation of this inverse problem leads to weighted X-ray transforms where the weight is a matrix. We show that such X-ray transforms are injective with very rough weights. Consequently, we can solve our quantum mechanical inverse problem in several settings, but many physically relevant problems we pose also remain open. We discuss the physical background of the proposed imaging method in detail. We give a rigorous mathematical treatment of a neutrino tomography method that has been previous…
Free boundary methods and non-scattering phenomena
2021
We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from t…
Robustness of Coherence: An Operational and Observable Measure of Quantum Coherence
2016
Quantifying coherence is an essential endeavour for both quantum foundations and quantum technologies. Here the robustness of coherence is defined and proven a full monotone in the context of the recently introduced resource theories of quantum coherence. The measure is shown to be observable, as it can be recast as the expectation value of a coherence witness operator for any quantum state. The robustness of coherence is evaluated analytically on relevant classes of states, and an efficient semidefinite program that computes it on general states is given. An operational interpretation is finally provided: the robustness of coherence quantifies the advantage enabled by a quantum state in a …
A new approach to fuzzy sets: Application to the design of nonlinear time-series, symmetry-breaking patterns, and non-sinusoidal limit-cycle oscillat…
2017
It is shown that characteristic functions of sets can be made fuzzy by means of the $\mathcal{B}_{\kappa}$-function, recently introduced by the author, where the fuzziness parameter $\kappa \in \mathbb{R}$ controls how much a fuzzy set deviates from the crisp set obtained in the limit $\kappa \to 0$. As applications, we present first a general expression for a switching function that may be of interest in electrical engineering and in the design of nonlinear time-series. We then introduce another general expression that allows wallpaper and frieze patterns for every possible planar symmetry group (besides patterns typical of quasicrystals) to be designed. We show how the fuzziness parameter…
Reply to Comment on "A no-go result for the quantum damped harmonic oscillator"
2019
In a recent paper, \cite{deguchi}, Deguchi and Fujiwara claim that our results in \cite{BGR} are wrong, and compute what they claim is the square integrable vacuum of their annihilation operators. In this brief note, we show that their vacuum is indeed not a vacuum, and we try to explain what is behind their mistake. We also consider a very simple example clarifying the core of the problem.
The complex Dirac Delta, Plemelj formula, and integral representations
2016
The extension of the Dirac Delta distribution (DD) to the complex field is needed for dealing with the complex-energy solutions of the Schr\"odinger equation, typically when calculating their inner products. In quantum scattering theory the DD usually arises as an integral representation involving plane waves of real momenta. We deal with the complex extension of these representations by using a Gaussian regularization. Their interpretation as distributions requires prescribing the integration path and a corresponding space of test functions. An extension of the Sokhotski-Plemelj formula is obtained. This definition of distributions is alternative to the historic one referred to surface int…
Exact, explicit and entire solutions to a nontrivial finite-difference equation and their applications
2003
Below, the explicit solution to a certain finite-difference equation is given and the required steps for derivation of these results are outlined. Everything is included as Mathematica formulae, so the notebook itself can be used for checking and improving the present results. Some important references for justifying some steps and crosschecking certain results have been included. Full references and derivations will be made available shortly. It should be noted that several applications for the solutions have been included at the end of the document. These include at least diagonalisation of certain infinite matrices, definition of isospectral operators with simple eigenvalues and alternat…