The Weyl–Mellin quantization map

2021 ◽  
Author(s):  
Alessandro Carotenuto ◽  
Fedele Lizzi ◽  
Flavio Mercati ◽  
Mattia Manfredonia
Keyword(s):  
Mellin Transform ◽  
Commutation Relations ◽  
Integrable Functions ◽  
Real Functions ◽  
Symmetric Operators ◽  
Square Integrable Functions

In this paper, we present a quantization of the functions of spacetime, i.e. a map, analog to Weyl map, which reproduces the [Formula: see text]-Minkowski commutation relations, and it has the desirable properties of mapping square integrable functions into Hilbert–Schmidt operators, as well as real functions into symmetric operators. The map is based on Mellin transform on radial and time coordinates. The map also defines a deformed ∗ product which we discuss with examples.

scholarly journals On Z -Invariant Self-Adjoint Extensions of the Laplacian on Quantum Circuits

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1047
Author(s):  
Aitor Balmaseda ◽  
Fabio Di Cosmo ◽  
Juan Manuel Pérez-Pardo
Keyword(s):  
Quantum Circuit ◽  
Beltrami Operator ◽  
Quantum Circuits ◽  
Invariance Properties ◽  
Integrable Functions ◽  
Symmetric Operators ◽  
Square Integrable Functions ◽  
Infinite Set ◽  
Group Of Symmetries

An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group G, criteria for the existence of G-invariant self-adjoint extensions of the Laplace–Beltrami operator over a Riemannian manifold are illustrated and critically revisited. These criteria are employed for characterising self-adjoint extensions of the Laplace–Beltrami operator on an infinite set of intervals, Ω , constituting a quantum circuit, which are invariant under a given action of the group Z . A study of the different unitary representations of the group Z on the space of square integrable functions on Ω is performed and the corresponding Z -invariant self-adjoint extensions of the Laplace–Beltrami operator are introduced. The study and characterisation of the invariance properties allows for the determination of the spectrum and generalised eigenfunctions in particular examples.

p-ADIC HILBERT SPACE REPRESENTATION OF QUANTUM SYSTEMS WITH AN INFINITE NUMBER OF DEGREES OF FREEDOM

1996 ◽  
Vol 10 (13n14) ◽  
pp. 1665-1673 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
ANDREW KHRENNIKOV
Keyword(s):  
Hilbert Space ◽  
Infinite Number ◽  
Degrees Of Freedom ◽  
Quantum Systems ◽  
Space Representation ◽  
Gaussian Measures ◽  
Infinite Dimensional ◽  
Integrable Functions ◽  
Symmetric Operators ◽  
Square Integrable Functions

Gaussian measures on infinite-dimensional p-adic spaces are introduced and the corresponding L2-spaces of p-adic valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in p-adic L2-spaces. p-adic Hilbert space representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. Many Hamiltonians with potentials which are too singular to exist as functions over reals are realized as bounded symmetric operators in L2-spaces with respect to a p-adic Gaussian measure.

On Heisenberg's Uncertainty Principle and the CCR

1993 ◽  
Vol 48 (3) ◽  
pp. 447-451 ◽  
Author(s):  
Reinhard Honegger
Keyword(s):  
Uncertainty Principle ◽  
Heisenberg Uncertainty Principle ◽  
Commutation Relations ◽  
Integrable Functions ◽  
Heisenberg Uncertainty ◽  
Heisenberg’S Inequality ◽  
Fock Representations ◽  
Heisenberg's Uncertainty Principle ◽  
Square Integrable Functions ◽  
Rigorous Mathematical Analysis

Abstract Realizing the canonical commutation relations (CCR) [N, Θ] = - i as N = - i d/dϑ and Θ to be the multiplication by ϑ on the Hilbert space of square integrable functions on [0, 2π], in the physical literature there seems to be some contradictions concerning the Heisenberg uncertainty principle ⟨ΔN⟨ ⟨ΔΘ⟨ ≥ 1/4. The difficulties may be overcome by a rigorous mathematical analysis of the domain of state vectors, for which Heisenberg's inequality is valid. It is shown that the exponentials exp {i t N} and exp{i sΘ} satisfy some commutation relations, which are not the Weyl relations. Finally, the present work aims at a better understanding of the phase and number operators in non-Fock representations.

scholarly journals A New Upper Bound for Sampling Numbers

2021 ◽  
Author(s):  
Nicolas Nagel ◽  
Martin Schäfer ◽  
Tino Ullrich
Keyword(s):  
Upper Bound ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sampling Procedure ◽  
Compact Embedding ◽  
Recovery Method ◽  
Grid Sampling ◽  
Integrable Functions ◽  
Random Draw ◽  
Square Integrable Functions

AbstractWe provide a new upper bound for sampling numbers $$(g_n)_{n\in \mathbb {N}}$$ ( g n ) n ∈ N associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants $$C,c>0$$ C , c > 0 (which are specified in the paper) such that $$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$ g n 2 ≤ C log ( n ) n ∑ k ≥ ⌊ c n ⌋ σ k 2 , n ≥ 2 , where $$(\sigma _k)_{k\in \mathbb {N}}$$ ( σ k ) k ∈ N is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding $$\mathrm {Id}:H(K) \rightarrow L_2(D,\varrho _D)$$ Id : H ( K ) → L 2 ( D , ϱ D ) . The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of $$H^s_{\text {mix}}(\mathbb {T}^d)$$ H mix s ( T d ) in $$L_2(\mathbb {T}^d)$$ L 2 ( T d ) with $$s>1/2$$ s > 1 / 2 . We obtain the asymptotic bound $$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$ g n ≤ C s , d n - s log ( n ) ( d - 1 ) s + 1 / 2 , which improves on very recent results by shortening the gap between upper and lower bound to $$\sqrt{\log (n)}$$ log ( n ) . The result implies that for dimensions $$d>2$$ d > 2 any sparse grid sampling recovery method does not perform asymptotically optimal.

scholarly journals A generalization of the Itô formula

2002 ◽  
Vol 31 (8) ◽  
pp. 477-496
Author(s):  
Said Ngobi
Keyword(s):  
Sobolev Space ◽  
White Noise ◽  
Itô Formula ◽  
Ito Formula ◽  
Integrable Functions ◽  
Square Integrable Functions ◽  
White Noise Space

The classical Itô formula is generalized to some anticipating processes. The processes we consider are in a Sobolev space which is a subset of the space of square integrable functions over a white noise space. The proof of the result uses white noise techniques.

scholarly journals Hermite Functions and Fourier Series

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 853
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano del Olmo
Keyword(s):  
Fourier Transform ◽  
Hermite Functions ◽  
Quantum Oscillator ◽  
Linear Combinations ◽  
Ladder Operators ◽  
Orthonormal Bases ◽  
Integrable Functions ◽  
The Fourier Transform ◽  
The One ◽  
Square Integrable Functions

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

scholarly journals Bound states of an inverse-cube singular potential: A candidate for electron-quadrupole binding

2019 ◽  
Vol 34 (28) ◽  
pp. 1950223 ◽  
Author(s):  
A. D. Alhaidari
Keyword(s):  
Bound States ◽  
Recursion Relation ◽  
Singular Potential ◽  
Electric Quadrupole ◽  
Exact Bound ◽  
Short Range Potential ◽  
Integrable Functions ◽  
Solution Energy ◽  
Expansion Coefficients ◽  
Square Integrable Functions

We use the Tridiagonal Representation Approach (TRA) to obtain exact bound states solution (energy spectrum and wave function) of the Schrödinger equation for a three-parameter short-range potential with [Formula: see text], [Formula: see text] and [Formula: see text] singularities at the origin. The solution is a finite series of square-integrable functions with expansion coefficients that satisfy a three-term recursion relation. The solution of the recursion is a non-conventional orthogonal polynomial with discrete spectrum. The results of this work could be used to study the binding of an electron to a molecule with an effective electric quadrupole moment which has the same [Formula: see text] singularity.

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