In this paper, we have given a new definition of continuous fractional wavelet transform in RN, namely the multidimensional fractional wavelet transform (MFrWT) and studied some of the basic ...properties along with the inner product relation and the reconstruction formula. We have also shown that the range of the proposed transform is a reproducing kernel Hilbert space and obtained the associated kernel. We have obtained the uncertainty principle like Heisenberg’s uncertainty principle, logarithmic uncertainty principle and local uncertainty principle of the multidimensional fractional Fourier transform (MFrFT). Based on these uncertainty principles of the MFrFT we have obtained the corresponding uncertainty principles i.e., Heisenberg’s, logarithmic and local uncertainty principles for the proposed MFrWT.
For Schrödinger equations with real quadratic Hamiltonians, it is known that the Wigner distribution of the solution at a given time equals, up to a linear coordinate transformation, the Wigner ...distribution of the initial condition. Based on Hardy's uncertainty principle for the joint time-frequency representation, we present a general uniqueness result for such Schrödinger equations, where the solution cannot have strong decay at two distinct times. This approach gives new proofs to known, sharp Hardy type estimates for the free Schrödinger equation, the harmonic oscillator and uniform magnetic potentials, as well as new uniqueness results.
ABSTRACT
The generalized and extended uncertainty principles affect the Newtonian gravity and also the geometry of the thermodynamic phase space. Under the influence of the latter, the ...energy–temperature relation of ideal gas may change. Moreover, it seems that the Newtonian gravity is modified in the framework of the Rényi entropy formalism motivated by both the long-range nature of gravity and the extended uncertainty principle. Here, the consequences of employing the generalized and extended uncertainty principles, instead of the Heisenberg uncertainty principle, on the Jeans mass are studied. The results of working in the Rényi entropy formalism are also addressed. It is shown that unlike the extended uncertainty principle and the Rényi entropy formalism that lead to the same increase in the Jeans mass, the generalized uncertainty principle can decrease it. The latter means that a cloud with mass smaller than the standard Jeans mass, obtained in the framework of the Newtonian gravity, may also undergo the gravitational collapse process.
For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is, a strip beyond the unitarity axis in which the Selberg zeta function has only finitely ...many zeroes. We make no assumption on the dimension δ of the limit set; in particular, we do not require the pressure condition δ ≤ 1/2. This is the first result of this kind for quantum Hamiltonians.
Our proof follows the strategy developed by Dyatlov and Zahl. The main new ingredient is the fractal uncertainty principle for δ-regular sets with δ < 1, which may be of independent interest.
We consider a new differential–difference operator Λ on the real line. We study the harmonic analysis associated with this operator. Next, we prove various mathematical aspects of the quantitative ...uncertainty principles, including Donoho–Stark’s uncertainty principle and variants of Heisenberg’s inequalities for its Hartley transform associated to the operator Λ.
The Generalized Uncertainty Principle and the related minimum length are normally considered in non-relativistic Quantum Mechanics. Extending it to relativistic theories is important for having a ...Lorentz invariant minimum length and for testing the modified Heisenberg principle at high energies. In this paper, we formulate a relativistic Generalized Uncertainty Principle. We then use this to write the modified Klein–Gordon, Schrödinger and Dirac equations, and compute quantum gravity corrections to the relativistic hydrogen atom, particle in a box, and the linear harmonic oscillator.
The aim of this paper is to prove a generalization of uncertainty principles for the continuous wavelet transform connected with the Riemann–Liouville operator in
L
p
-norm. More precisely, we ...establish the Heisenberg–Pauli–Weyl uncertainty principle, Donoho–Stark’s uncertainty principles and local Cowling-Price’s type inequalities. Finally, Pitt’s inequality and Beckner’s uncertainty principle are proved for this transform.
The optimal k-Wigner distribution Zhang, Zhichao; He, Yangfan; Zhang, Jianwei ...
Signal processing,
October 2022, 2022-10-00, Volume:
199
Journal Article
Peer reviewed
•We establish Heisenberg’s uncertainty principles for the kWD.•We discuss an application of the kWD in signal analysis.•We obtain the optimal parameter of the kWD corresponding to the highest ...time-frequency resolution.•We concern an application of the derived lower bounds in the estimation of bandwidths in kWD domains.
This study devotes to the optimal parameter of the k-Wigner distribution (kWD) in terms of the highest time-frequency resolution. We first disclose a relation between spreads in time-kWD and time domains, as well as those in Fourier transform (FT)-kWD and FT domains. Then we use them to set up an equivalence relation between the uncertainty product in time-kWD and FT-kWD domains and that in time and FT domains. Finally we separately deduce Heisenberg’s uncertainty inequalities of all signals and complex-valued signals for the kWD. As a result, the optimal kWD is none other than the ordinary Wigner distribution. Examples are carried out to verify the correctness of the theoretical result. An application of the derived uncertainty inequalities in the estimation of bandwidths in kWD domains is also given.
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