Related papers: Gabor Frame Decomposition of Evolution Operators a…
We investigate the sparsity of the Gabor-matrix representation of Fourier integral operators with a phase having quadratic growth. It is known that such an infinite matrix is sparse and well organized, being in fact concentrated along the…
We use the method of group contractions to relate wavelets analysis and Gabor analysis. Wavelets analysis is associated with unitary irreducible representations of the affine group while Gabor analysis is associated with unitary irreducible…
We obtain frame bounds estimates and the Gabor frame operator $S=S^{\alpha,\beta}$ for Gabor frames generated by the Cauchy kernel. In addition we find the explicit expression for the canonical dual window for all values of the lattice…
In a recent paper we have suggested that the finite temperature density matrix can be computed efficiently by a combination of polynomial expansion and iterative inversion techniques. We present here significant improvements over this…
We propose an expansion of the unitary evolution operator, associated to a given Schr\"odinger equation, in terms of a finite product of explicit unitary operators. In this manner, this unitary expansion can be truncated at the desired…
Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A…
We extend the Gabor analysis in \cite{GaSa} to a broad class of modulation spaces, allowing more general mixed quasi-norm estimates and weights in the definition of the modulation space quasi-norm. For such spaces we deduce invariance and…
For $m\in\mathbb{R}$ we introduce the symbol classes $S^m$, $m\in\mathbb{R}$, consisting of smooth functions $\sigma$ on $\mathbb{R}^{2d}$ such that $|\partial^\alpha \sigma(z)|\leq C_\alpha (1+|z|^2)^{m/2}$, $z\in\mathbb{R}^{2d}$, and we…
We exhibit the connection between the Wigner kernel and the Gabor matrix of a linear bounded operator T : $\mathcal{S}(\mathbb{R}^d) \to \mathcal{S}' (\mathbb{R}^d)$. The smoothing effect of the Gabor matrix is highlighted by basic…
We study sharp frame bounds of Gabor systems over rectangular lattices for different windows and integer oversampling rate. In some cases we obtain optimality results for the square lattice, while in other cases the lattices optimizing the…
We derive frame estimates for vector-valued Gabor systems with window functions belonging to Schwartz space. The main result provides frame bound estimates for windows composed of Hermite functions. The proof is based on a recently…
In this paper we give new estimates for the solution to the Schr\"odinger equation with quadratic and sub-quadratic potentials in the framework of modulation spaces.
Gabor functions have wide-spread applications in image processing and computer vision. In this paper, we prove that 2D Gabor functions are translation-invariant positive-definite kernels and propose a novel formulation for the problem of…
We analyze the many-particle Schrodinger equation for fermions in a thermal ensemble by introducing an exponential operator expansion, defined in the context of thermofield dynamics. The expansion is optimized variationally at each time…
In this paper we study the time evolution of an observable in the interacting fermion systems driven out of equilibrium. We present a method for solving the Heisenberg equations of motion by constructing excitation operators which are…
Recent results on the construction and applications of the transmutation (transformation) operators are discussed. Three new representations for solutions of the one-dimensional Schr\"odinger equation are considered. Due to the fact that…
Using the matrix representation of Fourier integral operators with respect to a Gabor frame, we study their compactness on weighted modulation spaces. As a consequence, we recover and improve some compactness results for pseudodifferential…
The transfer matrix ${\mathbf{M}}$ of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system with a time-dependent non-Hermitian Hamiltonian. This leads to a dynamical…
This work establishes new results on spectral theory and time evolution for matrix-valued discrete Schr\"odinger operators on the space of square-summable matrix sequences. The matrix-valued formalism is employed to streamline notation,…
We give estimates for positive solutions for the Schr\"odinger equation $(\Delta_\mu+W)u=0$ on a wide class of parabolic weighted manifolds $(M, d\mu)$ when $W$ decays to zero at infinity faster than quadratically. These can be combined…