Related papers: Metaplectic Quantum Time--Frequency Analysis, Oper…
We construct frames adapted to a given cover of the time-frequency or time-scale plane. The main feature is that we allow for quite general and possibly irregular covers. The frame members are obtained by maximizing their concentration in…
A periodic linear graph operator acts on states (functions) defined on the vertices of a graph equipped with a free translation action. Fourier transform with respect to the translation group reveals the central spectral objects, Bloch and…
A technique called analytic perturbation theory, which respects the required analytic properties, consistent with causality, is applied to the definition of the running coupling in the timelike region, to the description of inclusive…
We review quantum chaos on graphs. We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wavefunction statistics. This operator is the analogue of the classical evolution operator on…
For a quasi-split Satake diagram, we define a modified $q$-Weyl algebra, and show that there is an algebra homomorphism between it and the corresponding $\imath$quantum group. In other words, we provide a differential operator approach to…
We study the phenomena that arise when we combine the standard pseudodifferential operators with those operators that appear in the study of some sub-elliptic estimates, and on strongly pseudoconvex domains. The algebra of operators we…
The problem of time operator in quantum mechanics is revisited. The unsharp measurement model for quantum time based on the dynamical system-clock interaction, is studied. Our analysis shows that the problem of the quantum time operator…
In this paper we review a proposed geometrical formulation of quantum mechanics. We argue that this geometrization makes available mathematical methods from classical mechanics to the quantum frame work. We apply this formulation to the…
Consider an elliptic operator in divergence form with symmetric coefficients.If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is key to the spectral properties of the…
Discrepancy between periodic orbit theory and numerical calculation of a modified Kepler problem is cleared by a quantum mechanical calculation. The diagonal approximation already gives a good fit for the numerical calculation. A better…
The relation between disordered and chaotic systems is investigated. It is obtained by identifying the diffusion operator of the disordered systems with the Perron-Frobenius operator in the general case. This association enables us to…
We develop the method of similar operators to study the spectral properties of unbounded perturbed linear operators that can be represented by matrices of various kinds. The class of operators under consideration includes various…
Transfer operators have been used widely to study the long time properties of chaotic maps or flows. We describe quantum analogues of these operators, which have been used as toy models by the quantum chaos community, but are also relevant…
Using tools from quantum harmonic analysis, we show that the domain of the Laplacian of an operator is dense in the Toeplitz algebra over the Fock space $\mathcal{F}^2(\mathbb{C}^n)$. As an application, we provide a simplified treatment of…
Solving Partial Differential Equation (PDE) interface problems on varying domains is a critical task in design and optimization, yet it remains computationally prohibitive for traditional solvers. Although operator learning has shown…
We show that the theory of operator quantum error correction can be naturally generalized by allowing constraints not only on states but also on observables. The resulting theory describes the correction of algebras of observables (and may…
We propose a new problem of missing data reconstruction in the time-frequency plane. This problem called phase inpainting, consists in reconstructing a signal from time-frequency observations where all amplitudes and some phases are known…
Geometric methods have useful application for solving problems in a range of quantum information disciplines, including the synthesis of time-optimal unitaries in quantum control. In particular, the use of Cartan decompositions to solve…
The use of geometric and symmetry techniques in quantum and classical information processing has a long tradition across the physical sciences as a means of theoretical discovery and applied problem solving. In the modern era, the emergent…
We describe a multi-scale resolution approach to analyzing problems in Quantum Mechanics using Daubechies wavelet basis. The expansion of the wavefunction of the quantum system in this basis allows a natural interpretation of each basis…