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We revisit our study of general transport operator with general force field and general invariant measure by considering, in the $L^1$ setting, the linear transport operator $\T_H$ associated to a linear and positive boundary operator $H$…
We address some conjectures and open problems in "analysis of symmetries" which include the study of non-commutative harmonic analysis and discontinuous groups for reductive homogeneous spaces beyond the classical framework: (1) discrete…
Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary…
The purpose of these lectures is to give an accessible and self contained introduction to quantum scattering theory in one dimension. Part A defines the theoretical playground, and develops basic concepts of scattering theory in the time…
Using the basic ingredient of supersymmetry, we develop a simple alternative approach to perturbation theory in one-dimensional non-relativistic quantum mechanics. The formulae for the energy shifts and wave functions do not involve tedious…
Inspired by a recent work of Hyt\"onen and Naor, we solve a problem left open in our previous work joint with Mart\'{\i}nez and Torrea on the vector-valued Littlewood-Paley-Stein theory for symmetric diffusion semigroups. We prove a similar…
This thesis is dedicated to developing a dilation theory for semigroups of completely positive maps. The first part treats two-parameter semigroups, and contains also contributions to dilation theory of product system representations. The…
Unitarity is a cornerstone of quantum theory, ensuring the conservation of probability and information. Although non-Hermitian Hamiltonians are typically associated with open or dissipative systems, pseudo-Hermitian quantum mechanics shows…
The reconstruction theorem and the multilevel Schauder estimate have central roles in the analytic theory of regularity structures [17]. Inspired by [26], we provide elementary proofs for them by using the semigroup of operators.…
We propose an improved scheme of perturbation theory based on our exact solution [An Min Wang, quant-ph/0611216] in general quantum systems independent of time. Our elementary start-point is to introduce the perturbing parameter as late as…
These two lectures cover some of the advances that underpin recent progress in deriving continuum solutions from the exact renormalization group. We concentrate on concepts and on exact non-perturbative statements, but in the process will…
Complex extension of quantum mechanics and the discovery of pseudo-unitarily invariant random matrix theory has set the stage for a number of applications of these concepts in physics. We briefly review the basic ideas and present…
In order to enlarge the present arsenal of semiclassical toools we explicitly obtain here the Husimi distributions and Wehrl entropy within the context of deformed algebras built up on the basis of a new family of q-deformed coherent…
We present simple models which exhibit some of the remarkable features expected to hold for the as yet unknown non-perturbative formulation of string theories. Among these are: (a) the absence of a background or embedding space for the full…
In quantum mechanics and quantum field theory perturbation theory generically requires the inclusion of extra contributions non-perturbative in the coupling, such as instantons, to reproduce exact results. We show how full non-perturbative…
Determining the steady state of an open quantum system is crucial for characterizing quantum devices and studying various physical phenomena. Often, computing a single steady state is insufficient, and it is necessary to explore its…
We report a proof of the quantum Sanov Theorem by elementary application of basic facts about representations of the symmetric group, together with a complete characterization of the optimal error exponent in a situation where the null…
The formalism of abstracted quantum mechanics is applied in a model of the generalized Liar Paradox. Here, the Liar Paradox, a consistently testable configuration of logical truth properties, is considered a dynamic conceptual entity in the…
Recent work has shown that the hidden states of large language models contain signals useful for uncertainty estimation and hallucination detection, motivating a growing interest in efficient probe-based approaches. Yet it remains unclear…
Current quantum theories of an elementary free particle assume unitary space inversion and anti-unitary time reversal operators. In so doing robust classes of possible theories are discarded. The present work shows that consistent theories…