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We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We…
The capabilities of the functional-analytic and of the functional-integral approach for the construction of the Hamiltonian as a self-adjoint operator on Hilbert space are compared in the context of non-relativistic quantum mechanics.…
We consider the gap probability for the Bessel process in the single-time and multi-time case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of integrable kernels \`a la…
Quantum Mechanics and Signal Processing in the line R, are strictly related to Fourier Transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and…
It is proposed that the Schrodinger equation for a free point particle has non-linear corrections which depend on the mass of the particle. It is assumed that the corrections become extremely small when the mass is much smaller or much…
We use the Fokas method to analyze the derivative nonlinear Schr\"odinger (DNLS) equation $iq_t(x,t)=-q_{xx}(x,t)+(r q^2)_x$ on the interval $[0,L]$. Assuming that the solution $q(x,t)$ exists, we show that it can be represented in terms of…
We derive the large distance asymptotics of the Fredholm determinant of the so-called generalised sine kernel at the critical point. This kernel corresponds to a generalisation of the pure sine kernel arising in the theory of random…
We examine the dynamical behavior of matter coupled to gravity in the context of a linear Klein-Gordon equation coupled to a Friedman-Robertson-Walker metric. The resulting ordinary differential equations can be decoupled, the effect of…
The Schrodinger equation for non-relativistic quantum systems is derived from some classical physics axioms within an ensemble hamiltonian framework. Such an approach enables one to understand the structure of the equation, in particular…
We formulate the quantum mechanics of the solutions of a Klein-Gordon-type field equation: (\partial_t^2+D)\psi(t)=0, where D is a positive-definite operator acting in a Hilbert space \tilde H. We determine all the positive-definite inner…
We propose a new type of approximation to quantum determinants, ``quantum Fredholm determinant", and conjecture that, compared to the quantum Selberg zeta functions derived from Gutzwiller semiclassical trace formulas, such determinants…
A new differential equation is derived for an object ${\widehat S}(E,E^\prime,x)$, which when integrated over the appropriate range in $x$, yields the kernel $K(E,E^\prime)$ with which $n$-point correlation functions can be computed in a…
This is a review paper concerned with the global consistency of the quantum dynamics of non-commutative systems. Our point of departure is the theory of constrained systems, since it provides a unified description of the classical and…
The evolution of a two-level system subjected to stimulated transitions which is undergoing a sequence of measurements of the level occupation probability is evaluated. Its time correlation function is compared to the one obtained through…
The operator and the functional formulations of the dynamics of constrained systems are explored for determining unambiguously the quantum Hamiltonian of a nonrelativistic particle in a curved space.
Using a simple geometrical construction based upon the linear action of the Heisenberg--Weyl group we deduce a new nonlinear Schr\"{o}dinger equation that provides an exact dynamic and energetic model of any classical system whatsoever, be…
We establish a deep connection between the Prandtl equations linearised around a quadratic shear flow, confluent hypergeometric functions of the first kind, and the Schr\"odinger operator. Our first result concerns an ODE and a spectral…
In this article we review the Duistermaat-Heckman integration formula and the ensuing equivariant cohomology structure, in the finite dimensional case. In particular, we discuss the connection between equivariant cohomology and classical…
A quantum-field model of the conformally flat space is formulated using a standard field-theoretical technique, a probability interpretation and a way to establish the classical limit. The starting point is the following: after conformal…
The theoretical description of strongly correlated quantum systems out of equilibrium presents several challenges and a number of open questions persist. In this paper we focus on nonlinear electronic transport through a quantum dot…