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Algebraic approach to the integrability condition called shape invariance is briefly reviewed. Various applications of shape-invariance available in the literature are listed. A class of shape-invariant bound-state problems which represent…
We study a recently proposed modified Schr\"{o}dinger equation having an added nonlinear term, which gives rise to disentanglement. The process of quantum measurement is explored for the case of a pair of coupled spins. We find that the…
Is quantum mechanics about 'states'? Or is it basically another kind of probability theory? It is argued that the elementary formalism of quantum mechanics operates as a well-justified alternative to 'classical' instantiations of a…
We study the following problem: Is it possible to explain the quantum interference of probabilities in the purely corpuscular model for elementary particles? We demonstrate that (by taking into account perturbation effects of measurement…
We consider a scalar field interacting with a quantized metric varying on a submanifold (e.g. a scalar field interacting with a quantized gravitational wave). We explicitly sum up the perturbation series for the time-ordered vacuum…
The quantum statistical treatment of the Rutherford model, considering matter as a system of point charges (electrons and nuclei) is analyzed. First, in the historical context, the solutions of different fundamental problems, such as the…
We show how nonrelativistic many body techniques can be used to study quantum corrections to the classical limit, in particular of the $SU(2)$ Lipkin Model. We show that the quantum corrections are essentially of two types: unitary and…
A quantum computing circuit is presented that approximates a single spin wave quantum on a linear chain of spin 1/2 particles described by a Heisenberg Hamiltonian. The circuit is a product state where each qubit represents a spin. The spin…
We introduce a class of neural controlled differential equation inspired by quantum mechanics. Neural quantum controlled differential equations (NQDEs) model the dynamics by analogue of the Schr\"{o}dinger equation. Specifically, the hidden…
In this paper, I present a mapping between representation of some quantum phenomena in one dimension and behavior of a classical time-dependent harmonic oscillator. For the first time, it is demonstrated that quantum tunneling can be…
We report lowest-order series expansions for primary matrix functions of quantum states based on a perturbation theory for functions of linear operators. Our theory enables efficient computation of functions of perturbed quantum states that…
We present a new hydrodynamic analogy of nonrelativistic quantum particles in potential wells. Similarities between a real variant of the Schr\"odinger equation and gravity-capillary shallow water waves are reported and analyzed. We show…
In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the…
Quantum graphs are a paradigmatic model for quantum chaos as well as for spectral theory. We give a concise didactical introduction to quantum graphs, or Schr\"odinger Hamiltonians on metric graphs, with a focus on results related to…
In these lectures, we discuss different types of renormalization problems in QCD and their non-perturbative solution in the framework of the lattice formulation. In particular the recursive finite size methods to compute the…
A perturbative study of the Schr\"{o}dinger equation in a strong electromagnetic field with dipole approximation is accomplished in the Kramers-Henneberger frame. A prove that just odd harmonics appear in the spectrum for a linear polarized…
We demonstrate that perturbative expansions for quantum many-body systems can be rephrased in terms of tensor networks, thereby providing a natural framework for interpolating perturbative expansions across a quantum phase transition. This…
A new discrete model for energy relaxation of a quantum particle is described via a projection operator, causing the wave function collapse. Power laws for the evolution of the particle coordinate and momentum dispersions are derived. A new…
In this paper, we show how to use canonical perturbation theory for dissipative dynamical systems capable of showing limit cycle oscillations. Thus, our work surmounts the hitherto perceived barrier for canonical perturbation theory that it…
States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a K\"ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics…