The stochastic dynamics of the damped harmonic oscillator in a heat bath is simulated with an algorithm that is exact for time steps of arbitrary size. Exact analytical results are given for correlation functions and power spectra in the form they acquire when computed from experimental time-lapse recordings. Three applications are discussed: (i) The effects of finite sampling rate and time, de...

Abstract The interrelationship between the non-Markovian stochastic Schrödinger equations and the corresponding non-Markovian master equations is investigated in the finite temperature regimes. We show that the general finite temperature non-Markovian trajectories can be used to derive the corresponding non-Markovian master equations. A simple, yet important solvable example is the well-known d...

For x x1, x2, . . . , xn ∈ 0, 1 n and r ∈ {1, 2, . . . , n}, the symmetric function Fn x, r is defined as Fn x, r Fn x1, x2, . . . , xn; r ∑ 1≤i1<i2 ···<ir≤n ∏r j 1 1 xij / 1−xij , where i1, i2, . . . , in are positive integers. In this paper, the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of Fn x, r are discussed. As consequences, several inequalities are est...

In quantum physics the free particle and the harmonically trapped particle are arguably the most important systems a physicist needs to know about. It is little known that, mathematically, they are one and the same. This knowledge helps us to understand either from the viewpoint of the other. Here we show that all general time-dependent solutions of the free-particle Schrödinger equation can be...

We rigorously define the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ′-interaction, of strength β, centred at 0 (the bottom of the confining parabolic potential), by explicitly providing its resolvent. Our approach is based on a ‘coupling constant renormalization’, related to a technique originated in quantum field theory and implemented in the rigorous mathe...

The harmonic oscillator is a paradigm of predictability. The resonance response of a harmonic oscillator can be exploited to make clocks accurate to 1 part in 10. Harmonic oscillations are normally observed in systems operating near equilibrium and governed by linear or “Hooke’s law” restoring forces. Nonetheless, for small enough excursions from equilibrium, systems governed by nonlinear resto...

We obtain a class of oscillation modes with damping and absorption that are connected to the classical harmonic oscillator modes through the supersymmetric one-dimensional matrix procedure similar to relationships of the same type between Dirac and Schrödinger equations in particle physics. A single coupling parameter characterizes both the damping and absorption features of these modes, which ...

Discrete interaction models for the classical harmonic oscillator are used for introducing new mathematical generalizations in the usual continuous formalism. The inverted harmonic potential and generalized discrete hyperbolic and trigonometric functions are defined.

We generalize the previously considered cases of a harmonic oscillator subject to a random force (Brownian motion), or having random frequency, or random damping. We consider here a random mass which corresponds to an oscillator where the particles of the surrounding medium adhere to the oscillator for some (random) time after collision, thereby changing the oscillator mass. Such a model is app...