Related papers: Extremal property of a simple cycle
An adiabatic cycle of parameters in a quantum system can yield the quantum anholonomies, nontrivial evolution not just in phase of the states, but also in eigenvalues and eigenstates. Such exotic anholonomies imply that an adiabatic cycle…
We study the distances, called spacings, between pairs of neighboring energy levels for the quantum harmonic oscillator. Specifically, we consider all energy levels falling between E and E+1, and study how the spacings between these levels…
It has been proposed to investigate the equilibration properties of a small isolated quantum system by means of the matrix of asymptotic transition probabilities in some preferential basis. The trace $T$ of this matrix measures the degree…
This paper studies convergence to equilibrium for second-order Langevin dynamics under general growth conditions on the potential. Although we are principally motivated by the case when the potential is singular, e.g. when the dynamics has…
We investigate spectral properties of the operator describing a quantum particle confined to a planar domain $\Omega$ rotating around a fixed point with an angular velocity $\omega$ and demonstrate several properties of its principal…
By analyzing a paradigmatic example of the theory of dissipative systems -- the classical and quantum dissipative standard map -- we are able to explain the main features of the decay to the quantum equilibrium state. The classical…
We consider a dynamical system, possibly infinite dimensional or non-autonomous, with fast and slow time scales which is oscillatory with high frequencies in the fast directions. We first derive and justify the limit system of the slow…
The appearance of so-called exceptional points in the complex spectra of non-Hermitian systems is often associated with phenomena that contradict our physical intuition. One example of particular interest is the state-exchange process…
Spectral properties of bounded linear operators play a crucial role in several areas of mathematics and physics. For each self-adjoint, trace-class operator $O$ we define a set $\Lambda_n\subset \mathbb{R}$, and we show that it converges to…
We consider a class (convex set) of quantum states containing all finite rank states and infinite rank states with the sufficient rate of decreasing of eigenvalues (in particular, all Gaussian states). Quantum states from this class are…
We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous-time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a…
Systematic properties of the first excited state are presented for various ring sizes and spin quantum numbers which are only partly covered by the theorem of Lieb, Schultz and Mattis. For odd ring sizes the first excited energy eigenvalue…
We investigate a descent on simple graphs, starting with the complete graph on $n$ vertices and ending up with the cycle graph by removing one edge after another. We obtain quantitative results showing that graphs with large diameter must…
A limit cycle is a self-sustained periodic motion appearing in autonomous ordinary differential equations. As the period of the limit cycle is a-priori unknown, it is challenging to find them as stationary states of a rotating ansatz.…
It is possible that self-induced oscillations appear in reactors, and that their range does not reach the steady state, although such state exists. To prove this, a cascade of tank reactors coupled with mass recycle loop was tested…
Consider Ginibre's ensemble of $N \times N$ non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance $\frac{1}{N}$. As $N \uparrow \infty$ the normalized counting measure of the…
Suppose that an equilibrium is asymptotically stable when external inputs vanish. Then, every bounded trajectory which corresponds to a control which approaches zero and which lies in the domain of attraction of the unforced system, must…
We derive an accurate lower tail estimate on the lowest singular value $\sigma_1(X-z)$ of a real Gaussian (Ginibre) random matrix $X$ shifted by a complex parameter $z$. Such shift effectively changes the upper tail behaviour of the…
Exactly solvable models provide an opportunity to study different aspects of reduced quantum dynamics in detail. We consider the reduced dynamics of a single spin in finite XX and XY spin 1/2 chains. First we introduce a general expression…
This paper derives a differential contraction condition for the existence of an orbitally-stable limit cycle in an autonomous system. This transverse contraction condition can be represented as a pointwise linear matrix inequality (LMI),…