Related papers: On quantum Floquet theorem
We consider the semiclassical Schr\"odinger equation on $\mathbb R^d$ given by $$\mathrm{i} \hbar \partial_t \psi = \left(-\frac{\hbar^2}{2} \Delta + W_l(x) \right)\psi + V(t,x)\psi ,$$ where $W_l$ is an anharmonic trapping of the form…
Consider an operator equation $F(u)=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $F'(u)$ is not boundedly invertible, and well-posed otherwise. If $F$ is monotone $C^2_{loc}(H)$ operator, then we construct…
We consider the quasi-periodic Jacobi operator $H_{x,\omega}$ in $l^2(\mathbb{Z})$ $(H_{x,\omega}\phi)(n) = -b(x+(n+1)\omega)\phi(n+1) - b(x+n\omega)\phi(n-1) + a(x+n\omega)\phi(n) = E\phi(n),\ n\in\mathbb{Z},$ where $a(x),\ b(x)$ are…
We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged…
We find the explicit form of the basic monodromy operators for the case of the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_3))$. Expressions for the quantum Casimir elements of the quantum groups $\mathrm U_q(\mathfrak{gl}_3)$ and…
We prove that the weakly damped cubic Schr\"odinger flow in $L^2(\T)$ provides a dynamical system that possesses a global attractor. The proof relies on a sharp study of the behavior of the associated flow-map with respect to the weak $…
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad theory for open quantum systems. We deduce the density matrix…
The time-dependent Schroedinger equation with time-independent Hamiltonian matrix is a homogeneous linear oscillatory system in canonical form. We investigate whether any classical system that itself is linear, homogeneous, oscillatory and…
By coupling a Hamiltonian mechanical system with a linear Hamiltonian field theory one obtains an infinite-dimensional Hamiltonian system with regularizing nonlinearity, where the underlying phase space is given by the product of a…
In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $B_\varepsilon >0$. Coefficients of the operator $B_\varepsilon$ are periodic with respect to some lattice in $\mathbb{R}^d$ and depend on…
We study Schr\"odinger operators on $\mathbb{R}^2$ $$ H = \left(-\frac{\partial^2}{\partial x_1^2}\right)^{\alpha/2} + \left(-\frac{\partial^2}{\partial x_2^2}\right)^{\alpha/2} + V, $$ for $\alpha \in (0,2)$ and some sufficiently regular,…
We study the connections between operator moment sequences ${\mathcal T}=\displaystyle(T_n)_{n\in\mathbb{Z}_+}$ of self-adjoint operators on a complex Hilbert space $\mathcal{H}$ and the local moment sequences $\langle{\mathcal T}x,x\rangle…
In this manuscript, we introduce an exact expression for the response of a semi-classical two-level quantum system subject to arbitrary periodic driving. Determining the transition probabilities of a two-level system driven by an arbitrary…
We employ weak hypocoercivity methods to study the long-term behavior of operator semigroups generated by degenerate Kolmogorov operators with variable second-order coefficients, which solve the associated abstract Cauchy problem. We prove…
In this paper, we study the $L^{2}$-boundedness and $L^{2}$-compactness of a class of $h$-Fourier integral operators. These operators are bounded (respectively compact) if the weight of the amplitude is bounded (respectively tends to $0)$.
Quantum harmonic oscillators linearly coupled through coordinates and momenta, represented by the Hamiltonian $ {\hat H}=\sum^2_{i=1}\left( \frac{ {\hat p}^{2}_i}{2 m_i } + \frac{m_i \omega^2_i}{2} x^2_i\right) +{\hat H}_{int} $, where the…
We study the large-time behavior of global energy class ($H^1$) solutions of the one-dimensional nonlinear Schr\"odinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction…
We study the inverse spectral problem for periodic Schr\"odinger opera\-tors of kind $- \frac{1}{2} \hbar^2 \Delta_x + V(x)$ on the flat torus $\Bbb T^n := (\Bbb R / 2 \pi \Bbb Z)^n$ with potentials $V \in C^{\infty} (\Bbb T^n)$. We show…
We give a new formulation of the $T1$ theorem for compactness of Calder\'on-Zygmund singular integral operators. In particular, we prove that a Calder\'on-Zygmund operator $T$ is compact on $L^2(\mathbb{R}^n)$ if and only if $T1,T^*1\in…
Consider in $L^2(\R^2)$ the operator family $H(\epsilon):=P_0(\hbar,\omega)+\epsilon F_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector $\om$, $F_0$ a bounded pseudodifferential operator with symbol decreasing…