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Register a new userGrowth of Sobolev Norms in 1-d Quantum Harmonic Oscillator with Polynomial Time Quasi-periodic Perturbation
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We consider the one-dimensional quantum harmonic oscillator perturbed by a linear operator which is a polynomial of degree $2$ in $(x,-{rm i}partial_x)$, with coefficients quasi-periodically depending on time. By establishing the reducibility results, we describe the growth of Sobolev norms. In particular, the $t^{2s}-$polynomial growth of ${mathcal H}^s-$norm is observed in this model if the original time quasi-periodic equation is reduced to a constant Stark Hamiltonian.
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We prove that a linear d-dimensional Schr{o}dinger equation on $mathbb{R}^d$ with harmonic potential $|x|^2$ and small t-quasiperiodic potential $ipartial_t u -- Delta u + |x|^2 u + epsilon V (tomega, x)u = 0, x in mathbb{R}^d$ reduces to an autonomous system for most values of the frequency vector $omega in mathbb{R}^n$. As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in all Sobolev norms.
We consider operators of the form H+V where H is the one-dimensional harmonic oscillator and V is a zero-order pseudo-differential operator which is quasi-periodic in an appropriate sense (one can take V to be multiplication by a periodic function for example). It is shown that the eigenvalues of H+V have asymptotics of the form lambda_n(H+V)=lambda_n(H)+W(sqrt n)n^{-1/4}+O(n^{-1/2}ln(n)) as nto+infty, where W is a quasi-periodic function which can be defined explicitly in terms of V.
We consider the Cauchy problem for the nonlinear wave equation $u_{tt} - Delta_x u +q(t, x) u + u^3 = 0$ with smooth potential $q(t, x) geq 0$ having compact support with respect to $x$. The linear equation without the nonlinear term $u^3$ and potential periodic in $t$ may have solutions with exponentially increasing as $ t to infty$ norm $H^1({mathbb R}^3_x)$. In [2] it was established that adding the nonlinear term $u^3$ the $H^1({mathbb R}^3_x)$ norm of the solution is polynomially bounded for every choice of $q$. In this paper we show that $H^k({mathbb R}^3_x)$ norm of this global solution is also polynomially bounded. To prove this we apply a different argument based on the analysis of a sequence ${Y_k(ntau_k)}_{n = 0}^{infty}$ with suitably defined energy norm $Y_k(t)$ and $0 < tau_k <1.$
The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general Lizorkin--Triebel spaces based on mixed $L_p$-norms. In this context a Nikolskij--Plancherel--Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to spaces of the quasi-homogeneous type.
For an infinite penny graph, we study the finite-dimensional property for the space of harmonic functions, or ancient solutions of the heat equation, of polynomial growth. We prove the asymptotically sharp dimensional estimate for the above spaces.
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