اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA local quantum version of the Kolmogorov theorem
78
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Consider in $L^2 (R^l)$ the operator family $H(epsilon):=P_0(hbar,omega)+epsilon Q_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector $om$, $Q_0$ a bounded pseudodifferential operator with symbol holomorphic and decreasing to zero at infinity, and $epinR$. Then there exists $ep^ast >0$ with the property that if $|ep|<ep^ast$ there is a diophantine frequency $om(ep)$ such that all eigenvalues $E_n(hbar,ep)$ of $H(ep)$ near 0 are given by the quantization formula $E_alpha(hbar,ep)= {cal E}(hbar,ep)+laom(ep),alpharahbar +|om(ep)|hbar/2 + ep O(alphahbar)^2$, where $alpha$ is an $l$-multi-index.
قيم البحث
اقرأ أيضاً
We derive an adiabatic theory for a stochastic differential equation, $ varepsilon, mathrm{d} X(s) = L_1(s) X(s), mathrm{d} s + sqrt{varepsilon} L_2(s) X(s) , mathrm{d} B_s, $ under a condition that instantaneous stationary states of $L_1(s)$ are als
o stationary states of $L_2(s)$. We use our results to derive the full statistics of tunneling for a driven stochastic Schr{o}dinger equation describing a dephasing process.
We prove a local central limit theorem (LCLT) for the number of points $N(J)$ in a region $J$ in $mathbb R^d$ specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of $N(J)$ tends to infinity as
$|J| to infty$. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of the generating function for ${E(k;J)}$ --- the probabilities of there being exactly $k$ points in $J$ --- all lie on the negative real $z$-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE). A LCLT is also established for the probability density function of the $k$-th largest eigenvalue at the soft edge, and of the spacing between $k$-th neigbors in the bulk.
We study the many body quantum evolution of bosonic systems in the mean field limit. The dynamics is known to be well approximated by the Hartree equation. So far, the available results have the form of a law of large numbers. In this paper we go one
step further and we show that the fluctuations around the Hartree evolution satisfy a central limit theorem. Interestingly, the variance of the limiting Gaussian distribution is determined by a time-dependent Bogoliubov transformation describing the dynamics of initial coherent states in a Fock space representation of the system.
Given two Hilbert spaces, $mathcal{H}$ and $mathcal{K}$, we introduce an abstract unitary operator $U$ on $mathcal{H}$ and its discriminant $T$ on $mathcal{K}$ induced by a coisometry from $mathcal{H}$ to $mathcal{K}$ and a unitary involution on $mat
hcal{H}$. In a particular case, these operators $U$ and $T$ become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between $U$ and $T$ via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpinski lattice, which is pure point and has a Cantor-like structure.
Caratheodory showed that $n$ complex numbers $c_1,...,c_n$ can uniquely be written in the form $c_p=sum_{j=1}^m rho_j {epsilon_j}^p$ with $p=1,...,n$, where the $epsilon_j$s are different unimodular complex numbers, the $rho_j$s are strictly positive
numbers and integer $m$ never exceeds $n$. We give the conditions to be obeyed for the former property to hold true if the $rho_j$s are simply required to be real and different from zero. It turns out that the number of the possible choices of the signs of the $rho_j$s are {at most} equal to the number of the different eigenvalues of the Hermitian Toeplitz matrix whose $i,j$-th entry is $c_{j-i}$, where $c_{-p}$ is equal to the complex conjugate of $c_{p}$ and $c_{0}=0$. This generalization is relevant for neutron scattering. Its proof is made possible by a lemma - which is an interesting side result - that establishes a necessary and sufficient condition for the unimodularity of the roots of a polynomial based only on the polynomial coefficients. Keywords: Toeplitz matrix factorization, unimodular roots, neutron scattering, signal theory, inverse problems. PACS: 61.12.Bt, 02.30.Zz, 89.70.+c, 02.10.Yn, 02.50.Ga
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
