Gamow Vectors in a Periodically Perturbed Quantum System

We analyze the behavior of the wave function $\psi(x,t)$ for one dimensional time-dependent Hamiltonian $H=-\partial_x^2\pm2\delta(x)(1+2r\cos\omega t)$ where $\psi(x,0)$ is compactly supported. We show that $\psi(x,t)$ has a Borel summable expansion containing finitely many terms of the form \sum_{n=-\infty}^{\infty} e^{i^{3/2}\sqrt{-\lambda_{k}+n\omegai}|x|} A_{k,n} e^{-\lambda_{k}t+n\omega it}, where $\lambda_k$ represents the associated resonance. This expression defines Gamow vectors and resonances in a rigorous and physically relevant way for all frequencies and amplitudes in a time-dependent model. For small amplitude ($|r|\ll 1$) there is one resonance for generic initial conditions. We calculate the position of the resonance and discuss its physical meaning as related to multiphoton ionization. We give qualitative theoretical results as well as numerical calculations in the general case.Comment: 21 pages, 6 figure

Positivity for quantum cluster algebras from unpunctured orbifolds

We give the quantum Laurent expansion formula for the quantum cluster algebras from unpunctured orbifolds with arbitrary coefficients and quantization. As an application, positivity for such class of quantum cluster algebras is given. For technical reasons, it will always be assumed that the weights of the orbifold points are 2.Comment: 43 pages. arXiv admin note: substantial text overlap with arXiv:1807.0691

Modular Anomaly from Holomorphic Anomaly in Mass Deformed N=2 Superconformal Field Theories

We study the instanton partition functions of two well-known superconformal field theories with mass deformations. Two types of anomaly equations, namely, the modular anomaly and holomorphic anomaly, have been discovered in the literature. We provide a clean solution to the long standing puzzle about their precise relation, and obtain some universal formulas. We show that the partition function is invariant under the SL(2,Z) duality which exchanges theories at strong coupling with those of weak coupling.Comment: 5 pages. v2: journal versio

Gamow vectors and Borel summability

We analyze the detailed time dependence of the wave function $\psi(x,t)$ for one dimensional Hamiltonians $H=-\partial_x^2+V(x)$ where $V$ (for example modeling barriers or wells) and $\psi(x,0)$ are {\em compactly supported}. We show that the dispersive part of $\psi(x,t)$, its asymptotic series in powers of $t^{-1/2}$, is Borel summable. The remainder, the difference between $\psi$ and the Borel sum, is a convergent expansion of the form $\sum_{k=0}^{\infty}g_k \Gamma_k(x)e^{-\gamma_k t}$, where $\Gamma_k$ are the Gamow vectors of $H$, and $\gamma_k$ are the associated resonances; generically, all $g_k$ are nonzero. For large $k$, $\gamma_{k}\sim const\cdot k\log k +k^2\pi^{2}i/4$. The effect of the Gamow vectors is visible when time is not very large, and the decomposition defines rigorously resonances and Gamow vectors in a nonperturbative regime, in a physically relevant way. The decomposition allows for calculating $\psi$ for moderate and large $t$, to any prescribed exponential accuracy, using optimal truncation of power series plus finitely many Gamow vectors contributions. The analytic structure of $\psi$ is perhaps surprising: in general (even in simple examples such as square wells), $\psi(x,t)$ turns out to be $C^\infty$ in $t$ but nowhere analytic on \RR^+. In fact, $\psi$ is $t-$analytic in a sector in the lower half plane and has the whole of \RR^+ a natural boundary

Projected Euler method for stochastic delay differential equation under a global monotonicity condition

This paper investigates projected Euler-Maruyama method for stochastic delay differential equations under a global monotonicity condition. This condition admits some equations with highly nonlinear drift and diffusion coefficients. We appropriately generalized the idea of C-stability and B-consistency given by Beyn et al. [J. Sci. Comput. 67 (2016), no. 3, 955-987] to the case with delay. Moreover, the method is proved to be convergent with order $\frac{1}{2}$ in a succinct way. Finally, some numerical examples are included to illustrate the obtained theoretical results

Great enhancement of strong-field ionization in femtosecond-laser subwavelength-structured fused silica

A wavelength-degenerate pump-probe spectroscopy is used to study the ultrafast dynamics of strong-field ionization in femtosecond-laser subwavelength-structured fused silica. The comparative spectra demonstrate that femtosecond-laser subwavelength structuring always give rise to great enhancement for strong-field ionization as well as third-order nonlinear optical effects, which is the direct evidence of great local field enhancement in subwavelength apertures of fs-laser highly-excited surface. In short, the study shows the prominent subwavelength spatial effect of strong-field ionization in femtosecond-laser ablation of dielectrics, which greatly contributes to the well-known "incubation effect".Comment: 4 pages, 4 figure

#Cyberbullying in the Digital Age: Exploring People's Opinions with Text Mining

This study used text mining to investigate people's insights about cyberbullying. English-language tweets were collected and analyzed by R software. Our analysis demonstrated three major themes: (a) the major actions that needed to be taken into consideration (e.g. guiding parents and teachers to cyberbullying prevention, funding schools to fight cyberbullying), (b) certain events that were important to people (e.g. the Michigan cyberbullying law), and (c) people's major concerns in this regard (e.g. mental health issues among students). Parents and teachers have an important role in educating, informing, warning, preventing, and protecting against cyberbullying behaviors. The frequency of negative sentiments was almost 2.45 times more than positive sentiments

On the Distribution of Plasmoids In High-Lundquist-Number Magnetic Reconnection

The distribution function $f(\psi)$ of magnetic flux $\psi$ in plasmoids formed in high-Lundquist-number current sheets is studied by means of an analytic phenomenological model and direct numerical simulations. The distribution function is shown to follow a power law $f(\psi)\sim\psi^{-1}$, which differs from other recent theoretical predictions. Physical explanations are given for the discrepant predictions of other theoretical models.Comment: Accepted for publication in Phys. Rev. Let

Andreev bound states in iron pnictide superconductors

Recently, Andreev bound states in iron pnictide have been proposed as an experimental probe to detect the relative minus sign in the $s_\pm$-wave pairing. While previous theoretical investigations demonstrated the feasibility of the approach, the local density of states in the midgap regime is small, making the detection hard in experiments. We revisit this important problem from the Bogoliubov-de Gennes Hamiltonian on the square lattice with appropriate boundary conditions. Significant spectral weights in the midgap regime are spotted, leading to easy detection of the Andreev bound states in experiments. Peaks in the momentum-resolved local density of states appear and lead to enhanced quasiparticle interferences at specific momenta. We analyze the locations of these magic spots and propose they can be verified in experiments by the Fourier-transformed scanning tunneling spectroscopy.Comment: 5 pages, 3 figure

Categorification of sign-skew-symmetric cluster algebras and some conjectures on g-vectors

Using the unfolding method given in \cite{HL}, we prove the conjectures on sign-coherence and a recurrence formula respectively of ${\bf g}$-vectors for acyclic sign-skew-symmetric cluster algebras. As a following consequence, the conjecture is affirmed in the same case which states that the ${\bf g}$-vectors of any cluster form a basis of $\mathbb Z^n$. Also, the additive categorification of an acyclic sign-skew-symmetric cluster algebra $\mathcal A(\Sigma)$ is given, which is realized as $(\mathcal C^{\widetilde Q},\Gamma)$ for a Frobenius $2$-Calabi-Yau category $\mathcal C^{\widetilde Q}$ constructed from an unfolding $(Q,\Gamma)$ of the acyclic exchange matrix $B$ of $\mathcal A(\Sigma)$.Comment: 12 page