Nonlinear Schroedinger equation with two symmetric point interactions in one dimension

We consider a time-dependent one-dimensional nonlinear Schroedinger equation with a symmetric potential double well represented by two delta interactions. Among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the Hamiltonian. Then we establish the corresponding Strichartz-type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

Decay versus survival of a localized state subjected to harmonic forcing: exact results

We investigate the survival probability of a localized 1-d quantum particle subjected to a time dependent potential of the form $rU(x)\sin{\omega t}$ with $U(x)=2\delta (x-a)$ or $U(x)= 2\delta(x-a)-2\delta (x+a)$. The particle is initially in a bound state produced by the binding potential $-2\delta (x)$. We prove that this probability goes to zero as $t\to\infty$ for almost all values of $r$, $\omega$, and $a$. The decay is initially exponential followed by a $t^{-3}$ law if $\omega$ is not close to resonances and $r$ is small; otherwise the exponential disappears and Fermi's golden rule fails. For exceptional sets of parameters $r,\omega$ and $a$ the survival probability never decays to zero, corresponding to the Floquet operator having a bound state. We show similar behavior even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential

Central limit theorem for multiplicative class functions on the symmetric group

Hambly, Keevash, O'Connell and Stark have proven a central limit theorem for the characteristic polynomial of a permutation matrix with respect to the uniform measure on the symmetric group. We generalize this result in several ways. We prove here a central limit theorem for multiplicative class functions on symmetric group with respect to the Ewens measure and compute the covariance of the real and the imaginary part in the limit. We also estimate the rate of convergence with the Wasserstein distance.Comment: 23 pages; the mathematics is the same as in the previous version, but there are several improvments in the presentation, including a more intuitve name for the considered function

On Eigenvalues of the sum of two random projections

We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N are two N -by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P_N + Q_N is not universal in the usual sense.Comment: 14 page

Transmission Properties of the oscillating delta-function potential

We derive an exact expression for the transmission amplitude of a particle moving through a harmonically driven delta-function potential by using the method of continued-fractions within the framework of Floquet theory. We prove that the transmission through this potential as a function of the incident energy presents at most two real zeros, that its poles occur at energies $n\hbar\omega+\varepsilon^*$ ($0<Re(\varepsilon^*)<\hbar\omega$), and that the poles and zeros in the transmission amplitude come in pairs with the distance between the zeros and the poles (and their residue) decreasing with increasing energy of the incident particle. We also show the existence of non-resonant "bands" in the transmission amplitude as a function of the strength of the potential and the driving frequency.Comment: 21 pages, 12 figures, 1 tabl

Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process

Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d. complex Gaussian coefficients a_n. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent {0,1}-valued random variables X_k, where P(X_k=1)=r^{2k}. Moreover, the set of absolute values of the zeros of f has the same distribution as the set {U_k^{1/2k}} where the U_k are i.i.d. random variables uniform in [0,1]. The repulsion between zeros can be studied via a dynamic version where the coefficients perform Brownian motion; we show that this dynamics is conformally invariant.Comment: 37 pages, 2 figures, updated proof

Modal Series Expansions for Plane Gravitational Waves

[EN] Propagation of gravitational disturbances at the speed of light is one of the key predictions of the General Theory of Relativity. This result is now backed indirectly by the observations of the behavior of the ephemeris of binary pulsar systems. These new results have increased the interest in the mathematical theory of gravitational waves in the last decades, and severalmathematical approaches have been developed for a better understanding of the solutions. In this paper we develop a modal series expansion technique in which solutions can be built for plane waves from a seed integrable function. The convergence of these series is proven by the Raabe-Duhamel criteria, and we show that these solutions are characterized by a well-defined and finite curvature tensor and also a finite energy content.Acedo Rodríguez, L. (2016). Modal Series Expansions for Plane Gravitational Waves. Gravitation and Cosmology. 22(3):251-257. doi:10.1134/S0202289316030026S251257223A. Einstein and N. Rosen, Journal of the Franklin Institute 223, 43–54 (1937).N. Rosen, Gen. Rel. Grav. 10, 351–364 (1979).C. Sivaram, Bull. Astr. Soc. India 23, 77–83 (1995).J. M. Weisberg, D. J. Nice, and J. H. Taylor, Astroph. J. 722, 1030–1034(2010); arXiv: 1011.0718.B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016).J. B. Griffiths, Colliding waves in general relativity (Clarendon, Oxford, 1991).S. Chandrasekhar, The mathematical theory of black holes (Clarendon, Oxford, 1983).D. Bini, V. Ferrari and J. Ibañez, Nuovo Cim. B 103, 29–44 (1989).L. Acedo, G. González-Parra, and A. J. Arenas, Nonlinear Analysis: Real World Applications 11, 1819–1825 (2010).L. Acedo, G. González-Parra, and A. J. Arenas, Physica A 389, 1151–1157 (2010).G. González-Parra, L. Acedo, and A. J. Arenas, Numerical Algorithms, published online 2013. doi 10.1007/s11075-013-9776-xW. Rindler, Relativity: Special, General and Cosmological, 2nd ed. (Oxford Univ., New York, 2006).G. Arfken, Mathematical Methods for Physicists, 3rd. ed. (Academic, Orlando, Florida, 1985).L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 3rd ed. (Pergamon, New York, 1971).O. Costin, “Topological construction of transseries and introduction to generalized Borel summability,” in Analyzable Functions and Applications, Ed. by O. Costin, M. D. Kruskal, and A. Macintyre, Contemp. Math. 373 (Providence, RI, USA: Am. Math. Soc., 2005); arXiv: math/0608309.S. R. Coleman, Phys. Lett. B 70, 59–60 (1977).W. B. Campbell and T. A. Morgan, Phys. Lett. B 84, 87–88 (1979).A. S. Rabinowitch, Int. J. Adv. Math. Sciences 1 (3), 109–121 (2013).A. Feinstein and J. Ibañez, Phys. Rev. D 39 (2), 470–473 (1989)

Finite N Fluctuation Formulas for Random Matrices

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic $\sum_{j=1}^N (x_j - )$ is computed exactly and shown to satisfy a central limit theorem as $N \to \infty$. For the circular random matrix ensemble the p.d.f.'s for the linear statistics ${1 \over 2} \sum_{j=1}^N (\theta_j - \pi)$ and $- \sum_{j=1}^N \log 2|\sin \theta_j/2|$ are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem as $N \to \infty$.Comment: LaTeX 2.09, 11 pages + 3 eps figs (needs epsf.sty

Loss of Tumor Suppressor TMEM127 Drives Ret-Mediated Transformation Through Disrupted Membrane Dynamics

Internalization from the cell membrane and endosomal trafficking of receptor tyrosine kinases (RTKs) are important regulators of signaling in normal cells that can frequently be disrupted in cancer. The adrenal tumor pheochromocytoma (PCC) can be caused by activating mutations of the rearranged during transfection (RET) receptor tyrosine kinase, or inactivation of TMEM127, a transmembrane tumor suppressor implicated in trafficking of endosomal cargos. However, the role of aberrant receptor trafficking in PCC is not well understood. Here, we show that loss of TMEM127 causes wildtype RET protein accumulation on the cell surface, where increased receptor density facilitates constitutive ligand-independent activity and downstream signaling, driving cell proliferation. Loss of TMEM127 altered normal cell membrane organization and recruitment and stabilization of membrane protein complexes, impaired assembly, and maturation of clathrin-coated pits, and reduced internalization and degradation of cell surface RET. In addition to RTKs, TMEM127 depletion also promoted surface accumulation of several other transmembrane proteins, suggesting it may cause global defects in surface protein activity and function. Together, our data identify TMEM127 as an important determinant of membrane organization including membrane protein diffusability and protein complex assembly and provide a novel paradigm for oncogenesis in PCC where altered membrane dynamics promotes cell surface accumulation and constitutive activity of growth factor receptors to drive aberrant signaling and promote transformation

The Legacy of Hope Summit: A Consensus-Based Initiative and Report on Eating Disorders in the U.S. and Recommendations for the Path Forward

Background: Several unsuccessful attempts have been made to reach a cross-disciplinary consensus on issues fundamental to the field of eating disorders in the United States (U.S.). In January 2020, 25 prominent clinicians, academicians, researchers, persons with lived experience, and thought leaders in the U.S. eating disorders community gathered at the Legacy of Hope Summit to try again. This paper articulates the points on which they reached a consensus. It also: (1) outlines strategies for implementing those recommendations; (2) identifies likely obstacles to their implementation; and (3) charts a course for successfully navigating and overcoming those challenges. Methods: Iterative and consensual processes were employed throughout the Summit and the development of this manuscript. Results: The conclusion of the Summit culminated in several consensus points, including: (1) Eating disorder outcomes and prevention efforts can be improved by implementing creative health education initiatives that focus on societal perceptions, early detection, and timely, effective intervention; (2) Such initiatives should be geared toward parents/guardians, families, other caretakers, and frontline healthcare providers in order to maximize impact; (3) Those afflicted with eating disorders, their loved ones, and the eating disorders community as a whole would benefit from greater accessibility to affordable, quality care, as well as greater transparency and accountability on the part of in-hospital, residential, and outpatient health care providers with respect to their qualifications, methodologies, and standardized outcomes; (4) Those with lived experience with eating disorders, their loved ones, health care providers, and the eating disorders community as a whole, also would benefit from the establishment and maintenance of treatment program accreditation, professional credentialing, and treatment type and levels of care guidelines; and (5) The establishment and implementation of effective, empirically/evidence-based standards of care requires research across a diverse range of populations, adequate private and government funding, and the free exchange of ideas and information among all who share a commitment to understanding, treating, and, ultimately, markedly diminishing the negative impact of eating disorders. Conclusions: Widespread uptake and implementation of these recommendations has the potential to unify and advance the eating disorders field and ultimately improve the lives of those affected. A cross-disciplinary group of eating disorder professionals, thought leaders, and persons with lived experience have come together and reached a consensus on issues that are fundamental to the battle against the life-threatening and life-altering illnesses that are eating spectrum disorders. Those issues include: (1) the need for early detection, intervention, prevention, and evidenced-based standards of care; (2) the critical need to make specialized care more accessible and affordable to all those in need; (3) the importance of developing uniform, evidenced-based standards of care; (4) the need for funding and conducting eating spectrum disorder research; and (5) the indispensability of advocacy, education, and legislation where these illnesses are concerned. During the consensus process, the authors also arrived at strategies for implementing their recommendations, identified likely obstacles to their implementation, and charted a course for successfully navigating and overcoming those challenges. Above all else, the authors demonstrated that consensus in the field of eating spectrum disorders is possible and achievable and, in doing so, lit a torch of hope that is certain to light the path forward for years to come