From scattering theory to complex wave dynamics in non-hermitian PT-symmetric resonators

I review how methods from mesoscopic physics can be applied to describe the multiple wave scattering and complex wave dynamics in non-hermitian PT-symmetric resonators, where an absorbing region is coupled symmetrically to an amplifying region. Scattering theory serves as a convenient tool to classify the symmetries beyond the single-channel case and leads to effective descriptions which can be formulated in the energy domain (via Hamiltonians) and in the time domain (via time evolution operators). These models can then be used to identify the mesoscopic time and energy scales which govern the spectral transition from real to complex eigenvalues. The possible presence of magneto-optical effects (a finite vector potential) in multichannel systems leads to a variant (termed PTT' symmetry) which imposes the same spectral constraints as PT symmetry. I also provide multichannel versions of generalized flux-conservation laws.Comment: 10 pages, 5 figures, minireview for a theme issue, Philosophical Transactions of the Royal Society

Orbital fluctuations and strong correlations in quantum dots

In this lecture note we focus our attention to quantum dot systems where exotic strongly correlated behavior develops due to the presence of orbital or charge degrees of freedom. After giving a concise overview of the theory of transport and Kondo effect through a single electron transistor, we discuss how SU(4) Kondo effect develops in dots having orbitally degenerate states and in double dot systems, and then study the singlet-triplet transition in lateral quantum dots. Charge fluctuations and Matveev's mapping to the two-channel Kondo model in the vicinity of charge degeneracy point are also discussed.Comment: Lecture note to appear in Philosophical Magazin

Energy levels and their correlations in quasicrystals

Quasicrystals can be considered, from the point of view of their electronic properties, as being intermediate between metals and insulators. For example, experiments show that quasicrystalline alloys such as AlCuFe or AlPdMn have conductivities far smaller than those of the metals that these alloys are composed from. Wave functions in a quasicrystal are typically intermediate in character between the extended states of a crystal and the exponentially localized states in the insulating phase, and this is also reflected in the energy spectrum and the density of states. In the theoretical studies we consider in this review, the quasicrystals are described by a pure hopping tight binding model on simple tilings. We focus on spectral properties, which we compare with those of other complex systems, in particular, the Anderson model of a disordered metal.Comment: 15 pages including 19 figures. Review article, submitted to Phil. Ma

Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups

For the coinvariant rings of finite Coxeter groups of types other than H$_4$, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.Comment: 18 page

Functional central limit theorems for vicious walkers

We consider the diffusion scaling limit of the vicious walker model that is a system of nonintersecting random walks. We prove a functional central limit theorem for the model and derive two types of nonintersecting Brownian motions, in which the nonintersecting condition is imposed in a finite time interval $(0,T]$ for the first type and in an infinite time interval $(0,\infty)$ for the second type, respectively. The limit process of the first type is a temporally inhomogeneous diffusion, and that of the second type is a temporally homogeneous diffusion that is identified with a Dyson's model of Brownian motions studied in the random matrix theory. We show that these two types of processes are related to each other by a multi-dimensional generalization of Imhof's relation, whose original form relates the Brownian meander and the three-dimensional Bessel process. We also study the vicious walkers with wall restriction and prove a functional central limit theorem in the diffusion scaling limit.Comment: AMS-LaTeX, 20 pages, 2 figures, v6: minor corrections made for publicatio

The absolute position of a resonance peak

It is common practice in scattering theory to correlate between the position of a resonance peak in the cross section and the real part of a complex energy of a pole of the scattering amplitude. In this work we show that the resonance peak position appears at the absolute value of the pole's complex energy rather than its real part. We further demonstrate that a local theory of resonances can still be used even in cases previously thought impossible

Do acute elevations of serum creatinine in primary care engender an increased mortality risk?

Background: The significant impact Acute Kidney Injury (AKI) has on patient morbidity and mortality emphasizes the need for early recognition and effective treatment. AKI presenting to or occurring during hospitalisation has been widely studied but little is known about the incidence and outcomes of patients experiencing acute elevations in serum creatinine in the primary care setting where people are not subsequently admitted to hospital. The aim of this study was to define this incidence and explore its impact on mortality. Methods: The study cohort was identified by using hospital data bases over a six month period. Inclusion criteria: People with a serum creatinine request during the study period, 18 or over and not on renal replacement therapy. The patients were stratified by a rise in serum creatinine corresponding to the Acute Kidney Injury Network (AKIN) criteria for comparison purposes. Descriptive and survival data were then analysed. Ethical approval was granted from National Research Ethics Service (NRES) Committee South East Coast and from the National Information Governance Board. Results: The total study population was 61,432. 57,300 subjects with ‘no AKI’, mean age 64.The number (mean age) of acute serum creatinine rises overall were, ‘AKI 1’ 3,798 (72), ‘AKI 2’ 232 (73), and ‘AKI 3’ 102 (68) which equates to an overall incidence of 14,192 pmp/year (adult). Unadjusted 30 day survival was 99.9% in subjects with ‘no AKI’, compared to 98.6%, 90.1% and 82.3% in those with ‘AKI 1’, ‘AKI 2’ and ‘AKI 3’ respectively. After multivariable analysis adjusting for age, gender, baseline kidney function and co-morbidity the odds ratio of 30 day mortality was 5.3 (95% CI 3.6, 7.7), 36.8 (95% CI 21.6, 62.7) and 123 (95% CI 64.8, 235) respectively, compared to those without acute serum creatinine rises as defined. Conclusions: People who develop acute elevations of serum creatinine in primary care without being admitted to hospital have significantly worse outcomes than those with stable kidney function

On bulk singularities in the random normal matrix model

We extend the method of rescaled Ward identities of Ameur-Kang-Makarov to study the distribution of eigenvalues close to a bulk singularity, i.e. a point in the interior of the droplet where the density of the classical equilibrium measure vanishes. We prove results to the effect that a certain "dominant part" of the Taylor expansion determines the microscopic properties near a bulk singularity. A description of the distribution is given in terms of a special entire function, which depends on the nature of the singularity (a Mittag-Leffler function in the case of a rotationally symmetric singularity).Comment: This version clarifies on the proof of Theorem

Outcomes following autologous hematopoietic stem cell transplant for patients with relapsed Wilms' tumor: a CIBMTR retrospective analysis.

Despite the marked improvement in the overall survival (OS) for patients diagnosed with Wilms' tumor (WT), the outcomes for those who experience relapse have remained disappointing. We describe the outcomes of 253 patients with relapsed WT who received high-dose chemotherapy (HDT) followed by autologous hematopoietic stem cell transplant (HCT) between 1990 and 2013, and were reported to the Center for International Blood and Marrow Transplantation Research. The 5-year estimates for event-free survival (EFS) and OS were 36% (95% confidence interval (CI); 29-43%) and 45% (95 CI; 38-51%), respectively. Relapse of primary disease was the cause of death in 81% of the population. EFS, OS, relapse and transplant-related mortality showed no significant differences when broken down by disease status at transplant, time from diagnosis to transplant, year of transplant or conditioning regimen. Our data suggest that HDT followed by autologous HCT for relapsed WT is well tolerated and outcomes are similar to those reported in the literature. As attempts to conduct a randomized trial comparing maintenance chemotherapy with consolidation versus HDT followed by stem cell transplant have failed, one should balance the potential benefits with the yet unknown long-term risks. As disease recurrence continues to be the most common cause of death, future research should focus on the development of consolidation therapies for those patients achieving complete response to therapy

Microscopic derivation of multichannel Hubbard models for ultracold nonreactive molecules in an optical lattice

Recent experimental advances in the cooling and manipulation of bialkali-metal dimer molecules have enabled the production of gases of ultracold molecules that are not chemically reactive. It has been presumed in the literature that in the absence of an electric field the low-energy scattering of such nonreactive molecules (NRMs) will be similar to atoms, in which a single s -wave scattering length governs the collisional physics. However, Doçaj et al. [Phys. Rev. Lett. 116, 135301 (2016)] argued that the short-range collisional physics of NRMs is much more complex than for atoms and that this leads to a many-body description in terms of a multichannel Hubbard model. In this work we show that this multichannel Hubbard model description of NRMs in an optical lattice is robust against the approximations employed by Doçaj et al. to estimate its parameters. We do so via an exact, albeit formal, derivation of a multichannel resonance model for two NRMs from an ab initio description of the molecules in terms of their constituent atoms. We discuss the regularization of this two-body multichannel resonance model in the presence of a harmonic trap and how its solutions form the basis for the many-body model of Doçaj et al.. We also generalize the derivation of the effective lattice model to include multiple internal states (e.g., rotational or hyperfine). We end with an outlook to future research