Semiclassical energy formulas for power-law and log potentials in quantum mechanics

We study a single particle which obeys non-relativistic quantum mechanics in R^N and has Hamiltonian H = -Delta + V(r), where V(r) = sgn(q)r^q. If N \geq 2, then q > -2, and if N = 1, then q > -1. The discrete eigenvalues E_{n\ell} may be represented exactly by the semiclassical expression E_{n\ell}(q) = min_{r>0}\{P_{n\ell}(q)^2/r^2+ V(r)}. The case q = 0 corresponds to V(r) = ln(r). By writing one power as a smooth transformation of another, and using envelope theory, it has earlier been proved that the P_{n\ell}(q) functions are monotone increasing. Recent refinements to the comparison theorem of QM in which comparison potentials can cross over, allow us to prove for n = 1 that Q(q)=Z(q)P(q) is monotone increasing, even though the factor Z(q)=(1+q/N)^{1/q} is monotone decreasing. Thus P(q) cannot increase too slowly. This result yields some sharper estimates for power-potential eigenvlaues at the bottom of each angular-momentum subspace.Comment: 20 pages, 5 figure

Connection Conditions and the Spectral Family under Singular Potentials

To describe a quantum system whose potential is divergent at one point, one must provide proper connection conditions for the wave functions at the singularity. Generalizing the scheme used for point interactions in one dimension, we present a set of connection conditions which are well-defined even if the wave functions and/or their derivatives are divergent at the singularity. Our generalized scheme covers the entire U(2) family of quantizations (self-adjoint Hamiltonians) admitted for the singular system. We use this scheme to examine the spectra of the Coulomb potential $V(x) = - e^2 / | x |$ and the harmonic oscillator with square inverse potential $V(x) = (m \omega^2 / 2) x^2 + g/x^2$, and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra. We further show that, for any parity invariant singular potentials $V(-x) = V(x)$, the spectrum is determined solely by the eigenvalues of the characteristic matrix $U \in U(2)$.Comment: TeX, 18 page

Subsonic Boundary-Layer Wavefront Spectra for a Range of Reynolds Numbers

Aero-optic measurements of turbulent boundary layers were performed in wind tunnels at the University of Notre Dame and California Institute of Technology for heated walls at a range of Reynolds numbers. Temporally resolved measurements of wavefronts were collected at a range of Mach numbers between 0.03 and 0.4 and the range of Re_θ between 1,700 and 20,000. Wavefront spectra for both heated and un-heated walls were extracted and compared to demonstrate that wall heating does not noticeably alter the shape of wavefront spectra in the boundary layer. The effect of Reynolds number on the normalized spectra was also presented, and an empirical spectral model was modified to account for Reynolds number dependence. Measurements of OPD_(rms) for heated walls were shown to be consistent with results from prior experiments, and a method of estimating OPD_(rms) and other boundary layer statistics from wavefront measurements of heated-wall boundary layers was demonstrated and discussed

Supersonic plasma turbulence in the laboratory

The properties of supersonic, compressible plasma turbulence determine the behavior of many terrestrial and astrophysical systems. In the interstellar medium and molecular clouds, compressible turbulence plays a vital role in star formation and the evolution of our galaxy. Observations of the density and velocity power spectra in the Orion B and Perseus molecular clouds show large deviations from those predicted for incompressible turbulence. Hydrodynamic simulations attribute this to the high Mach number in the interstellar medium (ISM), although the exact details of this dependence are not well understood. Here we investigate experimentally the statistical behavior of boundary-free supersonic turbulence created by the collision of two laser-driven high-velocity turbulent plasma jets. The Mach number dependence of the slopes of the density and velocity power spectra agree with astrophysical observations, and supports the notion that the turbulence transitions from being Kolmogorov-like at low Mach number to being more Burgers-like at higher Mach numbers

Neoadjuvant chemoradiotherapy for locally advanced rectal cancer with peritumoral abscesses and fistulas

Purpose: Rectal cancer patients who present with peritumoral abscesses and fistulas at the time of diagnosis may be denied chemoradiotherapy (CRT) as the safety is unknown. The aim of this study was to investigate the safety of preoperative CRT in this patient group. Methods: We performed a retrospective nested case–control study to compare outcomes between patients with locally advanced rectal cancer with peritumoral abscesses and fistulas (study group) and patients with T4 locally advanced rectal cancer with no evidence of abscesses and fistulas (control group). These groups were matched by treatment center and radiotherapy delivery method. All patients received 50–54 Gy of conventionally fractionated RT with concurrent chemotherapy. Primary endpoint was grade 3–5 toxicity (by National Cancer Institute Common Toxicity Criteria for Adverse Events (NCI CTCAE). Secondary endpoints included postoperative morbidity, pathologic complete response (pCR), disease-free survival (DFS), and overall survival (OS) at 2 years. Results: A total of 33 patients were included in each group. Grade 3 toxicity was observed in 2 (6.1%) patients in the study group and 4 (12.1%) patients in the control group (p = 0.672). No patients developed grade 4–5 toxicity. Grade 3–4 Clavien–Dindo complications were observed in 5 (15.2%) patients in the study group and in 6 (18.2%) patients in the control group (p = 1.0). Pathologic CR was achieved in 3 (9.1%) and 5 (15.2%) patients, respectively (p = 0.708). Two-year OS was 78.3% vs. 81.8% (p = 0.944), 2‑year DFS was 62.8% vs. 69.7% (p = 0.693), respectively. Conclusion: The presence of peritumoral abscesses and fistulas in patients with locally advanced rectal cancer is not associated with increased toxicity or inferior clinical outcomes after preoperative CRT. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany

The first Russian experience in successfully using an implantable СardioWest TAH-t artificial heart system (SynCardia)

The paper describes the first Russian experience in using an implantable CardioWest TAH-t artificial heart (AH) system (SynCardia, USA) to treat critical heart failure. The AH system was implanted in a 60-year-old female patient with dilated cardiomyopathy, a limited probability of survival, and unoperated defects. The main medical and technical characteristics of the AH system and the patient’s clinical status in the preoperative and immediate and late (up to 238 days) postoperative periods are presented. Indications for and contraindication to implantation of the system and its possible application modes are considered. Equipment and a procedure for AH implantation, approaches to postoperative management, and treatment policy for postoperative complications are described in detail. The application of the AH system permitted the patient to wait for a donor heart. The latter was successfully transplanted in the patient