SUSY transformation of the Green function and a trace formula

An integral relation is established between the Green functions corresponding to two Hamiltonians which are supersymmetric (SUSY) partners and in general may possess both discrete and continuous spectra. It is shown that when the continuous spectrum is present the trace of the difference of the Green functions for SUSY partners is a finite quantity which may or may not be equal to zero despite the divergence of the traces of each Green function. Our findings are illustrated by using the free particle example considered both on the whole real line and on a half line

Convergence of expansions in Schr\"odinger and Dirac eigenfunctions, with an application to the R-matrix theory

Expansion of a wave function in a basis of eigenfunctions of a differential eigenvalue problem lies at the heart of the R-matrix methods for both the Schr\"odinger and Dirac particles. A central issue that should be carefully analyzed when functional series are applied is their convergence. In the present paper, we study the properties of the eigenfunction expansions appearing in nonrelativistic and relativistic $R$-matrix theories. In particular, we confirm the findings of Rosenthal [J. Phys. G: Nucl. Phys. 13, 491 (1987)] and Szmytkowski and Hinze [J. Phys. B: At. Mol. Opt. Phys. 29, 761 (1996); J. Phys. A: Math. Gen. 29, 6125 (1996)] that in the most popular formulation of the R-matrix theory for Dirac particles, the functional series fails to converge to a claimed limit.Comment: Revised version, accepted for publication in Journal of Mathematical Physics, 21 pages, 1 figur

A Tale of Two Democrats: How Authoritarianism Divides the Democratic Party

Authoritarianism has been predominantly used in American politics as a predictor of Republican identification and conservative policy preferences. We argue that this approach has neglected the role authoritarianism plays among Democrats and how it can operate within political parties regardless of their ideological orientation. Drawing from three distinct sets of data, we demonstrate the impact of authoritarianism in the 2016 Democratic Party’s primaries. Authoritarianism consistently predicts differences in primary voting among Democrats, particularly support for Hillary Clinton over Bernie Sanders. This effect is robust across various model specifications including controls for ideology, partisan strength, and other predispositions. These results highlight the potential of authoritarianism to shape leadership preferences within the Democratic Party. We advocate for a reconsideration of authoritarianism as a disposition with meaningful consequences for intraparty dynamics and conclude with practical implications regarding the future of the Democratic Party

Adaptive walks on time-dependent fitness landscapes

The idea of adaptive walks on fitness landscapes as a means of studying evolutionary processes on large time scales is extended to fitness landscapes that are slowly changing over time. The influence of ruggedness and of the amount of static fitness contributions are investigated for model landscapes derived from Kauffman's $NK$ landscapes. Depending on the amount of static fitness contributions in the landscape, the evolutionary dynamics can be divided into a percolating and a non-percolating phase. In the percolating phase, the walker performs a random walk over the regions of the landscape with high fitness.Comment: 7 pages, 6 eps-figures, RevTeX, submitted to Phys. Rev.

Inverse Spectral-Scattering Problem with Two Sets of Discrete Spectra for the Radial Schroedinger Equation

The Schroedinger equation on the half line is considered with a real-valued, integrable potential having a finite first moment. It is shown that the potential and the boundary conditions are uniquely determined by the data containing the discrete eigenvalues for a boundary condition at the origin, the continuous part of the spectral measure for that boundary condition, and a subset of the discrete eigenvalues for a different boundary condition. This result extends the celebrated two-spectrum uniqueness theorem of Borg and Marchenko to the case where there is also a continuous spectru

Solution of the Fokker-Planck equation with a logarithmic potential and mixed eigenvalue spectrum

Motivated by a problem in climate dynamics, we investigate the solution of a Bessel-like process with negative constant drift, described by a Fokker-Planck equation with a potential V(x) = - [b \ln(x) + a\, x], for b>0 and a<0. The problem belongs to a family of Fokker-Planck equations with logarithmic potentials closely related to the Bessel process, that has been extensively studied for its applications in physics, biology and finance. The Bessel-like process we consider can be solved by seeking solutions through an expansion into a complete set of eigenfunctions. The associated imaginary-time Schroedinger equation exhibits a mix of discrete and continuous eigenvalue spectra, corresponding to the quantum Coulomb potential describing the bound states of the hydrogen atom. We present a technique to evaluate the normalization factor of the continuous spectrum of eigenfunctions that relies solely upon their asymptotic behavior. We demonstrate the technique by solving the Brownian motion problem and the Bessel process both with a negative constant drift. We conclude with a comparison with other analytical methods and with numerical solutions.Comment: 21 pages, 8 figure

Tracing the Orphan Stream to 55 kpc with RR Lyrae Stars

We report positions, velocities and metallicities of 50 ab-type RR Lyrae (RRab) stars observed in the vicinity of the Orphan stellar stream. Using about 30 RRab stars classified as being likely members of the Orphan stream, we study the metallicity and the spatial extent of the stream. We find that RRab stars in the Orphan stream have a wide range of metallicities, from -1.5 dex to -2.7 dex. The average metallicity of the stream is -2.1 dex, identical to the value obtained by Newberg et al. (2010) using blue horizontal branch stars. We find that the most distant parts of the stream (40-50 kpc from the Sun) are about 0.3 dex more metal-poor than the closer parts (within ~30 kpc), suggesting a possible metallicity gradient along the stream's length. We have extended the previous studies and have mapped the stream up to 55 kpc from the Sun. Even after a careful search, we did not identify any more distant RRab stars that could plausibly be members of the Orphan stream. If confirmed with other tracers, this result would indicate a detection of the end of the leading arm of the stream. We have compared the distances of Orphan stream RRab stars with the best-fit orbits obtained by Newberg et al. (2010). We find that model 6 of Newberg et al. (2010) cannot explain the distances of the most remote Orphan stream RRab stars, and conclude that the best fit to distances of Orphan stream RRab stars and to the local circular velocity is provided by potentials where the total mass of the Galaxy within 60 kpc is M_{60}~2.7x10^{11} Msun, or about 60% of the mass found by previous studies. More extensive modelling that would consider non-spherical potentials and the possibility of misalignment between the stream and the orbit, is highly encouraged.Comment: Submitted to ApJ, 15 pages in emulateapj format, three tables in machine-readable format (download "Source" from "Other formats"

Exact propagators for SUSY partners

Pairs of SUSY partner Hamiltonians are studied which are interrelated by usual (linear) or polynomial supersymmetry. Assuming the model of one of the Hamiltonians as exactly solvable with known propagator, expressions for propagators of partner models are derived. The corresponding general results are applied to "a particle in a box", the Harmonic oscillator and a free particle (i.e. to transparent potentials).Comment: 31 page