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1. Logarithmic terms in atom-surface potentials: Limited applicability of rational approximations for intermediate distance

   https://doi.org/10.1103/PhysRevA.108.012815  

   Jentschura, U D; Moore, C (July 2023, Physical Review A)

   It is usually assumed that interaction potentials, in general, and atom-surface potential, in particular, can be expressed in terms of an expansion involving integer powers of the distance between the two interacting objects. Here, we show that, in the short-range expansion of the interaction potential of a neutral atom and a dielectric surface, logarithms of the atom-wall distance appear. These logarithms are accompanied with logarithmic sums over virtual excitations of the atom interacting with the surface in analogy to Bethe logarithms in quantum electrodynamics. We verify the presence of the logarithmic terms in the short-range expansion using a model problem with realistic parameters. By contrast, in the long-range expansion of the atom-surface potential, no logarithmic terms appear, and the interaction potential can be described by an expansion in inverse integer powers of the atom-wall distance. Several subleading terms in the large-distance expansion are obtained as a byproduct of our investigations. Our findings explain why the use of simple interpolating rational functions for the description of the atom-wall interaction in the intermediate regions leads to significant deviations from exact formulas. 

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2. The binary Darboux transformation revisited and KdV solitons on arbitrary short‐range backgrounds

   https://doi.org/10.1111/sapm.12436  

   Rybkin, Alexei (August 2021, Studies in Applied Mathematics)

   Abstract In the KdV context, we revisit the classical Darboux transformation in the framework of the vector Riemann–Hilbert problem. This readily yields a version of the binary Darboux transformation providing a short‐cut to explicit formulas for solitons traveling on a wide range of background solutions. Our approach also links the binary Darboux transformation and the double commutation method. 

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3. Incorporating multiscale methylation effects into nucleosome-resolution chromatin models for simulating mesoscale fibers

   https://doi.org/10.1063/5.0242199  

   Li, Zilong; Portillo-Ledesma, Stephanie; Janani, Moshe; Schlick, Tamar (March 2025, The Journal of Chemical Physics)

   Histone modifications play a crucial role in regulating chromatin architecture and gene expression. Here we develop a multiscale model for incorporating methylation in our nucleosome-resolution physics-based chromatin model to investigate the mechanisms by which H3K9 and H3K27 trimethylation (H3K9me3 and H3K27me3) influence chromatin structure and gene regulation. We apply three types of energy terms for this purpose: short-range potentials are derived from all-atom molecular dynamics simulations of wildtype and methylated chromatosomes, which revealed subtle local changes; medium-range potentials are derived by incorporating contacts between HP1 and nucleosomes modified by H3K9me3, to incorporate experimental results of enhanced contacts for short chromatin fibers (12 nucleosomes); for long-range interactions we identify H3K9me3- and H3K27me3-associated contacts based on Hi-C maps with a machine learning approach. These combined multiscale effects can model methylation as a first approximation in our mesoscale chromatin model, and applications to gene systems offer new insights into the epigenetic regulation of genomes mediated by H3K9me3 and H3K27me3. 

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4. Spectral properties of Schrödinger operators with locally $$H^{-1}$$ potentials

   https://doi.org/10.4171/JST/490  

   Lukić, Milivoje; Sukhtaiev, Selim; Wang, Xingya (May 2024, Journal of Spectral Theory)

   We study half-line Schrödinger operators with locally H^{-1} potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last-Simon-type description of the absolutely continuous spectrum and sufficient conditions for different spectral types. In the second part, we focus on potentials which are decaying in a local H^{-1} sense; we establish a spectral transition between short-range and long-range potentials and an ell^2 spectral transition for sparse singular potentials. The regularization procedure used to handle distributional potentials is also well suited for controlling rapid oscillations in the potential; thus, even within the class of smooth potentials, our results apply in situations which would not classically be considered decaying or even bounded. 

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5. A continuous analog of the binary Darboux transformation for the Korteweg–de Vries equation

   https://doi.org/10.1111/sapm.12578  

   Rybkin, Alexei (April 2023, Studies in Applied Mathematics)

   Abstract In the Korteweg–de Vries equation (KdV) context, we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann–Hilbert problem and yields a new explicit formula for perturbation of the negative spectrum of a wide class of step‐type potentials without changing the rest of the scattering data. This extends the previously known formulas for inserting/removing finitely many bound states to arbitrary sets of negative spectrum of arbitrary nature. In the KdV context, our method offers same benefits as the classical binary Darboux transformation does. 

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