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1. Sharp decay rate for eigenfunctions of perturbed periodic Schrödinger operators

   https://doi.org/10.1088/1361-6544/adc653  

   Liu, Wencai; Matos, Rodrigo; Treuer, John N (April 2025, Nonlinearity)

   Abstract This paper investigates uniqueness results for perturbed periodic Schrödinger operators on ${\mathbb{Z}}^d$. Specifically, we consider operators of the form $H=-\mathrm{\Delta }+V+v$, where Δ is the discrete Laplacian, $V:{\mathbb{Z}}^d\to \mathbb{R}$is a periodic potential, and $v:{\mathbb{Z}}^d\to \mathbb{C}$represents a decaying impurity. We establish quantitative conditions under which the equation $-\mathrm{\Delta }u+Vu+vu=\lambda u$, for $\lambda \in \mathbb{C}$, admits only the trivial solution $u\equiv 0$. Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes ofλ. 

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2. Characterizing the fundamental bending vibration of a linear polyatomic molecule for symmetry violation searches

   https://doi.org/10.1088/1367-2630/ace471  

   Jadbabaie, Arian; Takahashi, Yuiki; Pilgram, Nickolas H; Conn, Chandler J; Zeng, Yi; Zhang, Chi; Hutzler, Nicholas R (July 2023, New Journal of Physics)

   Abstract Polyatomic molecules have been identified as sensitive probes of charge-parity violating and parity violating physics beyond the Standard Model (BSM). For example, many linear triatomic molecules are both laser-coolable and have parity doublets in the ground electronic $\tilde{X}{}^2{\mathrm{\Sigma }}^{+}\left(010\right)$state arising from the bending vibration, both features that can greatly aid BSM searches. Understanding the $\tilde{X}{}^2{\mathrm{\Sigma }}^{+}\left(010\right)$state is a crucial prerequisite to precision measurements with linear polyatomic molecules. Here, we characterize the fundamental bending vibration of ${}^{174}$YbOH using high-resolution optical spectroscopy on the nominally forbidden $\tilde{X}{}^2{\mathrm{\Sigma }}^{+}\left(010\right)$ $\to \tilde{A}{}^2{\mathrm{\Pi }}_{1/2}\left(000\right)$transition at 588 nm. We assign 39 transitions originating from the lowest rotational levels of the $\tilde{X}{}^2{\mathrm{\Sigma }}^{+}\left(010\right)$state, and accurately model the state’s structure with an effective Hamiltonian using best-fit parameters. Additionally, we perform Stark and Zeeman spectroscopy on the $\tilde{X}{}^2{\mathrm{\Sigma }}^{+}\left(010\right)$state and fit the molecule-frame dipole moment to $D_{\mathrm{m}\mathrm{o}\mathrm{l}}=2.16\left(1\right)$Dand the effective electrong-factor to $g_S=2.07\left(2\right)$. Further, we use an empirical model to explain observed anomalous line intensities in terms of interference from spin–orbit and vibronic perturbations in the excited $\tilde{A}{}^2{\mathrm{\Pi }}_{1/2}\left(000\right)$state. Our work is an essential step toward searches for BSM physics in YbOH and other linear polyatomic molecules. 

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3. Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

   https://doi.org/10.1088/1361-6544/ac50bb  

   Chen, Qinbo; de la Llave, Rafael (March 2022, Nonlinearity)

   Abstract The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the followinga prioriunstable Hamiltonian system with a time-periodic perturbation ${\mathcal{H}}_{\epsilon }\left(p,q,I,\phi ,t\right)=h\left(I\right)+\sum _{i=1}^n\pm \left(\frac{1}{2}{p}_i^2+V_i\left(q_i\right)\right)+\epsilon H_1\left(p,q,I,\phi ,t\right),$where $\left(p,q\right)\in {\mathbb{R}}^n\times {\mathbb{T}}^n$, $\left(I,\phi \right)\in {\mathbb{R}}^d\times {\mathbb{T}}^d$withn,d⩾ 1,V_iare Morse potentials, andɛis a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbationsH₁. Indeed, the set of admissibleH₁isC^ωdense andC³open (a fortiori,C^ωopen). Our perturbative technique for the genericity is valid in theC^ktopology for allk∈ [3, ∞) ∪ {∞,ω}. 

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4. Bicycling geodesics are Kirchhoff rods

   https://doi.org/10.1088/1361-6544/acd613  

   Bor, Gil; Jackman, Connor; Tabachnikov, Serge (May 2023, Nonlinearity)

   Abstract A bicycle path is a pair of trajectories in ${\mathbb{R}}^n$, the ‘front’ and ‘back’ tracks, traced out by the endpoints of a moving line segment of fixed length (the ‘bicycle frame’) and tangent to the back track. Bicycle geodesics are bicycle paths whose front track’s length is critical among all bicycle paths connecting two given placements of the line segment. We write down and study the associated variational equations, showing that for $n⩾3$each such geodesic is contained in a 3-dimensional affine subspace and that the front tracks of these geodesics form a certain subfamily ofKirchhoff rods, a class of curves introduced in 1859 by Kirchhoff, generalizing the planar elastic curves of Bernoulli and Euler. 

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5. Validation of Elemental and Isotopic Abundances in Late-M Spectral Types with the Benchmark HIP 55507 AB System

   https://doi.org/10.3847/1538-4357/ad1243  

   Xuan, Jerry W.; Wang, Jason; Finnerty, Luke; Horstman, Katelyn; Grimm, Simon; Peck, Anne E.; Nielsen, Eric; Knutson, Heather A.; Mawet, Dimitri; Isaacson, Howard; et al (February 2024, The Astrophysical Journal)

   Abstract M dwarfs are common host stars to exoplanets but often lack atmospheric abundance measurements. Late-M dwarfs are also good analogs to the youngest substellar companions, which share similarT_eff∼ 2300–2800 K. We present atmospheric analyses for the M7.5 companion HIP 55507 B and its K6V primary star with Keck/KPIC high-resolution (R∼ 35,000)K-band spectroscopy. First, by including KPIC relative radial velocities between the primary and secondary in the orbit fit, we improve the dynamical mass precision by 60% and find $M_B={88.0}_{-3.2}^{+3.4}{M}_{\mathrm{Jup}}$, putting HIP 55507 B above the stellar–substellar boundary. We also find that HIP 55507 B orbits its K6V primary star with $a={38}_{-3}^{+4}$au ande= 0.40 ± 0.04. From atmospheric retrievals of HIP 55507 B, we measure [C/H] = 0.24 ± 0.13, [O/H] = 0.15 ± 0.13, and C/O = 0.67 ± 0.04. Moreover, we strongly detect¹³CO (7.8σsignificance) and tentatively detect ${\mathrm{H}}_2^{18}\mathrm{O}$(3.7σsignificance) in the companion’s atmosphere and measure ${}^{12}\mathrm{CO}{/}^{13}\mathrm{CO}={98}_{-22}^{+28}$and ${\mathrm{H}}_2^{16}\mathrm{O}/{\mathrm{H}}_2^{18}\mathrm{O}={240}_{-80}^{+145}$after accounting for systematic errors. From a simplified retrieval analysis of HIP 55507 A, we measure ${}^{12}\mathrm{CO}{/}^{13}\mathrm{CO}={79}_{-16}^{+21}$and ${\mathrm{C}}^{16}\mathrm{O}/{\mathrm{C}}^{18}\mathrm{O}={288}_{-70}^{+125}$for the primary star. These results demonstrate that HIP 55507 A and B have consistent¹²C/¹³C and¹⁶O/¹⁸O to the <1σlevel, as expected for a chemically homogeneous binary system. Given the similar flux ratios and separations between HIP 55507 AB and systems with young substellar companions, our results open the door to systematically measuring¹³CO and ${\mathrm{H}}_2^{18}\mathrm{O}$abundances in the atmospheres of substellar or even planetary-mass companions with similar spectral types. 

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