arXiv subjects

Bart Rosenzweig

Publications and source records attributed to Bart Rosenzweig.

5 recordsLinked to original sources

arXiv2026

On the fundamental solutions of two nonlocal parabolic equations related to logarithmic Laplacians

We answer in the affirmative a question posed by V. Maz'ya of whether one can continue as a meromorphic function of $t$ the series representation of the fundamental solution of a certain nonlocal parabolic equation associated to a logarithmic Laplacian on the circle, which arises in the study of boundary value problems associated to the ordinary Laplacian on domains with thin cavities. The $a\ln(n) +O(1)$ growth of the eigenvalues of the integral operator, together with explicit formulas for the eigenfunctions and the subleading asymptotic behavior of the eigenvalues, allows us to show that the fundamental solution is reminiscent of a sum of shifted Riemann zeta functions or polylogarithms, depending on the spatial variable. We show an analogous result for an operator related to a different logarithmic Laplacian on the interval, whose structure is similar. Along the way we are led to prove and to conjecture a number of curious identities involving Bell polynomials and Bernoulli numbers related to the exponential of the digamma function which are of independent interest.

math.CV↗

arXiv2025

An elementary approach to integral inequalities involving higher order derivatives

Motivated by previous work leveraging factorizations of second- and fourth-order differential operators, a general integral inequality involving higher order derivatives is proven by elementary means. It is then shown how this framework generalizes the notions of Hardy improving potentials and Bessel pairs. Numerous examples of inequalities both new and previously known in the literature are given that may be proven in this manner.

math.CA↗

arXiv2025

Maximally dissipative and self-adjoint extensions of $K$-invariant operators

We introduce the notion of $K$-invariant operators, $S$, (in a Hilbert space) with respect to a bounded and boundedly invertible operator $K$ defined via $K^*SK=S$. Conditions such that self-adjoint and maximally dissipative extensions of $K$-invariant symmetric operators are also $K$-invariant are investigated. In particular, the Friedrichs and Krein--von Neumann extensions of a nonnegative $K$-invariant symmetric operator are shown to always be $K$-invariant, while the Friedrichs extension of a $K$-invariant sectorial operator is as well. We apply our results to the case of Sturm--Liouville operators where $K$ is given by $(Kf)(x)=A(x)f(ϕ(x))$ under appropriate assumptions. Sufficient conditions on the coefficient functions for $K$-invariance to hold are shown to be related to Schröder's equation and all $K$-invariant self-adjoint extensions are characterized. Explicit examples are discussed including a Bessel-type Schrödinger operator satisfying a nontrivial $K$-invariance on the half-line.

math.SP↗

arXiv2021

Analysis of the Classical Trajectory Treatment of Photon Dynamics for Polaritonic Phenomena

Simulating photon dynamics in strong light-matter coupling situations via classical trajectories is proving to be powerful and practical. Here we analyze the performance of the approach through the lens of the exact factorization approach. Since the exact factorization enables a rigorous definition of the potentials driving the photonic motion it allows us to identify that the cause of the underestimation of photon number and intensities observed in earlier work is primarily due to an inadequate accounting of light-matter correlation in the classical Ehrenfest force rather than errors from treating the photons quasiclassically per se. The latter becomes problematic when the number of photons per mode begins to exceed a half.

quant-ph↗

arXiv2020

Case Studies of the Time-Dependent Potential Energy Surface for Dynamics in Cavities

The exact time-dependent potential energy surface driving the nuclear dynamics was recently shown to be a useful tool to understand and interpret the coupling of nuclei, electrons, and photons, in cavity settings. Here we provide a detailed analysis of its structure for exactly-solvable systems that model two phenomena: cavity-induced suppression of proton-coupled electron transfer and its dependence on the initial state, and cavity-induced electronic excitation. We demonstrate the inadequacy of simply using a weighted average of polaritonic surfaces to determine the dynamics. Such a weighted average misses a crucial term that redistributes energy between the nuclear and the polaritonic systems, and this term can in fact become a predominant term in determining the nuclear dynamics when several polaritonic surfaces are involved. Evolving an ensemble of classical trajectories on the exact potential energy surface reproduces the nuclear wavepacket quite accurately while evolving on the weighted polaritonic surface fails after a short period of time. The implications and prospects for application of mixed quantum-classical methods based on this surface are discussed.

quant-ph↗