arXiv subjects

Joe P. Chen

Publications and source records attributed to Joe P. Chen.

25 records · Page 2Linked to original sources

arXiv2016

Singularly continuous spectrum of a self-similar Laplacian on the half-line

We investigate the spectrum of the self-similar Laplacian, which generates the so-called "$pq$ random walk" on the integer half-line $\mathbb{Z}_+$. Using the method of spectral decimation, we prove that the spectral type of the Laplacian is singularly continuous whenever $p\neq \frac{1}{2}$. This serves as a toy model for generating singularly continuous spectrum, which can be generalized to more complicated settings. We hope it will provide more insight into Fibonacci and other weakly self-similar models.

math-ph↗

arXiv2015

Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces

We establish an asymptotic formulas for the eigenvalue counting function of the Schrödinger operator $-Δ+V$ for some unbounded potentials $V$ on several types of unbounded fractal spaces. We give sufficient conditions for Bohr's formula to hold on metric measure spaces which admit a cellular decomposition, and then verify these conditions for fractafolds and fractal fields based on nested fractals. In particular, we partially answer a question of Fan, Khandker, and Strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow-up of a Sierpinski gasket.

math-ph↗

arXiv2015

Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs

Consider the free field on a fractal graph based on a high-dimensional Sierpinski carpet (e.g. the Menger sponge), that is, a centered Gaussian field whose covariance is the Green's function for simple random walk on the graph. Moreover assume that a "hard wall" is placed at height zero so that the field stays positive everywhere. We prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph, thereby extending a result of Bolthausen, Deuschel, and Zeitouni for the free field on $\mathbb{Z}^d$, $d \geq 3$, to the fractal setting. Our proof utilizes the theory of transient regular Dirichlet forms, in conjunction with the relative entropy, coarse graining, and conditioning arguments introduced in the previous literature.

math.PR↗

arXiv2014

Heat kernels on 2d Liouville quantum gravity: a numerical study

We numerically compute the heat kernel on a square lattice torus equipped with the measure corresponding to Liouville quantum gravity (LQG). From the on-diagonal heat kernel we verify that the spectral dimension of LQG is 2. Furthermore, when diffusion is started from a high point of the underlying Gaussian free field, our numerics indicates superdiffusive space-time scaling with respect to the Euclidean metric in the small space-to-time regime. The implications of this result require further investigation, but seem to coincide with the notion that the Euclidean metric is not the right geodesic for characterizing the geometry of LQG.

math-ph↗

arXiv2013

Periodic billiard orbits of self-similar Sierpinski carpets

We identify a collection of periodic billiard orbits in a self-similar Sierpinski carpet billiard table. Based on a refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in a self-similar Sierpinski carpet, we construct what is called an eventually constant sequence of compatible periodic orbits of prefractal Sierpinski carpet billiard tables. The trivial limit of this sequence then constitutes a periodic orbit of a self-similar Sierpinski carpet billiard table. We also determine the corresponding translation surface for each prefractal billiard table, and show that the genera of a sequence of translation surfaces increase without bound. Various open questions and possible directions for future research are offered.

math.DS↗

arXiv2012

Statistical mechanics of Bose gas in Sierpinski carpets

We carry out a mathematically rigorous investigation into the equilibrium thermodynamics of massless and massive bosons confined in generalized Sierpinski carpets (GSCs), a class of infinitely ramified fractals having non-integer Hausdorff dimensions $d_h$. Due to the anomalous walk dimension $d_w>2$ associated with Brownian motion on GSCs, all extensive thermodynamic quantities are shown to scale with the spectral volume with dimension $d_s = 2(d_h/d_w)$ rather than the Hausdorff volume. We prove that for a low-temperature, high-density ideal massive Bose gas in an unbounded GSC, Bose-Einstein condensation occurs if and only if $d_s>2$, or equivalently, if the Brownian motion on the GSC is transient. We also derive explicit expressions for the energy of blackbody radiation in a GSC, as well as the Casimir pressure on the parallel plate of a fractal waveguide modelled after a GSC. Our proofs involve extensive use of the spectral zeta function, obtained via a sharp estimate of the heat kernel trace. We believe that our results can be verified through photonic and cold atomic experiments on fractal structures.

math-ph↗

arXiv2007

Quantum Theory of Cavity-Assisted Sideband Cooling of Mechanical Motion

We present a fully quantum theory describing the cooling of a cantilever coupled via radiation pressure to an illuminated optical cavity. Applying the quantum noise approach to the fluctuations of the radiation pressure force, we derive the opto-mechanical cooling rate and the minimum achievable phonon number. We find that reaching the quantum limit of arbitrarily small phonon numbers requires going into the good cavity (resolved phonon sideband) regime where the cavity linewidth is much smaller than the mechanical frequency and the corresponding cavity detuning. This is in contrast to the common assumption that the mechanical frequency and the cavity detuning should be comparable to the cavity damping.

cond-mat.mes-hall↗