Level Splitting in Association with the Multiphoton Bloch-Siegert Shift

We present a unitary equivalent spin-boson Hamiltonian in which terms can be identified which contribute to the Bloch-Siegert shift, and to the level splittings at the anticrossings associated with the Bloch-Siegert resonances. First-order degenerate perturbation theory is used to develop approximate results in the case of moderate coupling for the level splitting.Comment: 8 pages, 2 figure

Dissecting the hadronic contributions to (g−2)μ(g-2)_\mu by Schwinger's sum rule

The theoretical uncertainty of $(g-2)_\mu$ is currently dominated by hadronic contributions. In order to express those in terms of directly measurable quantities, we consider a sum rule relating $g-2$ to an integral of a photo-absorption cross section. The sum rule, attributed to Schwinger, can be viewed as a combination of two older sum rules: Gerasimov-Drell-Hearn and Burkhardt-Cottingham. The Schwinger sum rule has an important feature, distinguishing it from the other two: the relation between the anomalous magnetic moment and the integral of a photo-absorption cross section is linear, rather than quadratic. The linear property makes it suitable for a straightforward assessment of the hadronic contributions to $(g-2)_\mu$. From the sum rule we rederive the Schwinger $\alpha/2\pi$ correction, as well as the formula for the hadronic vacuum-polarization contribution. As an example of the light-by-light contribution we consider the single-meson exchange.Comment: 6 pages, 10 figures, published version extended by a few clarifying remarks and an Appendi

Solyanik estimates and local H\"older continuity of halo functions of geometric maximal operators

Let $\mathcal{B}$ be a homothecy invariant basis consisting of convex sets in $\mathbb{R}^n$, and define the associated geometric maximal operator $M_{\mathcal{B}}$ by $$M_{\mathcal{B}} f(x) :=\sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_R |f|$$ and the halo function $\phi_{\mathcal{B}}(\alpha)$ on $(1,\infty)$ by $$\phi_{\mathcal B}(\alpha) :=\sup_{E \subset \mathbb{R}^n :\, 0 < |E| < \infty}\frac{1}{|E|}|\{x\in \mathbb{R}^n : M_{\mathcal{B}} \chi_E (x) >1/\alpha\}|.$$ It is shown that if $\phi_{\mathcal{B}}(\alpha)$ satisfies the Solyanik estimate $\phi_{\mathcal B}(\alpha) - 1 \leq C (1 - \frac{1}{\alpha})^p$ for $\alpha\in(1,\infty)$ sufficiently close to 1 then $\phi_{\mathcal{B}}$ lies in the H\"older class $C^p(1,\infty)$. As a consequence we obtain that the halo functions associated with the Hardy-Littlewood maximal operator and the strong maximal operator on $\mathbb{R}^n$ lie in the H\"older class $C^{1/n}(1,\infty)$.Comment: 19 pages, 1 figure, minor typos corrected, incorporates referee's report, to appear in Adv. Mat

Multiphoton Bloch-Siegert shifts and level-splittings in spin-one systems

We consider a spin-boson model in which a spin 1 system is coupled to an oscillator. A unitary transformation is applied which allows a separation of terms responsible for the Bloch-Siegert shift, and terms responsible for the level splittings at anticrossings associated with Bloch-Siegert resonances. When the oscillator is highly excited, the system can maintain resonance for sequential multiphoton transitions. At lower levels of excitation, resonance cannot be maintained because energy exchange with the oscillator changes the level shift. An estimate for the critical excitation level of the oscillator is developed.Comment: 14 pages, 3 figure