Solvability of subprincipal type operators

In this paper we consider the solvability of pseudodifferential operators in the case when the principal symbol vanishes of order $k \ge 2$ at a nonradial involutive manifold $\Sigma_2$. We shall assume that the operator is of subprincipal type, which means that the $k$:th inhomogeneous blowup at $\Sigma_2$ of the refined principal symbol is of principal type with Hamilton vector field parallel to the base $\Sigma_2$, but transversal to the symplectic leaves of $\Sigma_2$ at the characteristics. When $k = \infty$ this blowup reduces to the subprincipal symbol. We also assume that the blowup is essentially constant on the leaves of $\Sigma_2$, and does not satisfying the Nirenberg-Treves condition (${\Psi}$). We also have conditions on the vanishing of the normal gradient and the Hessian of the blowup at the characteristics. Under these conditions, we show that $P$ is not solvable.Comment: Changed the formulation of Theorem 2.15, added an assuption. Corrected errors and clarified the arguments. Added reference

Gaussian tripartite entanglement in the simultaneous measurement of position and momentum

In this work, we prove the generation of genuine tripartite continuous-variable entanglement in the unitary dynamics of the simultaneous measurement process of position and momentum observables raised by Arthurs and Kelly, considering a measurement configuration where the system under examination is a rotated, displaced, and squeezed vacuum state. Under these assumptions, the measurement configuration is entirely described by a Gaussian state. Then, through the positive partial transpose criterion (PPT), we certify genuine tripartite entanglement by testing the non-separability of the three $\left(1~ \text{vs}~2\right)$-mode bipartitions of the system. This process allows us to classify the qualitative properties of the entanglement in the category of fully inseparable Gaussian states according to the classification exposed in [Giedke et al., \href{https://link.aps.org/doi/10.1103/PhysRevA.64.052303}{Phys. Rev. A \textbf{64}, 052303 (2001)}]. Besides, we determine the quantitative entanglement properties of the system using the residual tripartite R{\'e}nyi-2 entanglement as a quantifier measure.Comment: 15 pages, 8 figure

Study of Solid-State Radiolysis of Behenic, Fumaric, and Sebacic Acids for their Possible Use as Gamma Dosimeters Measured Via ATR-FT-IR Spectroscopy

The intensive use of ionizing radiation has promoted the constant investigation of adequate dosimetric systems in the measurement of doses applied in irradiated products. The objective of this work is to propose gamma dosimetric systems, using carboxylic acids in a solid state and measuring the change via infrared spectroscopy (carboxylic acid/ ATR-FT-IR1). We worked with three systems: (1) behenic acid/ATR-FT-IR, (2) sebacic acid/ATR-FT-IR, and (3) fumaric acid/ATR-FT-IR. The change in absorbance corresponding to the stretching vibration frequency of the carbonyl group to the absorbed dose (in the range of kGy) was measured. The results showed that the acid/ATR-FT-IR systems have a linear response with respect to the absorbed dose, for behenic acid/ATR-FT-IR from 0 to 122 kGy, for ATR-FT-IR sebacic acid from 0 to 61 kGy, and for fumaric acid/ATR-FT-IR from 0 to 34 kGy. The results indicated that the linear response of the absorbance dose in the three systems allows us to continue studying other variables to be able to propose them as chemical dosimeters