Vacuum Region Research Articles

Existence and stability of almost finite energy weak solutions to the quantum Euler-Maxwell system

In this paper we prove the existence of global in time, finite energy weak solutions to the quantum magnetohydrodynamics (QMHD) system, for a large class of initial data that are slightly more regular than being of finite energy. No restriction is given on their size or the uniform positivity of the mass density. The QMHD system is a well-established model in superconductivity due to Feynman, and is intimately related to a nonlinear Maxwell-Schrödinger (NLMS) system via the Madelung transformations. The lack of global well-posedness (GWP) results for the NLMS system in the energy norm provides an obstruction to stability of its solutions, and consequently to the use of approximation arguments to make rigorous the Madelung approach. We implement here a new strategy, that derives weak solutions to QMHD starting from weak solutions to NLMS, by exploiting its Duhamel formulation. This derivation is rigorously justified by means of suitable dispersive/smoothing estimates for almost finite energy weak solutions to NLMS. The corresponding weak solutions to QMHD satisfy a generalized irrotationality condition, that allows for the presence of non-trivial vorticity (e.g., vortex filaments) concentrated in the vacuum region. This is in sharp contrast with the Euler-Maxwell system where the dispersion is not able to deal with the vorticity transport, therefore almost GWP (for regular solutions that are small perturbations of constant states) holds only in a lifespan, reciprocal of the size of the initial vorticity. Moreover, we also prove a stability property in the energy norm of the weak solutions constructed by our approach. This result follows from new local smoothing estimates for NLMS that are also of independent interest. As a byproduct, we also show the stability for the electromagnetic Lorentz force.

Pulsar Nulling and Vacuum Radio Emission from Axion Clouds.

Nonrelativistic axions can be efficiently produced in the polar caps of pulsars, resulting in the formation of a dense cloud of gravitationally bound axions. Here, we investigate the interplay between such an axion cloud and the electrodynamics in the pulsar magnetosphere, focusing specifically on the dynamics in the polar caps, where the impact of the axion cloud is expected to be most pronounced. For sufficiently light axions m_{a}≲10^{-7} eV, we show that the axion cloud can occasionally screen the local electric field responsible for particle acceleration and pair production, inducing a periodic nulling of the pulsar's intrinsic radio emission. At larger axion masses, the small-scale fluctuations in the axion field tend to suppress the backreaction of the axion on the electrodynamics; however, we point out that the incoherent oscillations of the axion in short-lived regions of vacuum near the neutron star surface can produce a narrow radio line, which provides a complementary source of radio emission to the plasma-resonant emission processes identified in previous work. While this Letter focuses on the leading order correction to pair production in the magnetosphere, we speculate that there can exist dramatic deviations in the electrodynamics of these systems when the axion backreaction becomes nonlinear.

Numerical methods for scalar field dark energy in tabletop experiments and Lunar Laser Ranging

Numerous tabletop experiments have been dedicated to exploring the manifestations of screened scalar field dark energy, such as symmetron or chameleon fields. Precise theoretical predictions require simulating field configurations within the respective experiments. This paper focuses onto the less-explored environment-dependent dilaton field, which emerges in the strong coupling limit of string theory. Due to its exponential self-coupling, this field can exhibit significantly steeper slopes compared to symmetron and chameleon fields, and the equations of motion can be challenging to solve with standard machine precision. We present the first exact solution for the geometry of a vacuum region between two infinitely extended parallel plates. This solution serves as a benchmark for testing the accuracy of numerical solvers. By reparametrizing the model and transforming the equations of motion, we show how to make the model computable across the entire experimentally accessible parameter space. To simulate the dilaton field in one- and two-mirror geometries, as well as spherical configurations, we introduce a non-uniform finite difference method. Additionally, we provide an algorithm for solving the stationary Schrödinger equation for a fermion in one dimension in the presence of a dilaton field. The algorithms developed here are not limited to the dilaton field, but can be applied to similar scalar-tensor theories as well. We demonstrate such applications at hand of the chameleon and symmetron field. Our computational tools have practical applications in a variety of experimental contexts, including gravity resonance spectroscopy (q Bounce), Lunar Laser Ranging (LLR), and the upcoming Casimir and Non-Newtonian Force Experiment (cannex). A Mathematica implementation of all algorithms is provided.

Analytical prediction of the open‐circuit magnetic field for the V‐shaped interior PM machine by the improved subdomain method

AbstractThe open‐circuit magnetic field of the V‐shaped interior PM (IPM) machine is predicted by the analytical method. The method is implemented following the basic principle of the subdomain method. The machine model is divided into five different regions, including rotor slot, magnetic barrier, air gap, stator slot, and stator slot opening. To perform the analytical prediction process in the polar coordinate system, these regions should be transformed into the appropriate subdomains. The last four regions can be transformed into the subdomains with the required shape based on the existing method. The rotor slot cannot be treated as a single subdomain, for the shape of which is irregular. Herein, the PM and vacuum region in the rotor slot is divided into several fan‐shaped subdomains. The division method is described clearly. Then, the boundary conditions alongside the interface between adjacent subdomains are derived based on function transformation. On this basis, the open‐circuit magnetic field of the IPM machine can be predicted. The magnetic field of the 6p/36 s V‐shaped IPM machine is obtained by the proposed method, finite element method simulation, and experiment, respectively. Additionally, the comparison results verify the accuracy and efficiency of the proposed method.

High-throughput screening on optoelectronic properties of two-dimensional InN/GaN heterostructure from first principles.

A novel 2D InN/GaNlateral heterostructure (LHT) was simulated by stitching monolayer of 2D InN and monolayer of 2D GaN. The structural stability, electronic structure, and optical properties were systematically investigated using first-principle calculations and by considering the effects of strain. The results indicated that the designed heterostructure has a direct bandgap of 2.26eV which is further affected by applied biaxial strain. The bandgap of 2D InN/GaN lateral heterostructure decreases with the increase in biaxial strain, and tensile strain triggers a direct-to-indirect energy gap changeover at + 6%. Additionally, under compressive strain, heterostructure remains a direct bandgap semiconductor. Furthermore, the strain significantly affects the optical characteristics of lateral heterostructure. It has been noticed that the first optical absorption peak moves from 2.51eV (ɛ = - 4%) to 1.40eV (ɛ = 10%). Therefore, 2D InN/GaN lateral heterostructure provides an approachable way for utilizing in optoelectronic devices through the creation of in-plane lateral heterostructures. We performed all the computations using a self-consistent method based upon density functional theory. We used the PBEsol functional in the GGA to account for the exchange-correlation effects. We introduced a 10-Å vacuum region in the z-direction to avoid interaction between periodic images. We considered non-negligible weak dispersion correction in the lateral heterostructure using Grimme's DFT-D3 approach. In this study, we also computed the electrical and optical properties employing the local modified Becke-Johnson (lmBJ) exchange potential under meta-GGA functional to obtain more precise results.

Fermion localization on the brane in Ricci-inverse gravity

This study aims to analyze the effects of modified gravity via anti-curvature tensor on a gravitational model set up within a 5-dimensional space–time, focusing on brane-world scenarios. The anti-curvature tensor introduces innovative configuration that is equal to the inverse of the Ricci tensor. In this analysis, we consider f(R,A)=(R+αA), where the coupling parameter α influences both vacuum regions outside the brane and the core region within it. Through analysis, this study uncovers alterations in fermion localization, particularly impacting left-chirality fermion. To quantify fermion localization, probabilistic metrics such as Shannon entropy and relative probability are employed, demonstrating a heightened level of certainty as α values increase. These findings shed significant light on the dynamics of extra-dimensional models, providing valuable insights into the realms of particle physics and cosmology.

Investigating the optical properties and electronic structure of gallium phosphide nanotubes doped with arsenic via implementing first-principles calculations.

This study investigates the impact of arsenic doping on the optical characteristics and electronic structure of zigzag (8, 0) and armchair (4, 4) gallium phosphide nanotubes using first-principles calculations based on the GaP1-xAsx system, where x = 0, 0.25, 0.5, 0.75, and 1. The electronic calculations showed that doping more arsenic atoms reduces the energy band gap for zigzag and armchair GaPAs nanotubes. PDOS analysis indicates that Ga-4p and P-3p orbitals play a significant role in determining the electronic properties of the GaP nanotube. The dominance of Ga-4p and P-3p orbitals in both the valence and conduction bands indicates their importance across the energy spectrum of the material. The complex dielectric function and absorption coefficient of zigzag and armchair GaP1-xAsx nanotubes are calculated for incident radiation with energies ranging from 1 to 6.2eV. Optical results revealed that both zigzag and armchair GaPNTs exhibit strong absorption in the UV-visible regions due to electronic transitions between different Van Hove singularities. Also, due to quantum confinement effects, pure zigzag gallium phosphide nanotube exhibited two absorption edges at wavelengths (273 and 375nm). These edges stand from the energy level's quantization in the nanotube construction, affecting the absorption characteristics. Substitutional doping by arsenic atoms changes the absorption edge to the long wavelengths due to decreased bandgap energy. Investigating electronic structures and optical properties of nanotubes proposes several advantages, such as understanding the doping effects on the nanotube structure and contributing to the direction of the experimental studies. These computational studies play a key role in developing the applications of nanomaterials. Calculations of density functional theory (DFT) are achieved via the SIESTA package. SIESTA is a powerful and effective tool for executing DFT calculations on a large system of atoms. It generates numerous output files covering detailed information about the electronic structure, optical properties, total energy, optimized geometry, and other computed properties. The generalized gradient approximation GGA with Perdew-Burke-Ernzerhof PBE functional was used. A vacuum region of 10 A0 was applied to avoid the interactions of adjacent nanotubes.

Ultra-High Vacuum Cells Realized by Miniature Ion Pump Using High-Efficiency Plasma Source.

In recent years, there has been significant interest in quantum technology, characterized by the emergence of quantum computers boasting immense processing power, ultra-sensitive quantum sensors, and ultra-precise atomic clocks. Miniaturization of quantum devices using cold atoms necessitates the employment of an ultra-high vacuum miniature cell with a pressure of approximately 10-6 Pa or even lower. In this study, we developed an ultra-high vacuum cell realized by a miniature ion pump using a high-efficiency plasma source. Initially, an unsealed miniature ion pump was introduced into a vacuum chamber, after which the ion pump's discharge current, depending on vacuum pressures, was evaluated. Subsequently, a miniature vacuum cell was fabricated by hermetically sealing the miniature vacuum pump. The cell was successfully evacuated by a miniature ion pump down to an ultra-high vacuum region, which was derived by the measured discharge current. Our findings demonstrate the feasibility of achieving an ultra-high vacuum cell necessary for the operation of miniature quantum devices.

Well-Posedness of the Two-Dimensional Compressible Plasma-Vacuum Interface Problem

We consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolic-elliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre-Maxwell dynamics. At the free interface moving with the velocity of plasma particles, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, since it is driven by a given surface current which forces oscillations onto the system. We prove the local-in-time existence and uniqueness of solutions to this nonlinear free boundary problem, provided that at least one of the two magnetic fields, in the plasma or in the vacuum region, is non-zero at each point of the initial interface. The proof follows from the analysis of the linearized MHD equations in the plasma region and the elliptic system for the vacuum magnetic field, suitable tame estimates in Sobolev spaces for the full linearized problem, and a Nash–Moser iteration.

Numerical Analysis and Quantification of Transfer Efficiency Coupled with Capillary and Quadrupole Ion Guide in an API-MS System.

Ion trajectory simulation is a significant and useful tool for understanding ion transfer mechanisms within the first vacuum region of the atmospheric pressure ionization mass spectrometer (API-MS). However, the complex dynamic gas field and wide pressure range lead to inaccurate simulation and huge computational costs. In this work, a novel electrohydrodynamic simulation called the statistical diffusion-hard-sphere (SDHS) mixed collision model was developed for characterizing the ion trajectories. For the first time, the influence of the dynamic pressure on the ion trajectory is considered for simulation, which helps to avoid an intolerable computational cost. Comparing with the conventional Monte Carlo collision model, the SDHS method helps to improve the calculation accuracy of ion trajectories under the first vacuum region and reduce the computational cost for at least 12-folds. Simulation results showed that the maximum ion loss came from the gap of the electrodes. The distance of the capillary-quadrupole ion guide was also a non-negligible factor. The trend of quantitative experimental results matches the SDHS simulation results. The maximum ion transfer efficiencies of quantitative experiment and simulation were 55% and 52%, respectively. Moreover, three ions, caffeine, reserpine, and Ultramark 1621, were measured for evaluating the applicability of SDHS in real API-MS. The trend of experimental results showed good agreement with that of computation. And the results of caffeine further illustrated the reason that the small mass ion transfer efficiency decreased with increasing radio frequency voltage. SDHS method is expected to be useful in the design of ion guides for further improvement of the sensitivity of API-MS.

Joshi–Malafarina–Narayan singularity in weak magnetic field

The importance and significance of magnetic fields in the astrophysical scenario is well known. Many domains of astrophysical black hole physics such as polarized shadow image, high energy emitting processes and jet formation are dependent on the behavior of the magnetic fields in the vicinity of the compact objects. In light of this, we determine the master equation and master differential equation that determine the spatial behavior of the magnetic field inside a matter distribution or vacuum region, of general spherically symmetric metric, which is immersed in a test magnetic field. We also investigate here the case of JMN-1 singularity immersed in a uniform weak magnetic field and determine the behavior of magnetic fields by defining electromagnetic four potential vector. We find that the tangential component of the magnetic field is discontinuous at the matching surface of the JMN-1 singularity with the external Schwarzschild metric, resulting in surface currents. We define the covariant expression of surface current density in this scenario. We also analyze the behavior of center-of-mass energy of two oppositely charged particles in the geometry of the magnetized JMN-1 singularity. We briefly discuss the possible scenarios which would possess a discontinuous magnetic field and implications of the same and future possibilities in the realm of astrophysics are indicated.

Manipulating Non-Dissociative Transformations of Gaseous Ion Ensembles Prior to Ion-Mobility Separation.

Molecules with multiple sites capable of accepting protons form ensembles of protomers. The manifested protomer ratios in such ensembles are influenced by many experimental conditions. In a Synapt G2 ion mobility (IM)-enabled mass spectrometry system, there are several physical locations where ion population changes can be manifested. Using APCI-generated protomers of aminonaphthalenes, we investigated its intramolecular proton transfers from the N-protomer to the C-protomer. This lossless transformation of the N-protomer to the thermodynamically favored C-protomer can take place in the ion source itself. Initially, we learned that the cone gas slows down the transformation to the C-protomer. Gaseous ions are then accelerated in the first vacuum region, where ions undergo collisional activation (heating), which facilitates the transformation to the C-protomer. Afterward, the ions are mass selected and transferred to the pre-IM (Trap)-collision cell, where ions can also be transformed to the thermodynamically favored protomers. Trap accumulated ions are then released to the IM separator via a helium-filled entry cell. The role of helium is to minimize ion activation and scattering taking place upon entry to the high-pressure T-Wave IM separator (TWIMS). The helium cell is known to increase the IM peak resolution. However, we found that significant changes occur depending on the presence or absence of helium. Without helium, source-generated protomers rapidly changed to a predominantly thermodynamically favorable ensemble protomers. Apparently, the introduction of helium into the precell induced a dramatic decrease in collisional "heating" effect, which effectively slowed down the conversion rate of the amino-protomer into the more favorable ring-protomer. The final message is that mobilograms should not be considered as direct real-time, or intrinsic, representations of the protomer ratios in the ion source.

A Miniature Orthogonal Injection Ion Funnel (MO-IF) Providing Enhanced Performance for the Miniature Mass Spectrometer.

The sensitivity of the miniature mass spectrometer (mini-MS) is largely restricted by the ion transmission in rough vacuum region. Even though various "in-line" ion transfer devices have improved mini-MS sensitivity, the severe dynamic gas is still weakening the efficiency of ion transmission in this region. Inspired by the "off-axis" ion funnel design in the lab-scale mass spectrometers, a miniature orthogonal injection ion funnel (MO-IF) was developed in this study for the mini-MS with a continuous atmospheric pressure interface. Capable of directing injected ions by 90° and then transport them forward to the downstream skimmer, the MO-IF enabled the separation of ions from the dynamic gas flow jetted out of the inlet capillary. The key factors were optimized for the MO-IF, including the effects of RF amplitude, DC electric fields, and the position of the repeller. Under optimized conditions, the MO-IF minimized the negative effects of dynamic gas and improved the ion transmission efficiency by ∼2-fold in comparison with the in-line injection ion funnel. As a result, a lower limit of detection of 0.5 ng/mL were obtained with good linearity for hypaconitine. Additionally, the MO-IF further decreased the buffer gas pressure in the second vacuum chamber and improved the mass resolution by 1.1-1.5 times at different scan rates.

Meta-GGA study of 2D AlN/BN planer heterostructure and performance enhancement via strain engineering.

The 2D AlN/BN planer heterostructure is a promising wide band gap semiconductor, but systematic studies of its bandgap and optical characteristics under applied strain are scarce. Here, the engineering property of 2D AlN/BN comprising bandgap nature transition and optical absorption capability (from unstrained to strained) have been investigated using density functional theory calculations. The formation energy calculations confirm the stability of the simulated nanoheterostructure. The electronic band structure calculations demonstrate that nanoheterostructure is an indirect bandgap material with a large bandgap of 5.26 eV, which can be modified effectively by applying strain. According to the calculations, the transition from indirect to direct band gap behavior has been observed at +15% biaxial strain with 2.71 eV band gap energy. Meanwhile, calculations for optical absorption and dielectric function reveal that the system has significant absorption peaks in the ultraviolet region which are very sensitive to applied strain. As strain increases, the first absorption peaks are shifted towards a lower energy range from 5.73 eV (Ꜫ= 0 %) to 3.76 eV (Ꜫ = +15%), which features an enhancement of optical absorption for solar and solar-blind regions. Furthermore, we determined that the band edge positions in 2D AlN/BN straddled the water redox potential under strain, indicating its effectiveness as a proficient photocatalyst. These characteristics make 2D AlN/BN planer nanoheterostructure a promising candidate for applications in optoelectronics and photocatalytic water splitting performance. First principles computations based on density functional theory were employed to carry out all the calculations with a self-consistent approach. For solving the Kohn-Sham equations, the first principles dependent full-potential linearized augmented plane wave scheme were adopted. For addressing the exchange-correlation effects, the generalized gradient approximation of PBEsol functional was used. To prevent interaction between the periodic images, we have inserted a vacuum region of 10 Å in the z-direction. Non-negligible weak dispersion corrections in nanoheterostructure were considered by using the DFT-D3 method of Grimme's. The locally modified Becke-Johnson (lmBJ) exchange potential has also been applied to compute electronic and optical properties in this research to obtain more accurate information.

Ab-initio study of strain-tunable g-GaN/BN nanoheterostructure for optoelectronic and photocatalytic applications.

Two-dimensional (2D) nanoheterostructures of materials, integrating various phase or materials into a single nanosheet have stimulated large-scale research interest for designing novel two dimensional devices. In contemporary analysis present work, we examined the structural and electronic properties of the isolated 2D BN and GaN monolayers. We have investigated the structural stability and optoelectronic and photocatalytic response of the g-GaN/BN nanoheterostructure along with its response to strain. Nanoheterostructure g-GaN/BN is predicted to be a direct bandgap semiconductor with wide gap of 4.45eV, whose value can be effectively modulated by applied strain ( , ranging from 4.55 ( = - 4%) to 3.58eV ( = 8%). We also discovered that the tensile strain of 8% can substantially tune the direct bandgap of nanoheterostructure to indirect band gap nature. Even more important, the biaxial tensile strain engineering accentuates an enhancement of optical absorption in the UV region, broadening the light harvesting of the g-GaN/BN nanoheterostructure with the shifting of first absorption peak from 4.64 ( = - 4%) to 3.71eV ( = 8%). Furthermore, strain-tuned band edge potentials arrangement perfectly fits the water reduction and oxidation redox potentials. Our findings portend that the g-GaN/BN nanoheterostructure has application in prospective nanoscale optoelectronic devices and photocatalytic hydrogen evolution system. First principles calculations in this study are performed using density functional theory. Generalized gradient approximation within PBEsol functional employed to address the electron-electron exchange-correlation effects. For avoiding periodic interactions between the layers, we have inserted a vacuum region of thickness 10Å in the z-direction. For ensuring the convergence accuracy of the computed results, convergence criteria of the iteration process is set to be 0.0001eV. Local modified Becke-Johnson, a semi local functional, is applied for calculating electronic and optical properties for more accuracy of results. As in layered 2D nanoheterostructure, a factual depiction of the van der Waals interactions cannot be provided by conventional DFT techniques. Accordingly, in order to incorporate these interactions, we had employed the dispersion correction method of Grimme's.

Probabilistic locked mode predictor in the presence of a resistive wall and finite island saturation in tokamaks

We present a framework for estimating the probability of locking to an error field in a rotating tokamak plasma. This leverages machine learning methods trained on data from a mode-locking model, including an error field, resistive magnetohydrodynamics modeling of the plasma, a resistive wall, and an external vacuum region, leading to a fifth-order ordinary differential equation (ODE) system. It is an extension of the model without a resistive wall introduced by Akçay et al. [Phys. Plasmas 28, 082106 (2021)]. Tearing mode saturation by a finite island width is also modeled. We vary three pairs of control parameters in our studies: the momentum source plus either the error field, the tearing stability index, or the island saturation term. The order parameters are the time-asymptotic values of the five ODE variables. Normalization of them reduces the system to 2D and facilitates the classification into locked (L) or unlocked (U) states, as illustrated by Akçay et al., [Phys. Plasmas 28, 082106 (2021)]. This classification splits the control space into three regions: L̂, with only L states; Û, with only U states; and a hysteresis (hysteretic) region Ĥ, with both L and U states. In regions L̂ and Û, the cubic equation of torque balance yields one real root. Region Ĥ has three roots, allowing bifurcations between the L and U states. The classification of the ODE solutions into L/U is used to estimate the locking probability, conditional on the pair of the control parameters, using a neural network. We also explore estimating the locking probability for a sparse dataset, using a transfer learning method based on a dense model dataset.

Atomic circulation in evanescent of LG 10 mode with active dipole and quadrupole transitions

The total internal reflection of a light with phase singularity, such as Laguerre–Gaussian (LG) light, can generate an evanescent wave. The distribution of this type of light is firmly localized in the vicinity above the dielectric surface and displays a rotational character. Any atom approaching the surface from the vacuum region interacts with evanescent wave and produces interesting rotational effects. Recent studies have established the coupling of LG beams to both electric atomic transitions; dipole and quadrupole according to the selection rules of each, permitted and prohibited. Due to the small magnitude of the quadrupole interaction compared to the dipole interaction, we chose the two counter-propagating beams configuration to generate the doubling Rabi frequencies with more stable surface optical forces. We describe the essential rotational features and discuss how they can be used to influence atoms. Atom confinement and dynamics in the interaction of surface optical vortices with the two different atomic transitions are illustrated using typical experimentally accessible parameters for the case of the Cs atom interacts with the L G 10 mode.

A High Vacuum, High Electric Field Pulsed Power Interface Based on a Ceramic Insulator

Improving the flashover voltage in a vacuum interface between a pulsed-power system and a vacuum region has been a goal for many years. The interface problem is difficult because of the electrical, mechanical, and vacuum issues that must be satisfied simultaneously. In this paper, according to the theories of vacuum flashover and the design rules for high electric field insulators, a ceramic insulated vacuum interface is presented. The electrostatic field along the ceramic surface was obtained by ANSYS simulations. By optimizing the field shaper and shielding rings, the electric fields were well distributed and the fields around the cathode and anode triple junctions were effectively controlled. The high voltage test was carried out on a ~40 GW, ~100-ns pulser, and results show that the vacuum interface can work stably with the hold-off voltage of 600 kV. The experimental results agree with the theoretical and simulated results. Finally, the influence of the surface treatment on the ceramic flashover characteristics was also discussed.

The Pauli problem and wave function lifting: reconstruction of quantum states from physical observables

<p>In this paper we consider the problem to determine a unique quantum state from given distribution of position (density) and momentum density of particles, namely the so called Pauli problem in quantum physics. In the first part, we will review the method of wave function lifting developed in <sup>[<xref ref-type="bibr" rid="b4">4</xref>,<xref ref-type="bibr" rid="b5">5</xref>]</sup> to construct a complex wave function $ \psi\in H^s({{\mathbf R}}^d) $, $ s = 1, 2 $, associated to given density and momentum density. The second part is focused on the dynamical version of the wave function lifting, namely we study the relation between solutions to quantum fluid models and wave functions solving the nonlinear Schrödinger equation. The uniqueness of the lifted wave function is essentially related to the structure of vacuum regions of the position density.</p>

The Solution of the Einstein’s Equations in the Vacuum Region Surrounding a Spherically Symmetric Mass Distribution

The Solution of the Einstein’s Equations in the Vacuum Region Surrounding a Spherically Symmetric Mass Distribution