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\"With special attention to the Sturm-Liouville theory, this book discusses the full multiparameter theory as applied to second-order linear equations. It considers the spectral theory of these multiparameter problems in detail for both the regular and singular cases. The text covers eignencurves, the essential spectrum, eigenfunctions, oscillation theorems, the distribution of eigencurves, the limit point, limit circle theory, and more. This text is the culmination of more than two decades of research by F.V. Atkinson, one of the masters in the field, and his successors, who continued his work after he passed away in 2002\"-- Provided by publisher.

Here we present a novel stochastic Liouville equation with piecewisely correlated noises, in which the inter-piece correlation is rigorously incorporated by a convolution integral involving functional derivatives. Due to the feature of piecewise correlation, we can perform piecewise ensemble average and serve the average of the preceding interval as the initial condition of the subsequent propagation. This strategy avoids the long-time stochastic average and the statistical errors are saturated at long times. By doing so, we circumvent the intrinsic difficulty of the stochastic simulations caused by the fast increase in the variance of the quantum Brownian motion. Therefore, as demonstrated by the numerical examples, the proposed method enables us to simulate the long-time quantum dissipative dynamics with long memories in the non-perturbative regime.

In this paper, we prove that if u is a solution to the Liouville equation 0.1 Δ u + e 2 u = 0 in R 2 , then the diameter of R 2 under the conformal metric g = e 2 u δ is bounded below by π . Here δ is the Euclidean metric in R 2 . Moreover, we explicitly construct a family of solutions to ( 0.1 ) such that the corresponding diameters of R 2 range over [ π , 2 π ) . We also discuss supersolutions to ( 0.1 ). We show that if u is a supersolution to ( 0.1 ) and ∫ R 2 e 2 u d x < ∞ , then the diameter of R 2 under the metric e 2 u δ is less than or equal to 2 π . For radial supersolutions to ( 0.1 ), we use both analytical and geometric approaches to prove some inequalities involving conformal lengths and areas of disks in R 2 . We also discuss the connection of the above results with the sphere covering inequality in the case of Gaussian curvature bounded below by 1. Higher dimensional generalizations are also discussed.

NASA's Van Allen Probes observed significant, long‐lived fluxes of inner belt electrons up to ∼1 MeV after geomagnetic storms in March and June 2015. Reanalysis of Magnetic Electron Ion Spectrometer (MagEIS) data with improved background correction showed a clearer picture of the relativistic electron population that persisted through 2016 and into 2017 above the Fennell et al. (2015, https://doi.org/10.1002/2014gl062874) limit. The intensity and duration of these enhancements allow estimation of decay timescales for comparison with simulated decay rates and theoretical lifetimes. We compare decay timescales from these data and DREAM3D simulations based on them using geomagnetic activity‐dependent pitch angle diffusion coefficients derived from plasmapause‐indexed wave data (Malaspina et al., 2016, https://doi.org/10.1002/2016gl069982, 2018, https://doi.org/10.1029/2018gl078564) and phase space densities derived from MagEIS observations. Simulated decay rates match observed decay rates more closely than the theoretical lifetime due to significantly nonequilibrium pitch angle distributions in simulation and data. We conclude that nonequilibrium effects, rather than a missing diffusion or loss process, account for observed short decay rates. Plain Language Summary Earth's radiation belts are influenced by a wide variety of source and loss processes, so it is difficult to model and forecast its evolution or predict its effects on spaceborne assets. Decay timescales for loss processes are essential to understanding this balance, but the theoretical predictions for these timescales in the inner radiation belt can exceed the observed decay times by an order of magnitude or more. We have observed and simulated an exceptional period of radiation belt injection and decay to understand this discrepancy. We have found that changes in the wave properties due to geomagnetic activity can account for the difference: changes in the equilibrium distribution associated with the wave environment result in consistent refilling of non‐equilibrium modes that decay much faster than the equilibrium mode. Key Points DREAM3D simulations of Earth's inner electron belt, based on Van Allen Probes observations, are carried out to evaluate model decay rates Pitch angle diffusion using coefficients reflecting geomagnetic activity demonstrates realistic decay rates Decay rates extracted with a Random Sample Consensus‐based algorithm from modeled and observed fluxes agree, while theoretical lifetimes are too long

A node of a Sturm–Liouville problem is an interior zero of an eigenfunction. The aim of this paper is to present a simple and new proof of the result on sharp bounds of the node for the Sturm–Liouville equation with the Dirichlet boundary condition when the L 1 norm of potentials is given. Based on the outer approximation method, we will reduce this infinite-dimensional optimization problem to the finite-dimensional optimization problem.

We perform a phase-space analysis of strong-field enhanced ionisation in molecules, with emphasis on quantum-interference effects. Using Wigner quasi-probability distributions and the quantum Liouville equation, we show that the momentum gates reported in a previous publication (Takemoto and Becker 2011 Phys. Rev. A 84 023401) may occur for static driving fields, and even for no external field at all. Their primary cause is an interference-induced bridging mechanism that occurs if both wells in the molecule are populated. In the phase-space regions for which quantum bridges occur, the Wigner functions perform a clockwise rotation whose period is intrinsic to the molecule. This evolution is essentially non-classical and non-adiabatic, as it does not follow equienergy curves or field gradients. Quasi-probability transfer via quantum bridges is favoured if the electron's initial state is either spatially delocalised, or situated at the upfield molecular well. Enhanced ionisation results from the interplay of this cyclic motion, adiabatic tunnel ionisation and population trapping. Optimal conditions require minimising population trapping and using the bridging mechanism to feed into ionisation pathways along the field gradient.

We consider the equation − ( r ( x ) y ′ ( x ) ) ′ + q ( x ) y ( x ) = f ( x ) , x ∈ R , where f ∈ L p (ℝ), p ∈ (1, ∞) and r > 0 , 1 r ∈ L 1 loc ( R ) , q ∈ L 1 loc ( R ) . For particular equations of this form, we suggest some methods for the study of the question on requirements to the functions r and q under which the above equation is correctly solvable in the space L p (ℝ), p ∈ (1, ∞).

We show how to find the shape of an equilateral tree using the spectra of the Neumann and the Dirichlet problems generated by the Sturm–Liouville equation. In case of snowflake trees the spectra of the Neumann and Dirichlet problems uniquely determine the shape of the tree.

We are interested in the law of the first passage time of an Ornstein–Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these transformations appear in a realisation of a standard Ornstein–Uhlenbeck bridge. We provide three different proofs of this connection. The first is based on a similar result for Brownian motion, the second uses a generalisation of the so-called Gauss–Markov processes, and the third relies on the Lie group symmetry method. We investigate the properties of these transformations and study the algebraic and analytical properties of an involution operator which is used in constructing them. We also show that these transformations map the space of solutions of Sturm–Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we interpret our results through the method of images and give new examples of curves with explicit first passage time densities.

This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equations (RCSLEs) with quasi-rational solutions (q-RSs) were obtained by applying rational Rudjak–Zakhariev transformations (RRZTs) to the Jacobi equation re-written in the canonical form. The presented analysis was focused on the RRZTs leading to the canonical form of the Heun equation. It was demonstrated that the latter equation preserves its form under the second-order Darboux–Crum transformation. The associated Sturm–Liouville problems (SLPs) were formulated for the so-called ‘prime’ SLEs solved under the Dirichlet boundary conditions (DBCs). It was proven that one of the two X1-Jacobi DPSs composed of Heun polynomials contains both the X1-Jacobi orthogonal polynomial system (OPS) and the finite EOP sequence composed of the pseudo-Wronskian transforms of Romanovski–Jacobi (R-Jacobi) polynomials, while the second analytically solvable Heun equation does not have the discrete energy spectrum. The quantum-mechanical realizations of the developed formalism were obtained by applying the Liouville transformation to each of the SLPs formulated in such a way.