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Register a new userNon-variational computation of the eigenstates of Dirac operators with radially symmetric potentials
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We discuss a novel strategy for computing the eigenvalues and eigenfunctions of the relativistic Dirac operator with a radially symmetric potential. The virtues of this strategy lie on the fact that it avoids completely the phenomenon of spectral pollution and it always provides two-side estimates for the eigenvalues with explicit error bounds on both eigenvalues and eigenfunctions. We also discuss convergence rates of the method as well as illustrate our results with various numerical experiments.
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We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity --- but no smoothness --- of the augmented energy functional, we prove well-posedness of the method and convergence of the discrete approximations to a curve of steepest descent. In a smooth Hilbertian setting, classical theory would predict a convergence order of two in time, we prove convergence order of one-half in the general metric setting and under our weak hypotheses. Further, we illustrate these results with numerical experiments for gradient flows on a compact Riemannian manifold, in a Hilbert space, and in the $L^2$-Wasserstein metric.
In this paper, we investigate steady inviscid compressible flows with radial symmetry in an annulus. The major concerns are transonic flows with or without shocks. One of the main motivations is to elucidate the role played by the angular velocity in the structure of steady inviscid compressible flows. We give a complete classification of flow patterns in terms of boundary conditions at the inner and outer circle. Due to the nonzero angular velocity, many new flow patterns will appear. There exists accelerating or decelerating smooth transonic flows in an annulus satisfying one side boundary conditions at the inner or outer circle with all sonic points being nonexceptional and noncharacteristically degenerate. More importantly, it is found that besides the well-known supersonic-subsonic shock in a divergent nozzle as in the case without angular velocity, there exists a supersonic-supersonic shock solution, where the downstream state may change smoothly from supersonic to subsonic. Furthermore, there exists a supersonic-sonic shock solution where the shock circle and the sonic circle coincide, which is new and interesting.
We study solutions to $Lu=f$ in $Omegasubsetmathbb R^n$, being $L$ the generator of any, possibly non-symmetric, stable Levy process. On the one hand, we study the regularity of solutions to $Lu=f$ in $Omega$, $u=0$ in $Omega^c$, in $C^{1,alpha}$ domains~$Omega$. We show that solutions $u$ satisfy $u/d^gammain C^{varepsilon_circ}big(overlineOmegabig)$, where $d$ is the distance to $partialOmega$, and $gamma=gamma(L, u)$ is an explicit exponent that depends on the Fourier symbol of operator $L$ and on the unit normal $ u$ to the boundary $partialOmega$. On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the non-symmetric setting presents some new interesting features. Finally, we generalize the integration by parts identities in half spaces to the case of bounded $C^{1,alpha}$ domains. We do it via a new efficient approximation argument, which exploits the Holder regularity of $u/d^gamma$. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.
In this paper, we employ Meyer wavelets to characterize multiplier spaces between Sobolev spaces without using capacity. Further, we introduce logarithmic Morrey spaces $M^{t,tau}_{r,p}(mathbb{R}^{n})$ to establish the inclusion relation between Morrey spaces and multiplier spaces. By wavelet characterization and fractal skills, we construct a counterexample to show that the scope of the index $tau$ of $M^{t,tau}_{r,p}(mathbb{R}^{n})$ is sharp. As an application, we consider a Schrodinger type operator with potentials in $M^{t,tau}_{r,p}(mathbb{R}^{n})$.
In this paper, we classify the fundamental solutions for a class of Schrodinger operators.
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