ترغب بنشر مسار تعليمي؟ اضغط هنا

The action of pseudo-differential operators on functions harmonic outside a smooth hyper-surface

207   0   0.0 ( 0 )
 نشر من قبل Yves Colin de Verdiere
 تاريخ النشر 2012
  مجال البحث فيزياء
والبحث باللغة English




اسأل ChatGPT حول البحث

We consider a smooth hyper-surface Z of a closed Riemannian manifold X. Let P be the Poisson operator associating to a smooth function on Z its harmonic extension on XZ. If A is a pseudo-differential operator on X of degree <3, we prove that B=P^* A P is a pseudo-differential operator on Z and calculate the principal symbol of B.



قيم البحث

اقرأ أيضاً

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$ Au(x)=int_{mathbb{R}^n}int_{mathbb{R}^n}e^{i(x-y)cdotxi}sigma(x+tau(y-x),xi)u(y)dydxi, $$ where $tau:mathbb{R}^ntomathbb{R}^n$ is a general function. In particular, for the linear choices $tau(x)=0$, $tau(x)=x$, and $tau(x)=frac{x}{2}$ this covers the well-known Kohn-Nirenberg, anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions $tau$ and here we investigate the corresponding calculus in the model case of $mathbb{R}^n$. We also give examples of nonlinear $tau$ appearing on the polarised and non-polarised Heisenberg groups, inspired by the recent joint work with Marius Mantoiu.
124 - Hynek Kovav{r}ik 2021
We consider two-dimensional Pauli and Dirac operators with a polynomially vanishing magnetic field. The main results of the paper provide resolvent expansions of these operators in the vicinity of their thresholds. It is proved that the nature of the se expansions is fully determined by the flux of the magnetic field. The most important novelty of the proof is a comparison between the spatial asymptotics of the zero modes obtained in two different manners. The result of this matching allows to compute explicitly all the singular terms in the associated resolvent expansions. As an application we show how the magnetic field influences the time decay of the associated wave-functions quantifying thereby the paramagnetic and diamagnetic effects of the spin.
We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined subsymbol of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact order filtration of the Heisenberg calculus.
The article discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics (e.g., Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can obtain the correct spectrum by u sing the values of the quasimomentum running over the boundary of the (reduced) Brillouin zone only, rather than the whole zone? Or, do the edges of the spectrum occur necessarily at the set of ``corner high symmetry points? This is known to be true in 1D, while no apparent reasons exist for this to be happening in higher dimensions. In many practical cases, though, this appears to be correct, which sometimes leads to the claims that this is always true. There seems to be no definite answer in the literature, and one encounters different opinions about this problem in the community. In this paper, starting with simple discrete graph operators, we construct a variety of convincing multiply-periodic examples showing that the spectral edges might occur deeply inside the Brillouin zone. On the other hand, it is also shown that in a ``generic case, the situation of spectral edges appearing at high symmetry points is stable under small perturbations. This explains to some degree why in many (maybe even most) practical cases the statement still holds.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا