Sentence examples for vanishing potential from inspiring English sources

By vanishing potential we mean a potential that vanish on some bounded domain or become very close to zero at infinity.

In [16], Sato proposes a different kind of penalization in order to show the existence of multi-peak nodal solutions to a Schrödinger equation with a vanishing potential.

After our work has been finished, we found a very recent paper [18] in which the author uses a similar argumentation in order to prove the existence of a sequence of nodal multi-peak solutions which concentrate around the minimum points of a modified potential, associated to a vanishing potential.

showing that and are comparable in magnitude for vanishing potentials when the salt concentration is.

For (K x in L_{mathrm{loc}}^{infty}(mathbb{R}^{3})), does equation (1.1) admit a positive solution (uin E)? Denis and Carlo in [1] considered this problem with unbounded and vanishing potentials (V x)).

Stationary NLS problems with vanishing potentials were treated for instance by Bonheure et al. in [5, 6], where the authors obtained concentration of positive solutions around global minimum points of an auxiliary function and even around some lower dimensional spheres in (mathbb{R}^{N}).

Under these conditions, they obtained the existence of solutions for the following equations: textstylebegin{cases} -Delta u+V x u-Delta u^{2})u+V x u-Delta u^{2}}mathbb{R}^{N}, u=g u quad}(mbox{inR}mathbb{R}^{cases} (1.3) To the best of our kN}wledge, except for [23, 24] there is no paper dealing with the quinilinear SchröD^{1,2}equations (1.1) with vanishing potentials.

For the reactivity at the valence atomic level, or for some outer shell (n) considered at the atomic frontier, one may assume almost electronic free motion or at least electronic motion under almost vanishing nuclear potential V(r); this way the density (23a), while entering the quantum potential (10) recovers the negative kinetic energy by the virial identity (14).

The problem is closed imposing the condition of vanishing gravitational potential at infinity.

We showed that at the condition of vanishing chemical potential jump, the antitrapping term in the multicomponent diffusion equation remains unchanged with the case of binary dilute alloys.

To corroborate the existence of center sector transitions in gauge theories with matter, we study (at vanishing chemical potential) the interface tension in the three-dimensional Z2 gauge theory with Ising matter, the distribution of the Polyakov line in the four-dimensional SU 2 -Higgs model and deviSU 2 -Higgspe of order paramodel which is designed to detect center sector tranditions.