It is shown (by means of a perturbation series) that for class potentials $V(x)$ the stationary distribution solution $x(t)$ quantum Langevin equation approaches in weak-coupling limit ($f\ensuremath{\rightarrow}0$) mechanical canonical displacement oscillator, subject to potential $V(x)$, if and only $E(t)$ operator version purely random Gaussian process so that, particular, higher symmetrized averages ${〈E({t}_{1})\ensuremath{\cdots}E({t}_{n})〉}_{s}$ are expressible terms pair...

*The first author's work was partially supported by FONDECYT (Chile), project 0132-88, and a Summer Research Fellowship provided the Council of University Missouri-Columbia. He would like to thank Physics Department others at Universidad de Chile for their hospitality during his visit in April, 1990, when much this research completed. The second author part projects 0132-88 1238-90. Both authors also Fritz Gesztesy general comments encouragement.

It is shown that any Bäcklund transformation of a nonlinear differential equation integrable by the multichannel Schrödinger eigenvalue problem can be written in form V x = U′V - VU . This allows us to interpret formally as difference for which we immediately construct soliton solutions.

It is well known that ionized atoms cannot be both very negative and stable. The maximum ionization only one or two electrons, even for the largest atoms. reason this phenomenon examined critically it shown electrostatic considerations uncertainty principle account it. exclusion plays a crucial role. This by proving when Fermi statistics ignored, then degree of at least order $z$, nuclear charge, $z$ large.

We study the speed of propagation fronts for scalar reaction-diffusion equation ${u}_{t}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}u}_{\mathrm{xx}}+f(u)$ with $f(0)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}f(1)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$. give a new integral variational principle joining state $u\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ to $u\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$. No assumptions are made on reaction term...

It is shown that the sharp constant in Hardy-Sobolev-Maz'ya inequality on upper half space H 3 ⊂ R given by Sobolev constant.This achieved a duality argument relating problem to Hardy-Littlewood-Sobolev type whose determined as well.g(B(x, y)) ,

We prove the optimal lower bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda 2 minus lamda 1 greater-than-or-equal-to 3 pi squared slash d squared"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>λ<!-- λ --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>≥<!-- ≥ <mml:mn>3</mml:mn> <mml:msup> <mml:mi>π<!-- π </mml:msup>...

The authors investigate bounds for various combinations of the low eigenvalues Laplacian with Dirichlet boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^n $. These investigations continue and expand upon earlier work Payne, Pólya, Weinberger, Brands, Chiti, this present paper. In particular, generalize extend to n-dimensional setting Chiti examine their consequences interrelationships in detail. This includes comparing asymptotic forms as dimension n becomes large. also...

The authors consider bounds on the Neumann eigenvalues of Laplacian domains in $I\mathbb{R}^n $ light their recent results Dirichlet eigenvalues, particular, proof Payne-Pólya–Weinberger conjecture via spherical rearrangement. They prove bound ${1 / {\mu _1 }} + {1 _2 \geq {A {2\pi }}$ for first two nonzero an arbitrary bounded domain $\Omega dimensions and also stronger (and optimal) $\mu \leq \pi (j'_{1,1} )^{{2 A}} having a 4-fold rotational symmetry. (Here $(j'_{1,1} ) \approx 1.84118$...

The linear stability of a fluid bounded above by free deformable surface is studied numerically. When the heat flux fixed on and lower plane isothermic, oscillatory instabilities, which may occur at values Rayleigh number than critical value for onset steady convection, are found.

Let Ω be some domain in the hyperbolic space Hn (with n≥2), and let S1 a geodesic ball that has same first Dirichlet eigenvalue as Ω. We prove Payne-Pólya-Weinberger (PPW) conjecture for Hn, namely, second on is smaller than or equal to S1. also ratio of two eigenvalues balls decreasing function radius

A new, elementary proof of a recent result Laptev and Weidl [LW] is given.It sharp Lieb-Thirring inequality for one dimensional Schrödinger operators with matrix valued potentials.