ترغب بنشر مسار تعليمي؟ اضغط هنا

On the discrete spectrum of quantum layers

275   0   0.0 ( 0 )
 نشر من قبل Julie Rowlett
 تاريخ النشر 2011
  مجال البحث فيزياء
والبحث باللغة English




اسأل ChatGPT حول البحث

Consider a quantum particle trapped between a curved layer of constant width built over a complete, non-compact, $mathcal C^2$ smooth surface embedded in $mathbb{R}^3$. We assume that the surface is asymptotically flat in the sense that the second fundamental form vanishes at infinity, and that the surface is not totally geodesic. This geometric setting is known as a quantum layer. We consider the quantum particle to be governed by the Dirichlet Laplacian as Hamiltonian. Our work concerns the existence of bound states with energy beneath the essential spectrum, which implies the existence of discrete spectrum. We first prove that if the Gauss curvature is integrable, and the surface is weakly $kappa$-parabolic, then the discrete spectrum is non-empty. This result implies that if the total Gauss curvature is non-positive, then the discrete spectrum is non-empty. We next prove that if the Gauss curvature is non-negative, then the discrete spectrum is non-empty. Finally, we prove that if the surface is parabolic, then the discrete spectrum is non-empty if the layer is sufficiently thin.



قيم البحث

اقرأ أيضاً

We discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.
We study the spectrum of two kinds of operators involving a conical geometry: the Dirichlet Laplacian in conical layers and Schrodinger operators with attractive $delta$-interactions supported by infinite cones. Under the assumption that the cones ha ve smooth cross-sections, we prove that such operators have infinitely many eigenvalues accumulating below the threshold of the essential spectrum and we express the accumulation rate in terms of the eigenvalues of an auxiliary one-dimensional operator with a curvature-induced potential.
506 - Georgi Raikov 2015
We consider the Schrodinger operator $H_{eta W} = -Delta + eta W$, self-adjoint in $L^2({mathbb R}^d)$, $d geq 1$. Here $eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behav iour of the discrete spectrum of $H_{eta W}$ near the origin, and due to the irregular decay of $eta W$, we encounter some non semiclassical phenomena. In particular, $H_{eta W}$ has less eigenvalues than suggested by the semiclassical intuition.
63 - Julie Rowlett 2020
We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical descrip tion of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures.
Let $H_0$ be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and $V$ be a compact non-selfadjoint perturbation. We relate the regularity properties of $V$ to various spectral properties of the perturbed operator $H_0+V$. The structure of the discrete spectrum and the embedded eigenvalues are analysed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results scattered throughout the literature are recovered and extended. For illustrative purposes, the case of the one-dimensional discrete Laplacian is emphasized.
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