اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSpectral estimates for periodic fourth order operators
150
0
0.0
(
0
)
تأليف
Andrey Badanin
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the operator $H={d^4dt^4}+{ddt}p{ddt}+q$ with 1-periodic coefficients on the real line. The spectrum of $H$ is absolutely continuous and consists of intervals separated by gaps. We describe the spectrum of this operator in terms of the Lyapunov function, which is analytic on a two-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. We describe the spectrum of $H$ in terms of periodic, antiperiodic eigenvalues, and so-called resonances. We prove that 1) the spectrum of $H$ at high energy has multiplicity two, 2) the asymptotics of the periodic, antiperiodic eigenvalues and of the resonances are determined at high energy, 3) for some specific $p$ the spectrum of $H$ has an infinite number of gaps, 4) the spectrum of $H$ has small spectral band (near the beginner of the spectrum) with multiplicity 4 and its asymptotics are determined as $pto 0, q=0$.
قيم البحث
اقرأ أيضاً
We study Riesz means of eigenvalues of the Heisenberg Laplacian with Dirichlet boundary conditions on a cylinder in dimension three. We obtain an inequality with a sharp leading term and an additional lower order term.
We prove that the Fermi surface of a connected doubly periodic self-adjoint discrete graph operator is irreducible at all but finitely many energies provided that the graph (1) can be drawn in the plane without crossing edges (2) has positive couplin
g coefficients (3) has two vertices per period. If positive is relaxed to complex, the only cases of reducible Fermi surface occur for the graph of the tetrakis square tiling, and these can be explicitly parameterized when the coupling coefficients are real. The irreducibility result applies to weighted graph Laplacians with positive weights.
In this paper, we mainly discuss the analytic expression of exact copositivity of 4th order symmetric tensor defined by the special physical model. We first show that for the general 4th order 2-dimensional symmetric tensor, it can be transformed int
o solving the quadratic polynomials, and then we give a necessary and sufficient condition to test the copositivity of 4th order 2-dimensional symmetric tensor. Based on this, we consider a special 4th order 3-dimensional symmetric tensor defined by the vacuum stability for $mathbb{Z}_{3}$ scalar dark matter, and obtain the necessary and sufficient condition for its copositivity.
We consider the nonlinear equations obtained from soliton equations by adding self-consistent sources. We demonstrate by using as an example the Kadomtsev-Petviashvili equation that such equations on periodic functions are not isospectral. They defor
m the spectral curve but preserve the multipliers of the Floquet functions. The latter property implies that the conservation laws, for soliton equations, which may be described in terms of the Floquet multipliers give rise to conservation laws for the corresponding equations with self-consistent sources. Such a property was first observed by us for some geometrical flow which appears in the conformal geometry of tori in three- and four-dimensional Euclidean spaces (math/0611215).
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.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
