Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2308.06090

Lower bound for solutions to the secular equation in a direct sum of the Sobolev spaces of divided regions

Sohei Ashida
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper a lower bound for solutions to the secular equation of the Schrödinger equation with basis functions discontinuous on boundaries of divided regions is given. If the functions do not have the discontinuity, the bound reduces to that for the usual Rayleigh-Ritz method. Difference from the usual Rayleigh-Ritz method is bounded by the degree of discontinuity. The result can be regarded as a theoretical basis of the augmented plane wave method for the band structure calculations in solid state physics. The result would be useful also for other electronic eigenvalue problems in which the behavior of the eigenfunction near nuclei is very different from that in the interstitial region, because it allows us to use different basis functions in different regions in contrast with the usual Rayleigh-Ritz method in which we need to use basis functions with inappropriate behaviors in wide regions. The proof is based only on new general results about the Sobolev space, in particular, equivalence of two semi-norms concerned with discontinuity of the functions in a direct sum of the Sobolev spaces. Therefore, the result does not depend on specific forms of basis functions at all. The main ingredient of the proof is an estimate of elements of the orthogonal complement of $H^1_0(Ω)$ by their boundary values based on a characterization of them as weak solutions to an elliptic equation on a region $Ω$.

Explore related subjects

math.APmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sohei Ashida. 2023-12-07. Lower bound for solutions to the secular equation in a direct sum of the Sobolev spaces of divided regions. https://arxiv.org/abs/2308.06090

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Blow-Up Dynamics for the $L^2$ critical case of the $2$D Zakharov-Kuznetsov equation

We study blow-up dynamics for the $L^2$-critical cubic Zakharov--Kuznetsov equation in two dimensions, \[ \partial_tu+\partial_{x_1}(Δu+u^3)=0 \qquad\text{on }\mathbb R^2. \] For a class of localized $H^1$ perturbations of the ground state $Q$, we establish a trichotomy near the soliton manifold: exit from a small $L^2$-tube, global asymptotic stability, or finite-time blow-up. In the stable blow-up regime, the solution concentrates a single bubble and \[ λ(t)\sim \ell_0(T-t)^{1/(3-c)}, \] where $\ell_0>0$ depends on the initial datum and $c\in(1,2)$ is an explicit constant determined by the transverse tail of the first-order approximate profile. Consequently, \[ \|\nabla u(t)\|_{L^2} \sim \frac{\|\nabla Q\|_{L^2}} {\ell_0(T-t)^{1/(3-c)}}. \] After subtraction of the concentrating soliton, the radiation converges strongly in $L^p(\mathbb R^2)$ for every $2\leq p<\infty$ to a common nonzero profile $u^*$, while \[ u^*\notin H^s(\mathbb R^2) \qquad\text{for every }s\geq\frac c2. \] The stable blow-up branch is open in the relative $H^1$ topology of the localized class. Finally, every non-soliton datum in this class with non-positive energy blows up in finite time. Interval-arithmetic computer-assisted proofs certify the numerical inputs to the virial coercivity argument. They also yield a rigorous enclosure of $c$, justifying the polynomial moment of order $21$ imposed on the initial data.

math.AP↗
arXiv2026

Propagation of wave packets close to conical intersections

In this paper, we study the propagation of wave packets close to conical intersections with respect to a system of two Schr{ö}dinger equations presenting a codimension 2 crossing. We focus on the dynamics that occur when the wave packets pass through an area close to the crossing, and our main results provide an explicit formula for the outgoing wave packet in terms of the incoming one, with a complete description of its phase and of the classical trajectories it follows, including a drift.

math.AP↗
arXiv2026

A Volterra Calculus for Lie Groupoids

A pseudodifferential Volterra calculus for inverting parabolic differential equations on Lie groupoids is introduced. This enables the study of fundamental solutions of various cases of heat flows on singular manifolds with corners with non-resonant boundary indicial symbols, such as the $b$-manifolds, as well as other geometric bisection covariant heat flows. We also establish the short time asymptotic expansion for the heat kernel of a positive, elliptic differential operator on a Lie groupoid that acts on suitable Sobolev Hilbert modules and is positive definite with respect to the appropriate $L^2$ inner product.

math.AP↗