Recent zbMATH articles in MSC 34L40https://zbmath.org/atom/cc/34L402021-05-28T16:06:00+00:00WerkzeugGlobal bifurcation in nonlinear Dirac problems with spectral parameter in the boundary condition.https://zbmath.org/1459.340712021-05-28T16:06:00+00:00"Aliyev, Ziyatkhan S."https://zbmath.org/authors/?q=ai:aliyev.ziyatkhan-s"Manafova, Parvana R."https://zbmath.org/authors/?q=ai:manafova.parvana-rThe author consider nonlinear eigenvalue problems for a one-dimensional Dirac equation with spectral parameter in the boundary condition. They study local and global bifurcations of nontrivial solutions to these problems.
The existence of unbounded continua of nontrivial
solutions bifurcating from points and intervals of the line of trivial solutions
is given. Special cases for these problems have been paid much attention by various authors before.
So they extend some known results.
Reviewer: Hanying Feng (Shijiazhuang)Inverse resonance scattering for Dirac operators on the half-line.https://zbmath.org/1459.370582021-05-28T16:06:00+00:00"Korotyaev, Evgeny"https://zbmath.org/authors/?q=ai:korotyaev.evgeny-l"Mokeev, Dmitrii"https://zbmath.org/authors/?q=ai:mokeev.dmitrySummary: We consider massless Dirac operators on the half-line with compactly supported potentials. We solve the inverse problems in terms of Jost function and scattering matrix (including characterization). We study resonances as zeros of Jost function and prove that a potential is uniquely determined by its resonances. Moreover, we prove the following: (1) resonances are free parameters and a potential continuously depends on a resonance, (2) the forbidden domain for resonances is estimated, (3) asymptotics of resonance counting function is determined, (4) these results are applied to canonical systems.On the eigenvalues of spectral gaps of matrix-valued Schrödinger operators.https://zbmath.org/1459.650452021-05-28T16:06:00+00:00"Aljawi, Salma"https://zbmath.org/authors/?q=ai:aljawi.salma"Marletta, Marco"https://zbmath.org/authors/?q=ai:marletta.marcoSummary: This paper presents a method for calculating eigenvalues lying in the gaps of the essential spectrum of matrix-valued Schrödinger operators. The technique of dissipative perturbation allows eigenvalues of interest to move up the real axis in order to achieve approximations free from spectral pollution. Some results of the behaviour of the corresponding eigenvalues are obtained. The effectiveness of this procedure is illustrated by several numerical examples.On the direct and inverse transmission eigenvalue problems for the Schrödinger operator on the half line.https://zbmath.org/1459.340682021-05-28T16:06:00+00:00"Xu, Xiao-Chuan"https://zbmath.org/authors/?q=ai:xu.xiaochuan.1|xu.xiaochuanThe following transmission boundary value problem is considered:
\[
-\psi''+q(x)\psi=\lambda \psi,\; -\psi_0''=\lambda\psi_0,\; 0<x<1,
\]
\[
\psi'(0)-h\psi(0)=\psi'_0(0)-h\psi_0(0)=0,\;
\psi_0(1)=\psi(1),\; \psi_0'(1)=\psi'(1).
\]
The eigenvalue asymptotics is established, and the inverse problem of recovering
\(q\) from the spectrum is studied. Under additional assumptions the uniqueness
theorem is proved for this inverse problem.
Reviewer: Vjacheslav Yurko (Saratov)Scattering problem for a Dirac system with discontinuous boundary conditions.https://zbmath.org/1459.341922021-05-28T16:06:00+00:00"Guseĭnov, I. M."https://zbmath.org/authors/?q=ai:guseinov.i-m"Ragimova, G. S."https://zbmath.org/authors/?q=ai:ragimova.g-s"Khanmamedov, Ag. Kh."https://zbmath.org/authors/?q=ai:khanmamedov.agil-kh(no abstract)Sharp spectral bounds for complex perturbations of the indefinite Laplacian.https://zbmath.org/1459.341902021-05-28T16:06:00+00:00"Cuenin, Jean-Claude"https://zbmath.org/authors/?q=ai:cuenin.jean-claude"Ibrogimov, Orif O."https://zbmath.org/authors/?q=ai:ibrogimov.orif-oThe paper shows quantitative bounds for eigenvalues of complex perturbations of indefinite operators. In particular, the authors consider the operator
\[H_V:=\operatorname{sgn}(x)(-\partial_x^2+V)\quad \text{ in }\quad L^2(\mathbb R),\]
where \(\operatorname{sgn}\) is the sign function and \(V\) is a potential in an \(L^p\) space. If \(V\in L^1\), then it is proved that every eigenvalue \(\lambda\) of \(H_V\) satisfies that
\[\sqrt{2}|\lambda|\leq \sqrt{|\lambda|+|Re(\lambda)|}\ \|V\|_{L^1(\mathbb R)}.\]
This bound is shown to be sharp by explicit examples and it improves quantitatively previously known bounds. Other bounds for potentials in \(L^p\) are also presented and yield sharp spectral bounds on the imaginary parts of eigenvalues for all \(p\in[1,\infty)\).
Reviewer: Alberto Saldaña (Ciudad de México)