Gerald Teschl
We extend a result of Stolz and Weidmann on the approximation of isolated eigenvalues of singular Sturm-Liouville and Dirac operators by the eigenvalues of regular operators.
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We address the count of isolated and embedded eigenvalues in a generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues. The generalized eigenvalue problem determines spectral stability of nonlinear waves in a Hamiltonian dynamical system. The theory is based on the Pontryagin's Invariant Subspace theorem in an indefinite inner product space but it extends beyond the scope of earlier papers of Pontryagin, Krein, Grillakis, and others. Our main results are (i) the number of unstable and potentially unstab…
In this paper, we give Lieb-Thirring type inequalities for isolated eigenvalues of $d$-dimensional non-selfadjoint Schrödinger operators with complex-valued and dilation analytic potentials. In order to derive them, we prove that isolated eigenvalues and their multiplicities are invariant under complex dilation.
We work with the Friedrichs extension of a one dimensional Schrodinger whose potential has a certain type of regular singularity near one end point. We study the effect on the eigenvalues of shrinking the region slightly near the end point. In particular we calculate the rate at which the eigenvalues of the smaller region converge to the eigenvalues of the larger region as the region expands. A result of this type was needed by that author in another paper concerned with a domain perturbation problem on Riemannian manifolds. The problem here, however, is stated purely in terms of ordinary diff…
We prove inclusion theorems for both spectra and essential spectra as well as two-sided bounds for isolated eigenvalues for Klein-Gordon type Hamiltonian operators. We first study operators of the form $JG$, where $J$, $G$ are selfadjoint operators on a Hilbert space, $J = J^* = J^{-1}$ and $G$ is positive definite and then we apply these results to obtain bounds of the Klein-Gordon eigenvalues under the change of the electrostatic potential. The developed general theory allows applications to some other instances, as e.g. the Sturm-Liouville problems with indefinite weight.
We consider a quadratic operator pencil with a small periodic perturbation multiplied by the spectral parameter. It is motivated, in particular, by a one-dimensional Klein-Gordon equation with a time-parity-symmetric perturbation. We study in details the structure of the considered operator pencil. We show that its essential spectrum has a band structure and at certain thresholds, the bands bifurcate into small parabolas. We then study how the isolated limiting eigenvalues behave under the perturbation. We show that if zero is a limiting isolated eigenvalue, under the perturbation it remains a…
For a general ordinary differential operator $\mathcal{L}$ with periodic coefficients we prove that the characteristic polynomial of the Floquet matrix is irreducible over the field of meromorphic functions. We also consider a multipoint eigenvalue problem and show that its eigenspaces are spanned by pure or generalized Floquet solutions. Finally, at the end of the paper we mention some relevant conjectures and open questions.
We consider a $2d$ magnetic Schrödinger operator perturbed by a weak magnetic field which slowly varies around a positive mean. In a previous paper we proved the appearance of a `Landau type' structure of spectral islands at the bottom of the spectrum, under the hypothesis that the lowest Bloch eigenvalue of the unperturbed operator remained simple on the whole Brillouin zone, even though its range may overlap with the range of the second eigenvalue. We also assumed that the first Bloch spectral projection was smooth and had a zero Chern number. In this paper we extend our previous results to …
It is by now well known that the wave functions of rational solutions to the KP hierarchy (those which can be achieved as limits of the pure n-soliton solutions) satisfy an additional eigenvalue equation for ordinary differential operators in the spectral parameter. This property is known as ``bispectrality'' and has proved to be both interesting and useful. In this note, it is shown that certain (non-rational) soliton solutions of the KP hierarchy satisfy an eigenvalue equation for a non-local operator constructed by composing ordinary differential operators in the spectral parameter with tra…
We study the spectrum of the linear operator $L = - \partial_θ - ε\partial_θ (\sin θ\partial_θ)$ subject to the periodic boundary conditions on $θ\in [-π,π]$. We prove that the operator is closed in $L^2([-π,π])$ with the domain in $H^1_{\rm per}([-π,π])$ for $|ε| < 2$, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to z…
This work is devoted to the asymptotic behavior of eigenvalues of an elliptic operator with rapidly oscillating random coefficients on a bounded domain with Dirichlet boundary conditions. A sharp convergence rate is obtained for isolated eigenvalues towards eigenvalues of the homogenized problem, as well as a quantitative two-scale expansion result for eigenfunctions. Next, a quantitative central limit theorem is established for eigenvalue fluctuations; more precisely, a pathwise characterization of eigenvalue fluctuations is obtained in terms of the so-called homogenization commutator, in par…
Local form-subordination condition and Riesz basisness of root systems
针对带孤立单正特征值的无界自伴算子非对称扰动问题，引入形式局部从属方法，证明满足间隙与扰动强度条件时根系统含Riesz基且渐近公式成立，并在Schrödinger算子上验证了结果效力。
Motivated by the problems of analytic hypoellipticity, we show that a special family of compact non self-adjoint operators has a non-zero eigenvalue. We recover old results by Christ,Hanges, Himonas, Pham-The-Lai and Robert proved by using ordinary differential equations. We show our method applies to higher dimensional cases, giving in particular a new class of hypoelliptic but not analytic hypoelliptic operators.
On eigenvalues of self-adjoint extensions for defect larger than one
通过Krein公式，利用Q函数和另一解析函数刻画自伴扩张的特征值（孤立或非孤立），适用于Hilbert或Pontryagin空间中任意缺陷的对称算子，并引入广义Nevanlinna函数的广义值作为新工具。
In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension~$n\geq 2$. We use them to derive Cwikel-Lieb-Rozenblum inequalities and and Lieb-Thirring inequalities for the number of negative eigenvalues of fractional Schroedinger operators on noncommutative tori in any dimension~$n\geq 2$. The latter leads to a Sobolev inequality for noncommutative tori. On the way we establish a "borderline version" of the abstract Birman-Schwinger principle for the number of negative eigenvalues of relatively compact form perturbations of a non-negative semi-bounded operator wi…
For a class of zero order pseudodifferential operators we find the asymptotics of eigenvalues converging to a non-isolated tip of the essential spectrum.
In this paper, we consider spectral problem for the nth order ordinary differential operator with degenerate boundary conditions. We construct a nontrivial example of boundary value problem which has no eigenvalues.
Let $A(t)$ be a holomorphic family of self-adjoint operators of type (B) on a complex Hilbert space $\mathcal{H}$. Kato-Rellich perturbation theory says that isolated eigenvalues of $A(t)$ will be analytic functions of $t$ as long as they remain below the minimum of the essential spectrum of $A(t)$. At a threshold value $t_0$ where one of these eigenvalue functions hits the essential spectrum, the corresponding point in the essential spectrum might or might not be an eigenvalue of $A(t_0)$. Our results generalize a theorem of Simon to give a sufficient condition for the minimum of the essentia…
Eigenvalues and eigenfunctions of the non-local dispersal with Neumann-type boundary condition and symmetric kernel
针对非局部扩散算子在Neumann边界条件下的特征值问题，通过变分方法构造了位于连续谱之上的孤立特征值对应的非平凡特征函数，并给出了确保主特征值存在的充分几何条件。
Chebyshev collocation for linear, periodic ordinary and delay differential equations: a posteriori estimates
提出一种针对线性ODE和DDE问题的切比雪夫配置法，通过后验估计评估解精度，并推广到DDE系统以可靠估计稳定性，主要贡献是证明了单值算子特征值的后验估计定理。
We give an abstract definition of a one-dimensional Schrödinger operator with $δ'$-interaction on an arbitrary set~$Γ$ of Lebesgue measure zero. The number of negative eigenvalues of such an operator is at least as large as the number of those isolated points of the set~$Γ$ that have negative values of the intensity constants of the $δ'$-interaction. In the case where the set~$Γ$ is endowed with a Radon measure, we give constructive examples of such operators having an infinite number of negative eigenvalues.
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