Rowan Killip, Barry Simon
Necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a half-line Schrodinger operator with square integrable potential.
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We study the distribution of eigenvalues of the one-dimensional Schrödinger operator with a complex valued potential $V$. We prove that if $|V|$ decays faster than the Coulomb potential, then the series of imaginary parts of square roots of eigenvalues is convergent.
We consider Schrödinger operators $H=- \d^2/\d r^2+V$ on $L^2([0,\infty))$ with the Dirichlet boundary condition. The potential $V$ may be local or non-local, with polynomial decay at infinity. The point zero in the spectrum of $H$ is classified, and asymptotic expansions of the resolvent around zero are obtained, with explicit expressions for the leading coefficients. These results are applied to the perturbation of an eigenvalue embedded at zero, and the corresponding modified form of the Fermi golden rule.
This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schrödinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and slowly decaying potentials, for which the semiclassical rules are violated. Some new results are presented, concerning operators on product manifolds and graphs.
This paper is on magnetic Schrodinger operators in two dimensional domains with corners. Semiclassical formulas are obtained for the sum and number of eigenvalues. The obtained results extend former formulas for smooth domains in \cite{Fr, FK} to piecewise smooth domains.
A modified Fermi Golden Rule at threshold for 3D magnetic Schrödinger operators
本文研究了三维Schrödinger算子在阈值特征值处引入小磁场后转变为共振的现象，求得了共振虚部渐近展开的主项，并揭示了其与弱电场诱导共振的主要差异。
We investigate some bounds for the density of states in the pure point regime for the random Schrödinger operators $H^ω=-Δ+\displaystyle\sum_{n\in\mathbb{Z}^d}a_nq_n(ω)$, acting on $\ell^2(\mathbb{Z}^d)$, where $\{q_n\}$ are iid random variables and $a_n\simeq|n|^{-α}~~α>0$.
In this work we investigate the spectral statistics of random Schrödinger operators $H^ω=-Δ+\sum_{n\in\mathbb{Z}^d}(1+|n|^α)q_n(ω)|δ_n\rangle\langleδ_n|$, $α>0$ acting on $\ell^2(\mathbb{Z}^d)$ where $\{q_n\}_{n\in\mathbb{Z}^d}$ are i.i.d random variables distributed uniformly on $[0,1]$.
We prove upper and lower bounds for sums of eigenvalues of Lieb-Thirring type for non-self-adjoint Schrödinger operators on the half-line. The upper bounds are established for general classes of integrable potentials and are shown to be optimal in various senses by proving the lower bounds for specific potentials. We consider sums that correspond to both the critical and non-critical cases.
We introduce an algebra model to study higher order sum rules for orthogonal polynomials on the unit circle. We build the relation between the algebra model and sum rules, and prove an equivalent expression on the algebra side for the sum rules, involving a Hall-Littlewood type polynomial. By this expression, we recover an earlier result by Golinskii and Zlatǒs, and prove a new case - half of the Lukic conjecture in the case of a single critical point with arbitrary order.
In the first part of the paper we consider the Schrödinger operator $ -Δ-V(x),\quad V>0. $ We discuss the relation between the behavior of $V$ at the infinity and the properties of the negative spectrum of $H$. After that, we consider the case when $V$ changes its sign: $ V=V_+-V_-$, $2V_\pm=|V|\pm V. $ In this case, we treat $V$ and $-V$ symmetrically and study the relation between the behavior of $V$ at the infinity and the negative spectra of the operators $H_+=-Δ+V$ and $H_-=-Δ-V$.
Bounds for Eigenvalue Sums of Schrödinger Operators with Complex Radial Potentials
本文研究带复径向势的 Schrödinger 算子特征值和，利用 Schatten 空间预解式估计及 epsilon 消去方法，建立了基于势函数 $L^q$ 范数的特征值分布定量界。
Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrödinger operator with a complex-valued potential.
Lieb-Thirring-type inequalities for random Schrödinger operators with complex potentials
该研究综述带复势随机薛定谔算子特征值界的相关成果与证明，提出预解式的新Schatten范数估计方法，以此推导特征值和的界。
We consider Schrödinger operators in $\mathbb R^d$ with complex potentials supported on a hyperplane and show that all eigenvalues lie in a disk in the complex plane with radius bounded in terms of the $L^p$ norm of the potential with $d-1<p\leq d$. We also prove bounds on sums of powers of eigenvalues.
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