Evgeny Korotyaev
We consider the Stark operator perturbed by a compactly supported potentials on the real line. We determine forbidden domain for resonances, asymptotics of resonances at high energy and asymptotics of the resonance counting function for large radius.
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Many-body Stark resonances by the complex absorbing potential method
本文利用复吸收势方法刻画多体 Stark Hamilton 算子的共振，证明共振即二次复势系数趋于零时复离散本征值的极限点。
Semiclassical study of shape resonances in the Stark effect
该研究针对非全局解析势的斯塔克哈密顿量共振问题，引入锥外复畸变方法，通过逃逸函数法证明非俘获预解估计，建立泛函伪微分演算，证明形状共振模型的Weyl定律与传播子共振展开，为谱移函数研究提供工具。
In this paper we prove the existence of the Stark-Wannier quantum resonances for one-dimensional Schrodinger operators with smooth periodic potential and small external homogeneous electric field. Such a result extends the existence result previously obtained in the case of periodic potentials with a finite number of open gaps.
We study the time decay of magnetic Stark resonant states. As our main result we prove that for sufficiently large time these states decay exponentially with the rate given by the imaginary parts of eigenvalues of certain non-selfadjoint operator. The proof is based on the method of complex translations.
We study the weighted averages of resonance clusters for the hydrogen atom with a Stark electric field in the weak field limit. We prove a semiclassical Szego-type theorem for resonance clusters showing that the limiting distribution of the resonance shifts concentrates on the classical energy surface corresponding to a rescaled eigenvalue of the hydrogen atom Hamiltonian. This result extends Szego-type results on eigenvalue clusters to resonance clusters. There are two new features in this work: first, the study of resonance clusters requires the use of non self-adjoint operators, and second,…
Complex absorbing potential method for Stark resonances
本研究利用复吸收势方法刻画斯塔克哈密顿量的共振，证明吸收系数趋于零时斯塔克共振为带二次复吸收势的斯塔克哈密顿量复本征值的极限点，采用锥外复形变方法，支持含局部奇异性的势作为微扰。
We discuss the resonances of Hamiltonians with constant electric field in one dimension in the limit of small field. These resonances occur near the real axis, near zeros of the analytic continuation of a reflection coefficient for potential scattering, and near the line arg z = -2π/3. We calculate their asymptotics. In conclusion we make some remarks about the higher dimensional problem.
In this paper we study the influence of an electric field on a two dimen-sional waveguide. We show that bound states that occur under a geometrical deformation of the guide turn into resonances when we apply an electric field of small intensity having a nonzero component on the longitudinal direction of the system. MSC-2010 number: 35B34,35P25, 81Q10, 82D77.
We investigate the influence of an electric field on trapped modes arising in a two-dimensional curved quantum waveguide ${\bf Ω}$ i.e. bound states of the corresponding Laplace operator $-Δ\_{\bf Ω}$. Here the curvature of the guide is supposed to satisfy some assumptions of analyticity, and decays as $O(|s|^{-\varepsilon}), \varepsilon > 3$ at infinity. We show that under conditions on the electric field $ \bf F$, ${\bf H}(F):= -Δ\_{\bf Ω} + {\bf F}. {\bf x} $ has resonances near the discrete eigenvalues of $-Δ\_{\bf Ω}$.
Let $H^{\varepsilon}=-\frac{d^2}{dx^2}+\varepsilon x +V$, $\varepsilon\geq0$, on $L^2(\mathbf{R})$. Let $V=\sum_{k=1}^Nc_k|ψ_k\rangle\langleψ_k|$ be a rank $N$ operator, where the $ψ_k\in L^2(\mathbf{R})$ are real, compactly supported, and even. Resonances are defined using analytic scattering theory. The main result is that if $ζ_n$, ${\rm Im}ζ_n<0$, are resonances of $H^{\varepsilon_n}$ for a sequence $\varepsilon_n\downarrow0$ as $n\to\infty$ and $ζ_n\toζ_0$ as $n\to\infty$, ${\rm Im}ζ_0<0$, then $ζ_0$ is \emph{not} a resonance of $H^0$.
Semiclassical shape resonances for magnetic Stark Hamiltonians
研究磁斯托克尔哈密顿量在半经典极限下的形状共振，证明共振与参考算子的离散本征值一一对应，揭示共振数量的维德定律及共振实部的渐进行为。
On the asymptotic number of low-lying states in the two-dimensional confined Stark effect
本文研究了二维受限史特林效应中低能态的渐近数目，通过建立本征值的渐近展开式和韦伊型渐近性，揭示了低能态谱密度的弱渐近行为。
We study the stationary scattering theory for a perturbed 1-body Stark operator. We prove existence and completeness of the stationary wave operators, construct the associated generalized Fourier transforms, and characterize asymptotics of the generalized eigenfunctions of minimal growths in terms of the stationary scattering matrix.
We consider the vibrational energy levels of the first two electronic states of the molecule ion $H_2^+$. The Born-Oppenheimer method applied to the case of the Stark effect on a $H_2^+$-like molecule gives existence of sharp resonances localized in the same interval of energy of the vibrational levels.
A simple 1-D relativistic model for a diatomic molecule with a double point interaction potential is solved exactly in a constant electric field. The Weyl-Titchmarsh-Kodaira method is used to evaluate the spectral density function, allowing the correct normalization of continuum states. The boundary conditions at the potential wells are evaluated using Colombeau's generalized function theory along with charge conjugation invariance and general properties of self-adjoint extensions for point-like interactions. The resulting spectral density function exhibits resonances for quasibound states whi…
We consider the magnetic AC Stark effect for the quantum dynamics of a single particle in the plane under the influence of an oscillating homogeneous electric and a constant perpendicular magnetic field. We prove that the electron cyclotron resonance is insensitive to impurity potentials.
Asymptotics of the spectral data of perturbed Stark operators in the half-line with mixed boundary conditions
本文研究了半线扰动Stark算子的谱数据渐进行为，得出了特征值和规范常数的精确渐进行为，考虑了混合边界条件和特定函数空间中的扰动项。
On L 2 (R), we consider the Schrödinger operator (1.1) H ǫ = -- $\partial$ 2 $\partial$x 2 + v(x) -- ǫx, where v is a real analytic 1-periodic function and ǫ is a positive constant. This operator is a model to study a Bloch electron in a constant electric field ([1]). The parameter ǫ is proportional to the electric field. The operator (1.1) was studied both by physicists (see, e.g., the review [6]) and by mathematicians (see, e.g., [9]). Its spectrum is absolutely continuous and fills the real axis. One of main features of H ǫ is the existence of Stark-Wannier ladders. These are ǫ-periodic seq…
One-dimensional Stark operators in the half-line
本研究针对半直线上满足Dirichlet或Neumann边界条件的受扰一维Stark算子，推导了其谱数据的渐近公式，丰富了Stark算子的谱理论研究。
We study and develop the stationary scattering theory for a class of one-body Stark Hamiltonians with short-range potentials, including the Coulomb potential, continuing our study in [AIIS1,AIIS2]. The classical scattering orbits are parabolas parametrized by asymptotic orthogonal momenta, and the kernel of the (quantum) scattering matrix at a fixed energy is defined in these momenta. We show that the scattering matrix is a classical type pseudodifferential operator and compute the leading order singularities at the diagonal of its kernel. Our approach can be viewed as an adaption of the metho…
We consider in this Note resonances for a $h$-Pseudo-Differential Operator $H(x,hD_x;h)$ on $L^2(M)$ induced by a periodic orbit of hyperbolic type, as arises for Schrödinger operator with AC Stark effect when $M={\bf R}^n$, or the geodesic flow on an axially symmetric manifold $M$, extending Poincaré example of Lagrangian systems with 2 degrees of freedom. We generalize the framework of [GéSj], in the sense that we allow for hyperbolic and elliptic eigenvalues of Poincaré map, and look for so-called semi-excited resonances with imaginary part of magnitude $-h\log h$, or $h^s$, with $0<s<1$.
We consider a Stark Hamiltonian on a two-dimensional bounded domain with Dirichlet boundary conditions. In the strong electric field limit we derive, under certain local convexity conditions, a three-term asymptotic expansion of the low-lying eigenvalues. This shows that the excitation frequencies are proportional to the square root of the boundary curvature at a certain point determined by the direction of the electric field.
We provide a rigorous derivation of an asymptotic formula for perturbations in the resonance values caused by the presence of finite number of anisotropic imperfections of small shapes with constitutive parameters different from the background conductivity. The asymptotic expansion is carried out with respect to the size of the imperfections. The main feature of the method is to yield a robust procedure making it possible to recover information about the location, shape, and material properties of the anisotropic imperfections.
We study the asymptotic distribution of resonances for scattering by compactly supported potentials in hyperbolic space. We first establish an upper bound for the resonance counting function that depends only on the dimension and the support of the potential. We then establish the sharpness of this estimate by proving the a Weyl law for the resonance counting function holds in the case of radial potentials vanishing to some finite order at the edge of the support. As an application of the existence of potentials that saturate the upper bound, we derive additional resonance asymptotics that hol…
We study asymptotical behaviour of resonances for a quantum graph consisting of a finite internal part and external leads placed into a magnetic field, in particular, the question whether their number follows the Weyl law. We prove that the presence of a magnetic field cannot change a non-Weyl asymptotics into a Weyl one and vice versa. On the other hand, we present examples demonstrating that for some non-Weyl graphs the ``effective size'' of the graph, and therefore the resonance asymptotics, can be affected by the magnetic field.
Pseudo orbit expansion for the resonance condition on quantum graphs and the resonance asymptotics
本文介绍了推导量子图共振条件的伪轨道展开法，通过三个标准耦合示例演示其应用，同时针对非Weyl图阐释了删边法简化图结构的优势。
We consider a modification of the Winter model describing a quantum particle in presence of a spherical barrier given by a fixed generalized point interaction. It is shown that the three classes of such interactions correspond to three different types of asymptotic behaviour of resonances of the model at high energies.
Singularities and asymptotic distribution of resonances for Schrödinger operators in one dimension
研究一维 Schrödinger 算子共振态的高能分布，利用反射系数积分表示给出密度上界、扇形分布结果及多列共振态势的构造与渐近定位。
We study a hard regime of stimulation of two-frequency oscillations in the main resonance equation with a fast oscillating external force: $ \ve i ψ' + |ψ|^2ψ= \exp\big(it^2/ (2\ve)\big), 0<\ve\ll1$. This phenomenon is caused by resonance between an eigenmode and the external force. The asymptotic solution before, inside and after the resonance layer is studied in detail and matched.
Ultrasound wave propagation in a nonhomogeneous linearly elastic layer of constant thickness is considered. The resonances for the corresponding acoustic propagator are studied. It is shown that the distribution of the resonances depends on the smoothness of the coefficients. Namely, if the coefficients have jump discontinuities at the boundaries, then the resonances are asymptotically distributed along a straight line parallel to the real axis on the unphysical sheet of the complex frequency plane. In the contrary, if the coefficients are continuous, then it is shown that the resonances are a…
We compute resonance width asymptotics for the delta potential on the half-line, by deriving a formula for resonances in terms of the Lambert W function and applying a series expansion. This potential is a simple model of a thin barrier, motivated by physical problems such as quantum corrals and leaky quantum graphs.
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