Sylwia Kondej
We investigate the eigengenvalues problem for self-adjoint operators with the singular perturbations. The general results presented here includes weakly as well as strongly singular cases. We illustrate these results on two models which correspond to so-called additive strongly singular perturbations.
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The phenomenon "hypo-coercivity," i.e., the increased rate of contraction for a semi-group upon adding a large skew-adjoint part to the generator, is considered for 1D semigroups generated by the Schrödinger operators $-\partial^2_x + x^2 + iγ f (x)$ with a complex potential. For $f$ of the special form$ f (x) = 1/(1 + |x|^κ)$, it is shown using complex dilations that the real part of eigenvalues of the operator are larger than a constant times $|γ|^{2/(κ+2)}$.
We investigate the behaviour of the eigenvalues of two-dimensional Pauli operators with nonconstant magnetic fields perturbed by a sign-indefinite decaying electric potential V. We prove new eigenvalues asymptotics.
We present a survey of some recent results concerning the location and the Weyl formula for the complex eigenvalues of two non self-adjoint operators. We study the eigenvalues of the generator $G$ of the contraction semigroup $e^{tG}, \: t \geq 0,$ related to the wave equation in an unbounded domain $Ω$ with dissipative boundary conditions on $\partial Ω$. Also one examines the interior transmission eigenvalues (ITE) in a bounded domain $K$ obtaining a Weyl formula with remainder for the counting function $N(r)$ of complex (ITE). The analysis is based on a semi-classical approach.
Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.
We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with complex $\ell^p$-potentials for $1\leq p\leq\infty$. As a corollary, subsets of the essential spectrum free of embedded eigenvalues are determined for small $\ell^1$-potential. Further possible improvements and sharpness of the obtained spectral bounds are also discussed.
Recently, a trace formula for non-self-adjoint periodic Schrödinger operators in $L^2(\mathbb{R})$ associated with Dirichlet eigenvalues was proved in [9]. Here we prove a corresponding trace formula associated with Neumann eigenvalues. In addition we investigate Dirichlet and Neumann eigenvalues of such operators. In particular, using the Dirichlet and Neumann trace formulas we provide detailed information on location of the Dirichlet and Neumann eigenvalues for the model operator with the potential $Ke^{2ix}$, where $K\in\mathbb{C}$.
We develop a general technique for finding self-adjoint extensions of a symmetric operator that respect a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families o…
The purpose of this paper is to prove that the spectrum of the non-self-adjoint one-particle Hamiltonian proposed by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433--6443) has interior points. We do this by first recalling that the spectrum of this random operator is the union of the set of $\ell^\infty$ eigenvalues of all infinite matrices with the same structure. We then construct an infinite matrix of this structure for which every point of the open unit disk is an $\ell^\infty$ eigenvalue, this following from the fact that the components of the eigenvector are polynomials in the spect…
This note aims to give prominence to some new results on the absence and localization of eigenvalues for the Dirac and Klein-Gordon operators, starting from known resolvent estimates already established in the literature combined with the renowned Birman-Schwinger principle.
By developing the method of multipliers, we establish sufficient conditions which guarantee the total absence of eigenvalues of the Laplacian in the half-space, subject to variable complex Robin boundary conditions. As a further application of this technique, uniform resolvent estimates are derived under the same assumptions on the potential. Some of the results are new even in the self-adjoint setting, where we obtain quantum-mechanically natural conditions.
Physical realizations of non-standard singular self-adjoint extensions for one-dimensional Schrödinger operator in terms of the mass-jump are considered. It is shown that corresponding boundary conditions can be realized for the Hamiltonian with the position-dependent effective mass in two qualitatively different profiles of the effective mass inhomogeneity: the mass-jump and the mass-bump. The existence of quantized magnetic flux in a case of the mass-jump is proven by explicit demonstration of the Zeeman-like splitting for states with the opposite projections of angular momentum.
We consider nonlinear Schrödinger equations in $\R^3$. Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both non-resonant and resonant cases. In the resonant case, the linearized operators around the excited states are non-self adjoint perturbations to some linear Hamiltonians with embedded eigenvalues. Although self-adjoint perturbation turns embedded eigenvalues into resonances, this class of non-self adjoint perturbations turn an embedded eigenvalue into t…
For integers $m\geq 3$, we study the non-self-adjoint eigenvalue problems $-u^{\prime\prime}(x)+(x^m+P(x))u(x)=E u(x)$, $0\leq x<+\infty$, with the boundary conditions $u(+\infty)=0$ and $αu(0)+βu^{\prime}(0)=0$ for some $α, β\in\C$ with $|α|+|β|\not=0$, where $P(x)=a_1 x^{m-1}+a_2 x^{m-2}+...+a_{m-1} x$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $E_{n}$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues.
Eigenvalue bounds for non-self-adjoint Schrödinger operators and pseudodifferential generalizations
本文综述了确定性和随机Schrödinger算子在R^d和紧致流形上特征值界的理论，同时提出新定理将紧致流形上的结果推广到分数Laplacian，使用Cuenin和Sogge的方法，以L^p范数描述势函数。
We prove the absence of eigenvaues of the three-dimensional Dirac operator with non-Hermitian potentials in unbounded regions of the complex plane under smallness conditions on the potentials in Lebesgue spaces. Our sufficient conditions are quantitative and easily checkable.
Singular factorizations, self-adjoint extensions, and applications to quantum many-body physics
本文研究将二阶微分算子分解为一阶算子以定义自伴算子，发现该分解会引入奇异相互作用，并据此提出一种构造 Calogero-Sutherland 型精确可解量子多体系统的简便方法。
The so-called equation of motion method is useful to obtain the explicit form of the eigenvectors and eigenvalues of certain non self-adjoint bosonic Hamiltonians with real eigenvalues. These operators can be diagonalized when they are expressed in terms of pseudo-bosons, which do not behave as ordinary bosons under the adjoint transformation, but obey the Weil-Heisenberg commutation relations.
In this paper we develop certain aspects of perturbation theory for self-adjoint operators subject to small variations of their domains. We use the abstract theory of boundary triplets to quantify such perturbations and give the second order asymptotic analysis for resolvents, spectral projections, discrete eigenvalues of the corresponding self-adjoint operators. In particular, we derive explicit formulas for the first variation and the Hessian of the eigenvalue curves bifurcating from a discrete eigenvalue of an unperturbed operator. An application is given to a matrix valued Robin Laplacian …
In this work, we use regularized determinant approach to study the discrete spectrum generated by relatively compact non-self-adjoint perturbations of the magnetic Schrödinger operator $(-i\nabla - \textbf{\textup{A}})^{2} - b$ in dimension $3$ with constant magnetic field of strength $b>0$. The situation near the Landau levels $2bq$, $q \in \mathbb{N}$, is more interesting since they play the role of thresholds of the spectrum of the free operator. First, we obtain sharp upper bounds on the number of the complex eigenvalues near the Landau levels. Under appropriate hypothesis, we then prove t…
We study the eigenvalues of the non-self adjoint problem $-y^{\prime\prime}+V(x)y=E y$ on the half-line $0\leq x<+\infty$ under the Robin boundary condition at $x=0$, where $V$ is a monic polynomial of degree $\geq 3$. We obtain a Bohr-Sommerfeld-like asymptotic formula for $E_n$ that depends on the boundary conditions. Consequently, we solve certain inverse spectral problems, recovering the potential $V$ and boundary condition from the first $(m+2)$ terms of the asymptotic formula.
In this work we construct self-adjoint extensions of the Dirac operator associated to Hermitian matrix potentials with Coulomb decay and prove that the domain is maximal. The result is obtained by means of a Hardy-Dirac type inequality. In particular, we can work with some electromagnetic potentials such that both, the electric potential and the magnetic one, have Coulomb type singularity.
Two different self-adjoint Pauli extensions describing a spin-1/2 two-dimensional quantum system with singular magnetic field are studied. An Aharonov-Casher type formula is proved for the maximal Pauli extension and it is also checked that this extension can be approximated by operators corresponding to more regular magnetic fields.
We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter and on the vector of real physical parameters. We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of the parameters. Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unsta…
Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of A by the same formula. As an application the one-dimensional Schrödinger operator with generalized zero-range potential is considered in the Sobolev space W^p_2(\mathbb{R}), p\in\mathbb{N}.
For general non-symmetric operators $A$, we prove that the moment of order $γ\ge 1$ of negative real-parts of its eigenvalues is bounded by the moment of order $γ$ of negative eigenvalues of its symmetric part $H = {1/2} [A + A^*].$ As an application, we obtain Lieb-Thirring estimates for non self-adjoint Schrödinger operators. In particular, we recover recent results by Frank, Laptev, Lieb and Seiringer \cite{FLLS}. We also discuss moment of resonances of Schrödinger self-adjoint operators.
The Kaehler quotient of a complex reductive Lie group relative to the conjugation action carries a complex algebraic stratified Kaehler structure which reflects the geometry of the group. For the group SL(n,C), we interpret the resulting singular Poisson-Kaehler geometry of the quotient in terms of complex discriminant varieties and variants thereof.
The purpose of this paper is to study spectral properties of non-self-adjoint Schrödinger operators $-Δ-\frac{(n-2)^2}{4|x|^{2}}+V$ on $\mathbb{R}^n$ with complex-valued potentials $V\in L^{p,\infty}$, $p>n/2$. We prove Keller type inequalities which measure the radius of a disc containing the discrete spectrum, in terms of the $L^{p,\infty}$ norm of $V$. Similar inequalities also hold if the inverse-square potential is replaced by a large class of subcritical potentials with critical singularities at the origin. The main new ingredient in the proof is the uniform Sobolev inequality of Kenig-R…
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…
We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the non-self-adjoint Anderson model changes suddenly as one passes…
In the paper, we use $\mathrm{U}(2)$, the group of $2\times 2$ unitary matices, to parameterize the space of all self-adjoint boundary conditions for a fixed Sturm-Liouville equation on the interval $[0,1]$. The adjoint action of $\mathrm{U}(2)$ on itself naturally leads to a refined classification of self-adjoint boundary conditions--each adjoint orbit is a subclass of these boundary conditions. We give explicit parameterizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere $S^2$, and investigate the behavior of the $n$-th eigenvalue $λ_n$ as a function…
We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This result holds under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.
We prove concentration results for $\ell_p^n$ operator norms of rectangular random matrices and eigenvalues of self-adjoint random matrices. The random matrices we consider have bounded entries which are independent, up to a possible self-adjointness constraint. Our results are based on an isoperimetric inequality for product spaces due to Talagrand.
We consider Dirac, Pauli and Schrödinger quantum magnetic Hamiltonians of full rank in ${\rm L}^2 \big(\mathbb{R}^{2d} \big)$, $d \ge 1$, perturbed by non-self-adjoint (matrix-valued) potentials. On the one hand, we show the existence of non-self-adjoint perturbations, generating near each point of the essential spectrum of the operators, infinitely many (complex) eigenvalues. In particular, we establish point spectrum analogous of Bögli results [Bög17] obtained for non-magnetic Laplacians, and hence showing that classical Lieb-Thirring inequalities cannot hold for our magnetic models. On the …
We consider the graph whose vertex set is a conjugacy class ${\mathcal C}$ consisting of finite-rank self-adjoint operators on a complex Hilbert space $H$. The dimension of $H$ is assumed to be not less than $3$. In the case when operators from ${\mathcal C}$ have two eigenvalues only, we obtain the Grassmann graph formed by $k$-dimensional subspaces of $H$, where $k$ is the smallest dimension of eigenspaces. Classical Chow's theorem describes automorphisms of this graph for $k>1$. Under the assumption that operators from ${\mathcal C}$ have more than two eigenvalues we show that every automor…
Conditions for sectoriality and compactness of the resolvent for a non-self-adjoint Sturm--Liouville operator with singular distributional potential
本文研究带奇异分布复值势的非自伴Sturm--Liouville算子，利用广义原函数给出其预解式扇形性与紧性的充要条件。
We investigate the asymptotic behaviour of large eigenvalues for a class of finite difference self-adjoint operators with compact resolvent in $l^2$.
We consider singular self-adjoint extensions for the Schrödinger operator of spin-$1/2$ particle in one dimension. The corresponding boundary conditions at a singular point are obtained. There are boundary conditions with the spin-flip mechanism, i.e. for these point-like interactions the spin operator does not commute with the Hamiltonian. One of these extensions is the analog of zero-range $δ$-potential. The other one is the analog of so called $δ^{(1)}$-interaction. We show that in physical terms such contact interactions can be identified as the point-like analogues of Rashba Hamiltonian (…
Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac-Coulomb operators
针对兼具块分解与变分间隙性质的无界线性对称算子，提出仅需极弱假设的几何策略，定义带谱间隙的特异自伴延拓并通过极小极大原理刻画间隙内特征值，结果可应用于含变号势或多原子核的类Dirac-Coulomb哈密顿量。
We study the decay of eigenfunctions of the non self-adjoint matrix operator $\calH = (\begin{smallmatrix} -Δ+μ+U & W \W & Δ-μ-U \end{smallmatrix})$, for $μ>0$, corresponding to eigenvalues in the strip $-μ<\re E <μ$.
We completely determine the Brown measure of the sum of a self-adjoint element and an elliptic element, which is the limiting eigenvalue distribution of the random matrix \[Y_N+\sqrt{s-\frac{t}{2}}X_N+i\sqrt{\frac{t}{2}}X_N'\] where $Y_N$ is an $N\times N$ deterministic Hermitian matrix whose eigenvalue distribution converges as $N\to\infty$ and $X_N$ and $X_N'$ are independent Gaussian unitary ensembles. We also study various asymptotic behaviors of this Brown measure as the variance of the elliptic element approaches infinity.
We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic equation with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the $j$th eigenfunction can be approximated by a rescaled function that is constructed in terms of the $j$th eigenfunction of fourth or second order order effective opera…
Let $H := H_{0} + V$ and $H_{\perp} := H_{0,\perp} + V$ be respectively perturbations of the free Schrödinger operators $H_{0}$ on $L^{2}\big(\mathbb{R}^{2d+1}\big)$ and $H_{0,\perp}$ on $L^{2}\big(\mathbb{R}^{2d}\big)$, $d \geq 1$ with constant magnetic field of strength $b>0$, and $V$ is a complex relatively compact perturbation. We prove Lieb-Thirring type inequalities for the discrete spectrum of $H$ and $H_{\perp}$. In particular, these estimates give $a\, priori$ information on the distribution of the discrete eigenvalues around the Landau levels of the operator, and describe how fast se…
In this work, we review two methods used to approach singular Hamiltonians in (2+1) dimensions. Both methods are based on the self-adjoint extension approach. It is very common to find singular Hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modelled by point interactions. In general, it is possible to apply some kind of regularization procedure, as the vanishing of the wave function at the location of the singularity, ensuring that the wave function is square-integrable and then can be associated with a physical state.…
Let $A:D(A)\subseteq\H\to\H$ be an injective self-adjoint operator and let $τ:D(A)\to\X$, X a Banach space, be a surjective linear map such that $\|τϕ\|_\X\le c \|Aϕ\|_\H$. Supposing that \text{\rm Range}$ (τ')\cap\H' =\{0\}$, we define a family $A^τ_Θ$ of self-adjoint operators which are extensions of the symmetric operator $A_{|\{τ=0\}.}$. Any $ϕ$ in the operator domain $D(A^τ_Θ)$ is characterized by a sort of boundary conditions on its univocally defined regular component $\phireg$, which belongs to the completion of D(A) w.r.t. the norm $\|Aϕ\|_\H$. These boundary conditions are written in…
Several recent papers have focused their attention in proving the correct analogue to the Lieb-Thirring inequalities for non self-adjoint operators and in finding bounds on the distribution of their eigenvalues in the complex plane. This paper provides some improvement in the state of the art in this topic. Precisely, we address the question of finding quantitative bounds on the discrete spectrum of the perturbed Lamé operator of elasticity $-Δ^\ast + V$ in terms of $L^p$-norms of the potential. Original results within the self-adjoint framework are provided too.
We investigate the connections between Weyl-Titchmarsh-Kodaira theory for one-dimensional Schrödinger operators and the theory of $n$-entire operators. As our main result we find a necessary and sufficient condition for a one-dimensional Schrödinger operator to be $n$-entire in terms of square integrability of derivatives (w.r.t. the spectral parameter) of the Weyl solution. We also show that this is equivalent to the Weyl function being in a generalized Herglotz-Nevanlinna class. As an application we show that perturbed Bessel operators are $n$-entire, improving the previously known condition…
Given a self-adjoint operator $A:D(A)\subseteq\calH\to\calH$ and a continuous linear operator $τ:D(A)\to\X$ with Range$ τ'\cap\calH' ={0}$, $\X$ a Banach space, we explicitly construct a family $A^τ_Θ$ of self-adjoint operators such that any $A^τ_Θ$ coincides with the original $A$ on the kernel of $τ$. Such a family is obtained by giving a Kre\uın-like formula where the role of the deficiency spaces is played by the dual pair $(\X,\X')$; the parameter $Θ$ belongs to the space of symmetric operators from $\X'$ to $\X$. When $\X=\C$ one recovers the ``$\calH_{-2}$ -construction'' of Kiselev and …
A unified S-matrix framework of quantum singular interactions is presented for the comparison of self-adjoint extensions and physical renormalization. For the long-range conformal interaction the two methods are not equivalent, with renormalization acting as selector of a preferred extension and regulator of the unbounded Hamiltonian.
We study small, PT-symmetric perturbations of self-adjoint double-well Schrödinger operators in dimension $n\geq 1$. We prove that the eigenvalues stay real for a very small perturbation, then bifurcate to the complex plane as the perturbation gets stronger.
For a self--adjoint Laplace operator on a finite, not necessarily compact, metric graph lower and upper bounds on each of the negative eigenvalues are derived. For compact finite metric graphs Poincaré type inequalities are given.
We study the radial Schrödinger equation for a particle of mass $m$ in the field of the inverse-square potential $α/r^{2}$ in the medium-weak-coupling region, i.e., with $-1/4\leq2mα/\hbar^{2}\leq3/4$. By using the renormalization method of Beane \textit{et} \textit{al.,}with two regularization potentials, a spherical square well and a spherical $δ$ shell, we illustrate that the procedure of renormalization is independent of the choice of the regularization counterterm. We show that, in the aforementioned range of the coupling constant $α$, there exists at most one bound state, in complete agr…
We establish Lieb-Thirring type inequalities for non self-adjoint relatively compact perturbations of certain operators of mathematical physics. We apply our results to quantum Hamiltonians of Schr{ö}dinger and Pauli with constant magnetic field of strength $b\textgreater{}0$. In particular, we use these bounds to obtain some information on the distribution of the eigenvalues of the perturbed operators in the neighborhood of their essential spectrum.
We consider conditions under which an embedded eigenvalue of a self-adjoint operator remains embedded under small perturbations. In the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m < \infty we show that in favorable situations the set of small perturbations of a suitable Banach space which do not remove the eigenvalue form a smooth submanifold of co-dimension m.
Effective eigenvalue approximation from moments for self-adjoint trace-class operators
针对自伴迹类算子，提出仅依赖前n个矩的谱近似集合$\Lambda_n$，可超指数收敛到极值特征值并构造严格界，无需对角化，适用于量子信息等多领域。
In this note, we present some interesting observations on the Schiffer's conjecture, interior transmission eigenvalue problem and their connections to singular and nonsingular invisibility cloaking problems of acoustic waves.
We present a general counting result for the unstable eigenvalues of linear operators of the form $JL$ in which $J$ and $L$ are skew- and self-adjoint operators, respectively. Assuming that there exists a self-adjoint operator $K$ such that the operators $JL$ and $JK$ commute, we prove that the number of unstable eigenvalues of $JL$ is bounded by the number of nonpositive eigenvalues of~$K$. As an application, we discuss the transverse stability of one-dimensional periodic traveling waves in the classical KP-II (Kadomtsev--Petviashvili) equation. We show that these one-dimensional periodic wav…
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