Olivier Bourget, Diomba Sambou, Amal Taarabt
We consider some compact non-selfadjoint perturbations of fibered one-dimensional discrete Schrödinger operators. We show that the perturbed operator exhibits finite discrete spectrum under suitable\- regularity conditions.
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Let $H_0$ be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and $V$ be a compact non-selfadjoint perturbation. We relate the regularity properties of $V$ to various spectral properties of the perturbed operator $H_0+V$. The structure of the discrete spectrum and the embedded eigenvalues are analysed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results sca…
We analyze the perturbations $T+B$ of a selfadjoint operator $T$ in a Hilbert space $H$ with discrete spectrum $\{t_k \}$, $T ϕ_k = t_k ϕ_k$, as an extension of our constructions in arXiv: 0912.2722 where $T$ was a harmonic oscillator operator. In particular, if $t_{k+1}-t_k \geq c k^{α- 1}, \quad α> 1/2$ and $\| B ϕ_k \| = o(k^{α- 1})$ then the system of root vectors of $T+B$, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in $H$.
We study the spectrum of a quantum star graph with a non-selfadjoint Robin condition at the central vertex. We first prove that, in the high frequency limit, the spectrum of the Robin Laplacian is close to the usual spectrum corresponding to the Kirchhoff condition. Then, we describe more precisely the asymptotics of the difference in terms of the Barra-Gaspard measure of the graph. This measure depends on the arithmetic properties of the lengths of the edges. As a by-product, this analysis provides a Weyl Law for non-selfadjoint quantum star graphs and it gives the asymptotic behaviour of the…
A discrete analog of a skew selfadjoint canonical (Zakharov-Shabat or AKNS) system with a pseudo-exponential potential is introduced. For the corresponding Weyl function the direct and inverse problem are solved explicitly in terms of three parameter matrices. As an application explicit solutions are obtained for the discrete integrable nonlinear equation corresponding to the isotropic Heisenberg magnet model. State space techniques from mathematical system theory play an important role in the proofs.
Clusters of resonances for a non-selfadjoint multichannel discrete Schrödinger operator
针对非自伴多通道离散薛定谔算子，研究其在未受扰算子谱阈值附近的共振分布，通过分析紧扰动项，计算共振精确数量并精确描述其在复平面特殊点周围的簇状分布位置。
On the discrete spectrum of non-selfadjoint operators with applications to Schrödinger operators with complex potentials
本文研究非自伴算子的离散谱，通过Birman-Schwinger算子的偏迹给出半平面中特征值上限，推广了Cwikel-Lieb-Rozenblum不等式，并讨论了新获得的Lieb-Thirring型不等式的最优性。
We determine the Schatten class for the compact resolvent of Dirichlet realizations, in unbounded domains, of a class of non-selfadjoint differential operators. This class consists of operators that can be obtained via analytic dilation from a Schrödinger operator with magnetic field and a complex electric potential. As an application, we prove, in a variety of examples motivated by Physics, that the system of generalized eigenfunctions associated with the operator is complete, or at least the existence of an infinite discrete spectrum.
Consider in $L^2(R^d)$, $d\geq 1$, the operator family $H(g):=H_0+igW$. $\ds H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2$ is the quantum harmonic oscillator with rational frequencies, $W$ a $P$ symmetric bounded potential, and $g$ a real coupling constant. We show that if $|g|<ρ$, $ρ$ being an explicitly determined constant, the spectrum of $H(g)$ is real and discrete. Moreover we show that the operator $\ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1$ has real discrete spectrum but is not diagonalizable.
We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.
We study Weyl-type perturbation theorems in the context of linear closed relations. We establish general results on perturbations for dissipative relations. In the particular case of selfadjoint relations, we study finite-rank perturbations and carry out a detailed analysis of the corresponding changes in the spectrum.
We propose to build a combinatorial invariant, called the spectral monodromy, from the spectrum of a single non-selfadjoint h-pseudodifferential operator with two degrees of freedom in the semi-classical limit. Our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting selfadjoint h-pseudodifferential operators, given by S. Vu Ngoc. The first simple case that we treat in this work is a normal operator. In this case, the discrete spectrum can be identified with the joint spectrum of an integrable quantum system. The second more complex…
In this paper, we give a description of the spectrum of a class of non-selfadjoint perturbations of selfadjoint operators in dimension one and we show that it is given by Bohr-Sommerfeld quantization conditions. To achieve this, we make use of previous work by Michael Hitrik, Anders Melin and Johannes Sj{ö}strand. We also give an application of our result in the case of PT-symmetric pseudo-differential operators.
We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable window in the complex spectral plane, provided that the strength of the non-selfadjoint perturbation $\gg h$ (or sometimes $\gg h^…
Spectral multiplicity of selfadjoint Schroedinger operators on star-graphs with general interface conditions
该研究探讨星图上自伴薛定谔算子在一般界面条件下的谱重数，通过边界关系粘贴构造算子，并利用解耦算子的谱数据刻画奇异谱（连续谱与点谱）的性质。
We build a combinatorial invariant, called the spectral monodromy from the spectrum of a non-selfadjoint h -pseudodifferential operator with two degrees of freedom in the semi-classical limit. We treat small non-selfadjoint perturbation of selfadjoint h-pseudodifferential operators in two case: in the first, we assume that the classical flow of the unperturbed part is integrable; the second case, more interesting, when this flow is assumed to be quasi-integrable.
We construct quasimodes for some non-selfadjoint semiclassical operators at the boundary of the pseudo-spectrum using propagation of Hagedorn wave-packets. Assuming that the imaginary part of the principal symbol of the operator is non-negative and vanishes on certain points of the phase-space satisfying a subelliptic finite-type condition, we construct quasimodes that concentrate on these non-damped points. More generally, we apply this technique to construct quasimodes for non-selfadjoint semiclassical perturbations of the harmonic oscillator that concentrate on non-damped periodic orbits or…
Changing the discrete spectrum of the half-line matrix Schrödinger operator
本文综述了半直线矩阵薛定谔算子在保持连续谱不变时改变离散谱的方法，涵盖移除、添加及改变束缚态重数情形，并给出显式示例。
Spectral inclusion and spectral exactness for singular non-selfadjoint Sturm-Liouville problems
本文研究区间截断对奇异非自伴Sturm-Liouville算子谱的影响，针对Sims情形II和III给出谱包含与精确性结果，并为Sims情形I提出检测伪特征值的检验方法，数值实验验证其有效性。
This is a sequel of a recent article by Borichev-Golinskii-Kupin, where the authors obtain Blaschke-type conditions for special classes of analytic functions in the unit disk which satisfy certain growth hypotheses. These results were applied to get Lieb-Thirring inequalities for complex compact perturbations of a selfadjoint operator with a simply connected resolvent set. The first result of the present paper is an appropriate local version of the Blaschke-type condition from Borichev-Golinskii-Kupin. We apply it to obtain a similar condition for an analytic function in a finitely connected d…
We consider resolvents of operators taking the form ${\bf A}=Γ_1{\bf B}Γ_1$ where $Γ_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an operator that acts locally in real space. Such resolvents arise naturally when one wants to solve any of the large class of linear physical equations surveyed in Parts I, II, III, and IV that can be reformulated as problems in the extended abstract theory of composites. We review how $Q^*$-convex operators can be used to bound the spectrum of ${\bf A}$. Then, based on the Cherkaev-Gibiansky transformation and subsequent…
We discuss a functional model for multi--diagonal selfadjoint operators with almost periodic coefficients that generalizes the well known model for finite band Jacobi matrices. It give us an opportunity to construct examples of almost periodic operators with different spectral properties. Main result deals with an exact condition for the uniqueness of the model of the given type.
The Champagne bottle is one of the simplest and typical examples of Liouville integrable systems that exhibit a non-trivial classical monodromy. This geometrical invariant perturbs globally the existence of action-angle coordinates on the phrase space. However, our work shows a new way to detect the geometric modification on the phrase space by looking at the spectrum of a single operator, that is small non-selfadjoint perturbation of a selfadjoint semiclassical operator accepting the Hamiltonian of the Champagne bottle as its principal symbol.
We study Schrödinger operators $H=-Δ+V$ in $L^2(Ω)$ where $Ω$ is $\mathbb R^d$ or the half-space $\mathbb R_+^d$, subject to (real) Robin boundary conditions in the latter case. For $p>d$ we construct a non-real potential $V\in L^p(Ω)\cap L^{\infty}(Ω)$ that decays at infinity so that $H$ has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum $σ_{\rm ess}(H)=[0,\infty)$. This demonstrates that the Lieb-Thirring inequalities for selfadjoint Schrödinger operators are no longer true in the non-selfadjoint case.
We consider a class of one-dimensional nonselfadjoint semiclassical pseudo-differential operators, subject to small random perturbations, and study the statistical properties of their (discrete) spectra, in the semiclassical limit $h\to 0$. We compare two types of random perturbation: a random potential vs. a random matrix. Hager and Sjöstrand had shown that, with high probability, the local spectral density of the perturbed operator follows a semiclassical form of Weyl's law, depending on the value distribution of the principal symbol of our pseudodifferential operator. Beyond the spectral de…
We study some accurate semiclassical resolvent estimates for operators that are neither selfadjoint nor elliptic, and applications to the Cauchy problem. In particular we get a precise description of the spectrum near the imaginary axis and precise resolvent estimates inside the pseudo-spectrum. We apply our results to the Kramers-Fokker-Planck operator.
We work with small non-selfadjoint perturbations of a selfadjoint quantum Hamiltonian with two degrees of freedom, assuming that the principal symbol of the selfadjoint part is (classically) a nearly integrable system, together with a globally non-degenerate condition. We define a monodromy directly from the spectrum of such an operator, in the semiclassical limit. Moreover, this spectral monodromy allows to detect the topological modification on the dynamics of the nearly integrable system. It can be identified with the monodromy for KAM invariant tori of the nearly integrable system.
We study the operator theory associated with such infinite graphs $G$ as occur in electrical networks, in fractals, in statistical mechanics, and even in internet search engines. Our emphasis is on the determination of spectral data for a natural Laplace operator associated with the graph in question. This operator $Δ$ will depend not only on $G$, but also on a prescribed positive real valued function $c$ defined on the edges in $G$. In electrical network models, this function $c$ will determine a conductance number for each edge. We show that the corresponding Laplace operator $Δ$ is automati…
We study some classes of symmetric operators for the discrete series representations of the quantum algebra U_q(su_{1,1}), which may serve as Hamiltonians of various physical systems. The problem of diagonalization of these operators (eigenfunctions, spectra, overlap coefficients, etc.) is solved by expressing their overlap coefficients in terms of the known families of q-orthogonal polynomials. We consider both bounded and unbounded operators. In the latter case they are not selfadjoint and have deficiency indices (1,1), which means that they have infinitely many selfadjoint extensions. We fi…
This mini-course of 20 lectures aims at highlights of spectral theory for self-adjoint partial differential operators, with a heavy emphasis on problems with discrete spectrum. Part I: Discrete Spectrum (ODE preview, Laplacian - computable spectra, Schroedinger - computable spectra, Discrete spectral theorem via sesquilinear forms, Laplace eigenfunctions, Natural boundary conditions, Magnetic Laplacian, Schroedinger in confining well, Variational characterizations, Monotonicity of eigenvalues, Weyl's asymptotic, Polya's conjecture, Reaction-diffusion stability, Thin fluid film stability) Part …
Aim of this paper is trying to show the possible significance, and usefulness, of various non-selfadjoint operators for suitable Observables in non-relativistic and relativistic quantum mechanics, and in quantum electrodynamics: More specifically, this work starts dealing with: (i) the hermitian (but not selfadjoint) Time operator in non-relativistic quantum mechanics and in quantum electrodynamics; with (ii) idem, with the introduction of Time and Space operators; and with (iii) the problem of the four-position and four-momentum operators, each one with its hermitian and anti-hermitian parts,…
We study the momentum operator defined on the disjoint union of two intervals. Even in one dimension, the question of two non-empty open and non-overlapping intervals has not been worked out in a way that extends the cases of a single interval and gives a list of the selfadjoint extensions. Starting with zero boundary conditions at the four endpoints, we characterize the selfadjoint extensions and undertake a systematic and complete study of the spectral theory of the selfadjoint extensions. In an application of our extension theory to harmonic analysis, we offer a new family of spectral pairs…
We study spectral properties of two-dimensional canonical systems $y'(t)=zJH(t)y(t)$, $t\in[a,b)$, where the Hamiltonian $H$ is locally integrable on $[a,b)$, positive semidefinite, and Weyl's limit point case takes place at $b$. We answer the following questions explicitly in terms of $H$: Is the spectrum of the associated selfadjoint operator discrete ? If it is discrete, what is its asymptotic distribution ? Here asymptotic distribution means summability and limit superior conditions relative to comparison functions growing sufficiently fast. Making an analogy with complex analysis, this co…
A semiclassical argument is used to show that the low-lying spectrum of a selfadjoint operator, the so-called spectral localizer, determines the number of Dirac or Weyl points of an ideal semimetal. Apart from the IMS localization procedure, an explicit computation for the local toy models given by a Dirac or Weyl point is the key element of proof. The argument has numerous similarities to Witten's reasoning leading to the strong Morse inequalities. The same techniques allow to prove a spectral localization for the Callias operator in terms of a multi-parameter spectral flow of selfadjoint Fre…
In this paper we consider a mathematical model for the inverse beta decay in a uniform magnetic field. With this model we associate a Hamiltonian with cutoffs in an appropriate Fock space. No infrared regularization is assumed. The Hamiltonian is selfadjoint and has a ground state. We study its essential spectrum and determine its spectrum. Conditions for uniqueness of ground state are given. The coupling constant is supposed suffciently small.
The spectrum of a selfadjoint quadratic operator pencil of the form $λ^2M-λG-A$ is investigated where $M\geq 0$, $G\geq 0$ are bounded operators and $A$ is selfadjoint bounded below is investigated. It is shown that in the case of rank one operator $G$ the eigenvalues of such a pencil are of two types. The eigenvalues of one of these types are independent of the operator $G$. Location of the eigenvalues of both types is described. Examples for the case of the Sturm-Liouville operators $A$ are given.
A class of non-selfadjoint, $\PT$-symmetric operators is identified similar to a self-adjoint one, thus entailing the reality of the spectrum. The similarity transformation is explicitly constructed through the method of the quantum normal form, whose convergence (uniform with respect to the Planck constant) is proved. Further consequences of the uniform convergence of the quantum normal form are the establishment of an exact quantization formula for the eigenvalues.
We consider $C=A+B$ where $A$ is selfadjoint with a gap $(a,b)$ in its spectrum and $B$ is (relatively) compact. We prove a general result allowing $B$ of indefinite sign and apply it to obtain a $(δV)^{d/2}$ bound for perturbations of suitable periodic Schrodinger operators and a (not quite)Lieb-Thirring bound for perturbations of algebro-geometric almost periodic Jacobi matrices.
This work continues \cite{bib1} where the construction of Hamiltonian $H$ for the system of three quantum particles is considered. Namely the system consists of two fermions with mass $1$ and another particle with mass $m>0$. In the present paper, like in \cite{bib1}, we study the part $T_{l=1}$ of auxilliary operator $T = \oplus_{l=0}^{\infty} T_l$ involving the construction of the resolvent for the operator $H$. In this work together with the previous one two constants $0<m_1<m_0<\infty$ were found such that: 1) for $m>m_0$ the operator $T_{l=1}$ is selfadjoint but for $m \leqslant m_0$ it h…
This work is a continuation and extension of the note published in the Russian Math Surveys 1997 n 6. For any pair of solutions of the spectral problem for the second order selfadjoint real Schrodinger Operator on the graph their Symplectic Wronskian is a 1-cycle in the graph. This is a vector-valued symplectic 2-form on the space of solutions. This construction was applied to the Scattering Theory on the graphs with finite number of tails. The asymptotic values of solutions is a Lagrangian Plane of half dimension. This property determines all unitary properties of Scattering Matrix, which is …
Making use of recent techniques in the theory of selfadjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a 3-D bounded domain with regular boundary. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator togheter with the corresponding related eigenfunctions. These operators are used to build up a model-system where the dynamics of a quantum particle depends on the state of a quantum bit.
We consider the analytic continuation of the transfer function associated with a 2x2 operator matrix having unbounded couplings into unphysical sheets of its Riemann surface. We construct a family of non-selfadjoint operators which factorize the transfer function and reproduce certain parts of its spectrum including the nonreal (resonance) spectrum situated in the unphysical sheets neighboring the physical sheet.
In dimension $d\geq 3$, a variational principle for the size of the pure point spectrum of (discrete) Schrödinger operators $H(\mathfrak{e},V)$ on the hypercubic lattice $\mathbb{Z}^{d}$, with dispersion relation $\mathfrak{e}$ and potential $V$, is established. The dispersion relation $\mathfrak{e}$ is assumed to be a Morse function and the potential $V(x)$ to decay faster than $|x|^{-2(d+3)}$, but not necessarily to be of definite sign. Our estimate on the size of the pure-point spectrum yields the absence of embedded and threshold eigenvalues of $H(\mathfrak{e},V)$ for a class ot potentials…
Under a perturbation by a decaying electric potential, the Landau Hamiltonian acquires some discrete eigenvalues between the Landau levels. We study the perturbation by an "expanding" electric potential $V(t^{-1}x)$, $t>0$, and derive a quasi-classical formula for the counting function of the discrete spectrum as $t\to \infty$.
The Schrödinger equation is considered on the half line with a selfadjoint boundary condition when the potential is real valued, integrable, and has a finite first moment. It is proved that the potential and the two boundary conditions are uniquely determined by a set of spectral data containing the discrete eigenvalues for a boundary condition at the origin, the continuous part of the spectral measure for that boundary condition, and a subset of the discrete eigenvalues for a different boundary condition. This result provides a generalization of the celebrated uniqueness theorem of Borg and M…
Spectral triples describe and generalize Riemannian spin geometries by converting the geometrical information into algebraic data, which consist of an algebra $A$, a Hilbert space $H$ carrying a representation of $A$ and the Dirac operator $D$ (a selfadjoint operator acting on $H$). The gravitational action is described by the trace of a suitable function of $D$. In this paper we examine the (path-integral-) quantization of such a system given by a finite dimensional commutative algebra. It is then (in concrete examples) possible to construct the moduli space of the theory, i.e. to divide the …
We discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.
We discuss criteria for a self-adjoint operator on L^2(X) to have empty essential spectrum. We state a general result for the case of a locally compact abelian group X and give examples for X=R^n.
We consider non-selfadjoint operators of the kind arising in linearized NLS and prove dispersive bounds for the time-evolution without assuming that the edges of the essential spectrum are regular. Our approach does not depend on any specific properties of NLS. Rather, it is axiomatic on the linear level, and our results are obtained from four assumptions (which are of course motivated by NLS). This work is in three dimensions.
We prove Cwikel-Lieb-Rosenbljum and Lieb-Thirring type bounds on the discrete spectrum of a two-body pair operator and calculate spectral asymptotics for the eigenvalue moments and the local spectral density in the pseudo-relativistic limit.
In this paper, we develop spectral analysis of a discrete non-Hermitian quantum system that is a discrete counterpart of some continuous quantum systems on a complex contour. In particular, simple conditions for discreteness of the spectrum are established.
We prove that a positive-definite measure in $\mathbb{R}^n$ with uniformly discrete support and discrete closed spectrum, is representable as a finite linear combination of Dirac combs, translated and modulated. This extends our recent results where we proved this under the assumption that also the spectrum is uniformly discrete. As an application we obtain that Hof's quasicrystals with uniformly discrete diffraction spectra must have a periodic diffraction structure.
We proved recently that a measure on R, whose support and spectrum are both uniformly discrete sets, must have a periodic structure. Here we show that this is not the case if the support and the spectrum are just discrete closed sets.
Supersymmetric (SUSY) transformation operators corresponding to complex factorization constants are analyzed as operators acting in the Hilbert space of functions square integrable on the positive semiaxis. Obtained results are applied to Hamiltonians possessing spectral singularities which are non-Hermitian SUSY partners of selfadjoint operators. A new regularization procedure for the resolution of the identity operator in terms of continuous biorthonormal set of the non-Hermitian Hamiltonian eigenfunctions is proposed. It is also shown that the continuous spectrum eigenfunction has zero bino…
We study totally ergodic quantum dynamical systems with quasi--discrete spectrum. We investigate the classification problem for such systems in terms of algebraic invariants. The results are noncommutative analogs of (a part of) the theory of Abramov.
A non-perturbative anti-normal quantization of relativistic Yang-Mills fields with a compact semisimple gauge group entails an infinite discrete bosonic energy-mass spectrum of gauge bosons in the framework of Gelfand nuclear triples. The quantum spectrum is bounded from below and has a positive mass gap. The spectrum is both Poincare and gauge invariant.
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