arXiv subjects
Publications and source records attributed to Nurulla Azamov.
In this paper I give an explicit construction of an analogue of eigenspace for points of the singular spectrum of a self-adjoint operator. This construction is based on an abstract version of homogeneous Lippmann-Schwinger equation.
Given a self-adjoint operator $H_0$ and a relatively compact self-adjoint perturbation $V$, we study in some detail the spectral properties of the product $(H_0-z)^{-1}V$. For some numbers $r_z,$ the eigenvalues of $(H_0 + s V - z)^{-1}V$ are $(s - r_z)^{-1}$, where $s$ is any complex number. We study the root spaces of the eigenvalues $(s - r_z)^{-1}$ and complex-analytic properties of the functions $r_z$ such as branching points. In particular, for a generic case, we give a variety of necessary and sufficient conditions for branching. The functions $r_z,$ called coupling resonances, are important in the spectral analysis of $H_0 + r V$ for any real number $r.$ For instance, they afford a description of the spectral shift function (SSF) of the pair $H_0$ and $V,$ as well as the absolutely continuous and singular parts of the SSF. A thorough study of real-valued coupling resonances $r_λ$ for real $λ$ outside of the essential spectrum was carried out in a recent work by the first author. Here we extend this study to the complex domain, motivated by the fact, which is well known in the case of a rank-one perturbation, that the behaviour of coupling resonances $r_z$ near the essential spectrum provides valuable information about the latter.
There are classical theorems of analysis which, given certain conditions on a perturbation, assert stability of the essential and absolutely continuous components of the spectrum of a self-adjoint operator. Whereas the singular component is known to be highly volatile under the weakest of all possible perturbations, -- rank one. In this note I announce a theorem which asserts that, nevertheless, the singular component of spectrum in an open interval is in a sense \emph{relatively stable} provided the limiting absorption principle (LAP) holds in the interval. One of the benefits of this result is the provision of a method for disproving LAP where it is suspected to fail.
Let $H_0$ be a self-adjoint operator on a Hilbert space $\mathcal H$ endowed with a rigging $F,$ which is a zero-kernel closed operator from $\mathcal H$ to another Hilbert space $\mathcal K$ such that the sandwiched resolvent $F (H_0 - z)^{-1}F^*$ is compact. Assume that $H_0$ obeys the limiting absorption principle (LAP) in the sense that the norm limit $F (H_0 - λ- i0)^{-1}F^*$ exists for a.e.~$λ.$ Numbers~$λ$ for which such limit exists we call $H_0$-regular. A number~$λ$ we call semi-regular, if the limit $F (H_0 + F^*JF - λ- i0)^{-1}F^*$ exists for at least one bounded self-adjoint operator $J$ on $\mathcal K;$ otherwise we call~$λ$ essentially singular. In this paper I discuss essentially singular points. In particular, I give different conditions which ensure that a real number~$λ$ is essentially singular, and discuss their relation to eigenvalues of infinite multiplicity which are known examples of essentially singular points.
In this note I prove the following property of Herglotz functions, which to my knowledge is new: For a Herglotz function $h(z)$ and a real number $r \in \mathbb R$ define a Herglotz function $g_r(z) = (r - h(z))^{-1}.$ Let $μ_r^{(s)}$ be the singular part of the measure $μ_r$ which corresponds to $g_r(z)$ via the Herglotz representation theorem. Then the measure $\int_0^1 μ_r^{(s)}\,dr$ is absolutely continuous, its density is integer-valued a.e., and moreover the density takes values $0$ or $1$ a.e.
In this note the following theorem is proved. Let $\mathcal H$ and $\mathcal K$ be Hilbert spaces. Let $H_0$ be a self-adjoint operator on $\mathcal H,$ $F \colon \mathcal H \to \mathcal K$ be a closed $|H_0|^{1/2}$-compact operator, and $J \colon \mathcal K \to \mathcal K$ be a bounded self-adjoint operator. If the operator $$ F (H_0 - λ- iy)^{-1} F^* $$ has norm limit as $y \to 0^+$ for a.e.~$λ,$ then so does the operator $$ F (H_0 + F^*JF - λ- iy)^{-1} F^*. $$ An invariant operator ideal version of this result is also discussed.
Let $H_0$ and $V$ be self-adjoint operators, such that $V$ admits a factorisation $V = F^*JF$ with bounded self-adjoint $J$ and $|H_0|^{1/2}$-compact $F.$ Coupling resonance functions, $r_j(z),$ of the pair $H_0$ and $V$ can be defined as $r_j(z) = - σ_j(z) ^{-1},$ where $σ_j(z)$ are eigenvalues of the compact-operator valued holomorphic function $F(H_0-z)^{-1} F^*J.$ Taken together, the functions $r_j(z)$ form an infinite-valued holomorphic function on the resolvent set of~$H_0.$ These functions contain a lot of information about the pair $H_0, V$ (this is well-known in the case of rank one $V$). A point $z_0$ of the resolvent set we call \emph{absorbing} if some $r_j(z)$ goes to $\infty$ as $z \to z_0$ along some half-interval. In this note I present some partial results concerning absorbing points of coupling resonances.
Given a self-adjoint operator $H_0$ and a relatively $H_0$-compact self-adjoint operator $V,$ the functions $r_j(z) = - σ_j^{-1}(z),$ where $σ_j(z)$ are eigenvalues of the compact operator $(H_0-z)^{-1}V,$ bear a lot of important information about the pair $H_0$ and $V.$ We call them coupling resonances. In case of rank one (and positive) perturbation $V,$ there is only one coupling resonance function, which is a Herglotz function. This case has been studied in depth in the literature, and appears in different situations, such as Sturm-Liouville theory, random Schrödinger operators, harnomic and spectral analyses, etc. The general case is complicated by the fact that the resonance functions are no longer single valued holomorphic functions, and potentially can have quite an erratic behaviour, typical for infinitely-valued holomorphic functions. Of special interest are those coupling resonance functions $r_z$ which approach a real number $r_{λ+i0}$ from the interval $[0,1]$ as the spectral parameter $z=λ+iy$ approaches a point $λ$ of the essential spectrum, since they are responsible for spectral flow through $λ$ inside essential spectrum when $H_0$ gets deformed to $H_1 = H_0+V$ via the path $H_0 + rV, r \in [0,1].$ In this paper it is shown that if the pair $H_0,$ $V$ satisfies the limiting absorption principle, then the coupling resonance functions are well-behaved near the essential spectrum in the following sense. Let $I$ be an open interval inside the essential spectrum of $H_0$ and $ε>0.$ Then there exists a compact subset~$K$ of~$I$ such that $| I \setminus K | < ε,$ and $K$ has a "non-tangential" neighbourhood in the upper complex half-plane, such that any coupling resonance function is either single-valued in the neighbourhood, or does not take a real value in the interval $[0,1].$
This paper is a continuation of the study of spectral flow inside essential spectrum initiated in \cite{AzSFIES}. Given a point $λ$ outside the essential spectrum of a self-adjoint operator $H_0,$ the resonance set, $\mathcal R(λ),$ is an analytic variety which consists of self-adjoint relatively compact perturbations $H_0+V$ of $H_0,$ for which $λ$ is an eigenvalue. One may ask for criteria for the vector $V$ to be tangent to the resonance set. Such criteria were given in \cite{AzSFnRI}. In this paper we study similar criteria for the case of $λ$ inside the essential spectrum of $H_0.$ For the case $λ\in σ_{ess}(H_0)$ the resonance set is defined in terms of the well-known limiting absorption principle. Among the results of this paper is that the resonance set contains plenty of straight lines, moreover, given any regular relatively compact perturbation $V$ there exists a finite rank self-adjoint operator, $\tilde V,$ such that the straight line $H_0 + \mathbb R(V-\tilde V)$ belongs to the resonance set. Another result of this paper is that inside the essential spectrum there exist plenty of transversal to the resonance set perturbations $V$ which have order $\geq 2,$ in contrast to what happens outside the essential spectrum, \cite{AzSFnRI}.
In this short note we demonstrate that the definition of the density of states of a Schrödinger operator with bounded potential in general depends on the choice of the domain undergoing the thermodynamic limit.
In this work we describe a simple MATLAB based language which allows to create randomized multiple choice questions with minimal effort. This language has been successfully tested at Flinders University by the author in a number of mathematics topics including Numerical Analysis, Abstract Algebra and Partial Differential Equations. The open source code of Spike is available at: https://github.com/NurullaAzamov/Spike. Enquiries about Spike should be sent to azamovnurulla@gmail.com
It is a well-known result of T.\,Kato that given a continuous path of square matrices of a fixed dimension, the eigenvalues of the path can be chosen continuously. In this paper, we give an infinite-dimensional analogue of this result, which naturally arises in the context of the so-called unitary spectral flow. This provides a new approach to spectral flow, which seems to be missing from the literature. It is the purpose of the present paper to fill in this gap.
A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrödinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators --- the density of states. In this paper we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances.
Given a self-adjoint operator H, a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure is defined such that for all points of this set the wave and the scattering matrices can be defined and constructed unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This new abstract scattering theory allows to prove that for any trace class perturbations of arbitrary self-adjoint operators the singular part of the spectral shift function is an almost everywhere integer-valued function.
In this note it is shown that for trace-class perturbations of self-adjoint operators the singular part of the spectral shift function is additive.
In this paper we prove for rank one perturbations that negative two times reciprocal of the imaginary part of resonance point is equal to the rate of change of the scattering phase as a function of the coupling constant, where the coupling constant is equal to the real part of the resonance point. This equality is in agreement with Breit-Wigner formula from quantum scattering theory. For general relatively trace class perturbations, we also give a formula for the spectral shift function in terms of resonance points, non-real and real.
With the essential spectrum of a self-adjoint operator given a relatively trace class perturbation one can associate an integer-valued invariant which admits different descriptions as the singular spectral shift function, total resonance index, and singular $μ$-invariant. In this paper we give a direct proof of the equality of the total resonance index and singular $μ$-invariant assuming only the limiting absorption principle. The proof is based on an application of the argument principle to the poles and zeros of the analytic continuation of the scattering matrix considered as a function of the coupling parameter.
Let $H_0 = -Δ+ V_0(x)$ be a Schroedinger operator on $L_2(\mathbb{R}^ν),$ $ν=1,2,$ or 3, where $V_0(x)$ is a bounded measurable real-valued function on $\mathbb{R}^ν.$ Let $V$ be an operator of multiplication by a bounded integrable real-valued function $V(x)$ and put $H_r = H_0+rV$ for real $r.$ We show that the associated spectral shift function (SSF) $ξ$ admits a natural decomposition into the sum of absolutely continuous $ξ^{(a)}$ and singular $ξ^{(s)}$ SSFs. This is a special case of an analogous result for resolvent comparable pairs of self-adjoint operators, which generalises the known case of a trace class perturbation while also simplifying its proof. We present two proofs -- one short and one long -- which we consider to have value of their own. The long proof along the way reframes some classical results from the perturbation theory of self-adjoint operators, including the existence and completeness of the wave operators and the Birman-Krein formula relating the scattering matrix and the SSF. The two proofs demonstrate the equality of the singular SSF with two a priori different but intrinsically integer-valued functions: the total resonance index and the singular $μ$-invariant.