Finite Perturbations Research Articles

Spectral dissection of finite rank perturbations of normal operators

Finite rank perturbations T=N+K of a bounded normal operator N acting on a separable Hilbert space are studied thanks to a natural functional model of T; in its turn the functional model solely relies on a perturbation matrix/characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of T. Under mild geometric conditions on the spectral measure of N and some smoothness constraints on K we show that the operator T admits invariant subspaces, or even it is decomposable.

The statistics of spectral shifts due to finite rank perturbations

This article is dedicated to the following class of problems. Start with an N × N Hermitian matrix randomly picked from a matrix ensemble—the reference matrix. Applying a rank-t perturbation to it, with t taking the values 1 ⩽ t ⩽ N, we study the difference between the spectra of the perturbed and the reference matrices as a function of t and its dependence on the underlying universality class of the random matrix ensemble. We consider both, the weaker kind of perturbation which either permutes or randomizes t diagonal elements and a stronger perturbation randomizing successively t rows and columns. In the first case we derive universal expressions in the scaled parameter τ = t/N for the expectation of the variance of the spectral shift functions, choosing as random-matrix ensembles Dyson’s three Gaussian ensembles. In the second case we find an additional dependence on the matrix size N.

Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

Finite rank perturbations of Toeplitz products on the Bergman space

In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution ⋄ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if Fj,Gj (1≤j≤N) are polynomials of z and z¯ then ∑j=1NTFjTGj−TH is a finite rank operator for some L1-function H if and only if ∑j=1NFj⋄Gj belongs to L1 and H=∑j=1NFj⋄Gj. In the case Fj's are holomorphic and Gj's are conjugate holomorphic, it is shown that H is a solution to a system of first order partial differential equations with a constraint.

Finite rank perturbations of complex symmetric operators

A bounded linear operator A acting on a complex Hilbert space H is called complex symmetric if there exists a conjugation C on H such that CAC=A⁎. In this paper, we prove that a large class of rank-one perturbations of normal operators are complex symmetric. We show also that every operator A can be approximated by non-complex symmetric operators of the form A+F where F is a finite rank operator whose range can be controlled.

Matrix measures and finite rank perturbations of self-adjoint operators

Matrix-valued measures provide a natural language for the theory of finite rank perturbations. In this paper we use this language to prove some new perturbation theoretic results. Our main result is a generalization of the Aronszajn–Donoghue theorem about the mutual singularity of the singular parts of the spectrum for rank one perturbations to the case of finite rank perturbations. Simple direct sum type examples indicate that an exact generalization is not possible. However, in this paper we introduce the notion of vector mutual singularity for the matrix-valued measures and show that if we use this notion, the mutual singularity still holds for the finite rank perturbations. As for the scalar spectral measures and the classical mutual singularity, we show that the singular parts are mutually singular for almost all perturbations. One of the ways to prove that is to use a generalization of the Aleksandrov’s spectral averaging to the matrix-valued measures, which is also one of the main results of this paper. Finally, the spectral representation of the perturbed operator is obtained. The matrix Muckenhoupt A_2 condition appears naturally there, and it plays an important role in establishing the vector mutual singularity of the spectral measures.

Effect of nonlinear temperature and concentration profiles on the stability of a layer of fluid with chemical reaction

Investigation of the onset of thermosolutal convection with chemical reaction is carried out for different types of basic temperature and concentration gradients using linear theory and energy method. An unconditional nonlinear stability with exponential decay of finite amplitude perturbations is achieved and the Galerkin technique is utilized to solve the resulting eigenvalue problem obtained from linear and nonlinear analysis. The numerical scheme is validated with existing results and the results are obtained for linear, parabolic, inverted parabolic, piecewise linear, oscillatory, and step-function profiles of temperature and concentration gradients. The linear and nonlinear results show the existence of subcritical instability.

Susceptibility of transient chimera states.

Chaotic dynamics of a dynamical system is not necessarily persistent. If there is (without any active intervention from outside) a transition towards a (possibly nonchaotic) attractor, this phenomenon is called transient chaos, which can be observed in a variety of systems, e.g., in chemical reactions, population dynamics, neuronal activity, or cardiac dynamics. Also, chimera states, which show coherent and incoherent dynamics in spatially distinct regions of the system, are often chaotic transients. In many practical cases, the control of the chaotic dynamics (either the termination or the preservation of the chaotic dynamics) is desired. Although the self-termination typically occurs quite abruptly and can so far in general not be properly predicted, previous studies showed that in many systems a 'terminal transient phase" (TTP) prior to the self-termination existed, where the system was less susceptible against small but finite perturbations in different directions in state space. In this study, we show that, in the specific case of chimera states, these susceptible directions can be related to the structure of the chimera, which we divide into the coherent part, the incoherent part and the boundary in between. That means, in practice, if self-termination is close we can identify the direction of perturbation which is likely to maintain the chaotic dynamics (the chimera state). This finding improves the general understanding of the state space structure during the TTP, and could contribute also to practical applications like future control strategies of epileptic seizures which have been recently related to the collapse of chimera states.

Dynamics of Perturbations under Diffusion in a Porous Medium

We consider the dynamics of finite perturbations of a plane phase transition surface in the problem of evaporation of a fluid inside a low-permeability layer of a porous medium. In the case of a nonwettable porous medium, the problem has two stationary solutions, each containing a discontinuity. These discontinuities correspond to plane stationary phase transition surfaces located inside the low-permeability porous layer. One of these surfaces is unstable with respect to long-wavelength perturbations, while the other is stable. We study the evolution of perturbations of the stable plane phase transition surface. It is known that when two phase transition surfaces are located close enough to each other, the dynamics of a weakly nonlinear and weakly unstable wave packet is described by the Kolmogorov–Petrovskii–Piskunov (KPP) diffusion equation. As traveling wave solutions, this equation has heteroclinic solutions with either oscillating or monotonic structure of the front. The boundary value problem in the full statement, which should be considered if the distance between the stable and unstable plane phase transition surfaces is not small, also has similar solutions. We formulate a sufficient condition for the decrease of finite perturbations of the stable plane phase transition surface. This condition depends on their position with respect to the standing wave type and traveling front type solutions of the model equations in the model description when the KPP equation holds.

Optimal perturbations and transition energy thresholds in boundary layer shear flows

Subcritical transition to turbulence in spatially developing boundary layer flows can be triggered efficiently by finite amplitude perturbations. In this work, we employ adjoint-based optimization to identify optimal initial perturbations in the Blasius boundary layer, culminating in the computation of the subcritical transition critical energy threshold and the associated fully localized critical optimum in a spatially extended configuration, the so called minimal seed. By dynamically rescaling the variables with the local boundary layer thickness, we show that the identified edge trajectory approaches the same attracting phase space region as previously reported edge trajectories, and reaches the region more efficiently.

Transition Threshold for the 2-D Couette Flow in a Finite Channel

In this paper, we study nonlinear stability of the 3D plane Couette flow $(y,0,0)$ at high Reynolds number ${Re}$ in a finite channel $\mathbb{T}\times [-1,1]\times \mathbb{T}$. It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}$ for some $c_0>0$ independent of $Re$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow in the $L^\infty$ sense, and rapidly converges to a streak solution for $t\gg Re^{\frac 13}$ due to the mixing-enhanced dissipation effect. This result confirms the transition threshold conjecture proposed by Trefethen et al.(Science, 261(1993), 578-584). To this end, we develop the resolvent estimate method to establish the space-time estimates for the full linearized Navier-Stokes system around the flow $(V(t,y,z), 0,0)$, where $V(t,y,z)$ is a small perturbation(but independent of $Re$) of the Couette flow $y$.

Perturbations of Christoffel–Darboux Kernels: Detection of Outliers

Two central objects in constructive approximation, the Christoffel–Darboux kernel and the Christoffel function, encode ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel–Darboux kernel in $$\mathbb {C}^d$$ , with emphasis on the perturbation of Christoffel functions and their level sets with respect to perturbations of small norm or low rank. The statistical notion of leverage score provides a quantitative criterion for the detection of outliers in large data. Using the refined theory of Bergman orthogonal polynomials, we illustrate the main results, including some numerical simulations, in the case of finite atomic perturbations of area measure of a 2D region. Methods of function theory of a complex variable and (pluri) potential theory are widely used in the derivation of our perturbation formulas.

Morphological stability of rod-shaped continuous phases

Morphological transition of a rod-shaped phase into a string of spherical particles is commonly observed in the microstructures of alloys during solidification (Ratke and Mueller, 2006). This transition phenomenon can be explained by the classic Plateau-Rayleigh theory which was derived for fluid jets based on the surface area minimization principle. The quintessential work of Plateau-Rayleigh considers tiny perturbations (amplitude much less than the radius) to the continuous phase and for large amplitude perturbations, the breakup condition for the rod-shaped phase is still a knotty issue. Here, we present a concise thermodynamic model based on the surface area minimization principle as well as a non-linear stability analysis to generalize Plateau-Rayleigh’s criterion for finite amplitude perturbations. Our results demonstrate a breakup transition from a continuous phase via dispersed particles towards a uniform-radius cylinder, which has not been found previously, but is observed in our phase-field simulations. This new observation is attributed to a geometric constraint, which was overlooked in former studies. We anticipate that our results can provide further insights on microstructures with spherical particles and cylinder-shaped phases.

Stability of finite perturbations of the phase transition interface for one problem of water evaporation in a porous medium

We study global dynamics of phase transition evaporation interfaces in horizontally extended domains of porous layers where a water located over a vapor. The derivation of the model equation describing the secondary structures, which bifurcate from the ground state in a small neighborhood of the instability threshold in the case of a quasi-stationary approach to the description of the diffusion process, is presented. The resulting equation is reduced to the equation in the form of Kolmogorov-Petrovsky-Piskounov. The obtained approximate equation predicts the existence of stationary solutions in the full problem. To verify the obtained results, the numerical solution of the problem of the motion of the phase transition interface is performed using the original program code developed by the authors. The results of numerical simulation are used to verify the possibility of using stationary solutions obtained in the weakly nonlinear approximation to determine the scenario for the development of the initial localized finite amplitude perturbation. It is shown that the obtained approximate stationary solutions accurately predict the behavior of the perturbation in the vicinity of the turning point of the bifurcation diagram. A modification of the formulas describing an approximate stationary soliton-like solution is proposed in the case when the perturbation amplitude is comparable with the height of a low-permeable layer of a porous medium in which the phase transition interface is located. By numerical simulation it is shown that this modified approximate solution is in good agreement with the results of numerical calculation for the full problem.

Stability of One-Dimensional Steady Flows with Detonation Wave in a Channel of Variable Cross-Sectional Area

The stability of one-dimensional steady flows of an ideal (inviscid and non-heat-conducting) gas in channels of variable cross section with combustion in a structurally stable detonation wave is studied. The detonation wave represents a discontinuity surface propagating normally to the axis of the channel. It was previously established that, in this formulation, steady flows with combustion in a Chapman–Jouguet detonation wave are always unstable, in contrast to combustion flows in an overcompressed detonation wave. The stability analysis of such flows is reduced to the numerical solution of an initial-boundary value problem describing the evolution of finite flow perturbations between the moving detonation wave and the minimal nozzle cross section (in the case of a sudden contraction) or the exit nozzle cross section. The problem is solved by applying a modified Godunov scheme of higher order accuracy. More specifically, a Riemann solver with switching (between the overcompressed and Chapman–Jouguet detonation waves) with an explicitly represented detonation wave is created. Examples of stable flows and flows collapsing due to large initial perturbations are given, and their dynamics with transitions to a Chapman–Jouguet detonation wave and back are computed.

Torsion stability of a cylinder with circular and elliptical section under finite perturbations

In this paper, the stability of a cylinder with circular and elliptical section made of compressible and incompressible nonlinear elastic materials under finite perturbations is considered. A set of computational experiments was performed. The permissible boundaries of the region with respect to final and initial perturbations for given parameters of loading and structures are established. Finite sequences of bifurcation points are constructed, confirming, in contrast to stability at small perturbations, the existence of a hierarchy of stable equilibrium states. Classical linearized stability theories are evaluated. New phenomena and characteristic effects are established.

Critical Evolution of Finite Perturbations of a Water Evaporation Surface in Porous Media

It is shown that the approximate steady-state solutions, which satisfy the model dissipative equation that describes the process of water evaporation in the neighborhood of the instability threshold of a phase transition interface, determine localized damped finite-amplitude perturbations when a certain condition is fulfilled. These steady-state solutions can be used for forecasting the scenario of the development of a perturbation with sufficient accuracy if this perturbation has no common points with any steady-state solution. If the initial position of the phase transition front is located between the spectrally stable solution and any of the steady-state solutions, this front damps. If the initial position of the front is located above at least one of the spectrally unstable steady-state solutions, then the solution is catastrophically restructured.

Reconstructing real symmetric matrices from eigenvalues of finite dimensional perturbations

An algorithm for the reconstruction of a symmetric matrix from the eigenvalues of finite dimensional perturbations is given. The sufficient condition of determining the finite symmetric matrices is derived.

On the effect of flow restrictions on the nucleation behavior of molecules in tubular flow Nucleators

This paper discusses theoretical and experimental considerations of organic molecules nucleating inside tubular reactors or nucleators. Temperature evolution of these liquid systems is experimentally shown for different flow rates as a function of distance when these nucleators are immersed into a water bath set at spontaneous nucleating conditions. When different restrictions in the flow path are introduced before the cooling phase of the liquid; important differences on the nucleation rates are observed. For this study, Aspirin was dissolved in a blend of water and ethanol in a 50/50 vol%. At a concentration of 200 mg/mL and a nucleation temperature of 10 °C, demonstrated to be close to the metastable zonewidth, these flow restrictions show an antagonistic effect on the nucleation rate. One restriction, placed right before the nucleator enters the cooling bath, induces a reduction in nucleation rate. Putting more restrictions into the flow path with an equal separation of 5 cm in between, the nucleation recovers back to its initial value when a second restriction of an expansion ratio 2 is applied. Restrictions with an expansion ratio of 4 exceed this nucleation rate up to an order of magnitude when more than 2 restrictions are put in place. At a higher kinetic driving force defined KDF, at a concentration of 300 mg/mL of Aspirin, the influence of the restriction becomes invariant as a function of the expansion ratio. For all experiments, the nucleation rates is highly increased with the number of restrictions introduced into the flow path. A thermal gradient difference by using the restrictions on the cooling rate of the liquid flowing inside the tubing was not observed experimentally. Therefore only hydrodynamic changes of the flow seems a plausible cause for this nucleation rate - restriction dependence as it is expected that finite amplitude perturbations, amplified by the use of restrictions, cause vortex shedding in current experimental setup.

On some extensions of the A-model

The A-model for finite rank singular perturbations of class \(\mathfrak{H}_{-m-2}\setminus\mathfrak{H}_{-m-1}\), \(m \in \mathbb{N}\), is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces \((\mathfrak{H}_n)_{n\in\mathbb{Z}}\) admit an orthogonal decomposition \(\mathfrak{H}^-_n \oplus \mathfrak{H}^+_n\), with the corresponding projections satisfying \(P^{\pm}_{n+1}\subseteq P^{\pm}_n\), nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces.