Revised version. Misprints corrected. New examples and a digression on a possible application to the electrodynamics of a point particle added. Accepted for publication in Journal of Functional Analysis
Given, on the Hilbert space $\H_0$, the self-adjoint operator $B$ and the skew-adjoint operators $C_1$ and $C_2$, we consider, on the Hilbert space $\H\simeq D(B)\oplus\H_0$, the skew-adjoint operator $$W=[\begin{matrix} C_2&\uno -B^2&C_1\end{matrix}]$$ corresponding to the abstract wave equation $\ddotϕ-(C_1+C_2)\dotϕ=-(B^2+C_1C_2)ϕ$. Given then an auxiliary Hilbert space $\fh$ and a linear map $τ:D(B^2)\to\fh$ with a kernel $\K$ dense in $\H_0$, we explicitly construct skew-adjoint operators $W_Θ$ on a Hilbert space $\H_Θ\simeq D(B)\oplus\H_0\oplus \fh$ which coincide with $W$ on $\N\simeq\K\oplus D(B)$. The extension parameter $Θ$ ranges over the set of positive, bounded and injective self-adjoint operators on $\fh$.
In the case $C_1=C_2=0$ our construction allows a natural definition of negative (strongly) singular perturbations $A_Θ$ of $A:=-B^2$ such that the diagram $$ \begin{CD} W @>>> W_Θ@AAA @VVV A@>>> A_Θ\end{CD} $$ is commutative.
The present work is devoted to the study of a boundary value problem for second order linear differential equation set on singular cylindrical domain. This problem can be regarded via a natural change of variables as an elliptic abstract differential equation with variable operators coefficients subject to some anti-periodic conditions. The complete study of this abstract version allows us to establish some interesting regularity results for our problem. The study is performed in the framework of Hölder spaces.
We consider the fractional powers of singular (point-like) perturbations of the Laplacian, and the singular perturbations of fractional powers of the Laplacian, and we compare such two constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains. In application to the linear and non-linear Schrödinger equations for the corresponding operators we outline a programme of relevant questions that deserve being investigated.
Boundary value problem for complete second order elliptic equation is considered in Banach space. The equation and boundary conditions involve a small and spectral parameter. The uniform L_{p}-regularity properties with respect to space variable and parameters are established. Here, the explicit formula for the solution is given and behavior of solution is derived when the small parameter approaches zero. It used to obtain singular perturbation result for abstract elliptic equation
We consider the singular limit problem in a real Hilbert space for abstract second order evolution equations with a parameter $\varepsilon \in (0,1]$. We first give an alternative proof of the celebrated results due to Kisynski (1963) from the viewpoint of the energy method. Next we derive a more precise asymptotic profile as $\varepsilon \to +0$ of the solution itself depending on $\varepsilon$ under rather high regularity assumptions on the initial data.
We qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts. We also qualify the norms of the resulting singular fractional Sobolev spaces and their mutual control with the corresponding classical Sobolev norms.
In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.
In this paper, we study the separability and spectral properties of singular degenerate elliptic equations in vector valued spaces. We prove that a realization operator by this equation with some boundary conditions is separable and Fredholm in Lebesque spaces. The leading part of the associated differential operator is not self-adjoint. The sharpe estimate of the resolvent, discreetness of spectrum and completeness of root elements of this operator is obtained
Let $G$ be a compact, connected simple Lie group and $\mathfrak{g}$ its Lie algebra. It is known that if $μ$ is any $G$-invariant measure supported on an adjoint orbit in $\mathfrak{g}$, then for each integer $k$, the $k$% -fold convolution product of $μ$ with itself is either singular or in $% L^{2}$. This was originally proven by computations that depended on the Lie type of $\mathfrak{g}$, as well as properties of the measure. In this note, we observe that the validity of this dichotomy is a direct consequence of the Duistermaat-Heckman theorem from symplectic geometry and that, in fact, an…
The paper discusses nonlinear singular perturbations of delta type of the fractional Schrödinger equation $\imath\partial_tψ=\left(-\triangle\right)^sψ$, with $s\in(\frac{1}{2},1]$, in dimension one. Precisely, we investigate local and global well posedness (in a strong sense), conservations laws and existence of blow-up solutions and standing waves.
We derive a description of the family of canonical selfadjoint extensions of the operator of multiplication in a de Branges space in terms of singular rank-one perturbations using distinguished elements from the set of functions associated with a de Branges space. The scale of rigged Hilbert spaces associated with this construction is also studied from the viewpoint of de Branges's theory.
In order to successfully explore quantum systems which are perturbations of simple models, it is essential to understand the complexity of perturbation bounds. We must ask ourselves: How quantum many-body systems can be artificially engineered to produce the needed behavior. Therefore, it is convenient to make use of abstract framework to better understand classical and quantum systems. Thus, our investigation's purpose is to explore stability and perturbation bounds of positive $C_0$-semigroups on abstract state spaces using the Dobrushin's ergodicity coefficient. Consequently, we obtain a li…
In this paper we investigate the one-dimensional harmonic oscillator with a singular perturbation concentrated in one point. We describe all possible selfadjoint realizations and we show that for certain conditions on the perturbation exactly one negative eigenvalues can arise. This eigenvalue tends to $-\infty$ as the perturbation becomes stronger.
We provide a new perturbation theorem for substochastic semigroups on abstract AL spaces extending Kato's perturbation theorem to non-densely defined operators. We show how it can be applied to piecewise deterministic Markov processes and transport equations with abstract boundary conditions. We give particular examples to illustrate our results.
This paper can be considered as the sequel of [6], where the authors have proposed an abstract construction of Hardy spaces H^1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarify the use of Hardy spaces. Then with the help of the bilinear interpolation theory, we then give applications to study bilinear operators on Lebesgue spaces. These ideas permit us to study singular operators with singularities similar to those of bilinear Calderon-Zygmund operators in a far more abs…
The existence of a ground state of the Nelson Hamiltonian with a perturbation is considered. The self-adjointness of the Hamiltonian and the existence of a ground state are proven for arbitrary values of coupling constants.
Using dual perturbation theory in a non-sun-reflexive context, we establish a correspondence between 1. a class of nonlinear abstract delay differential equations (DDEs) with unbounded linear part and an unknown taking values in an arbitrary Banach space and 2. a class of abstract weak* integral equations of convolution type involving the sun-star adjoint of a translation-like strongly continuous semigroup. For this purpose we also characterize the sun dual of the underlying state space. More generally we consider bounded linear perturbations of an arbitrary strongly continuous semigroup and w…
A conjecture from a paper by J. Bierkens and A.C.M. Ran concerning the location of eigenvalues of rank one perturbations of singular M-matrices is shown to be false in dimension four and higher, but true for dimension two, as well as for dimension three with an additional condition on the perturbation.
We present an abstract result on removing regularization for singular potentials which are not semibounded from below. The relation between ``right'' regularizations and ``right'' self-adjoint extensions of the perturbed Schrödinger operators is examined.
When a solution to an abstract inverse linear problem on Hilbert space is approximable by finite linear combinations of vectors from the cyclic subspace associated with the datum and with the linear operator of the problem, the solution is said to be a Krylov solution, i.e., it belongs to the Krylov subspace of the problem. Krylov solvability of the inverse problem allows for solution approximations that, in applications, correspond to the very efficient and popular Krylov subspace methods. We study here the possible behaviours of persistence, gain, or loss of Krylov solvability under suitable…
The work is devoted to the study of Laplace operator when the potential is a singular generalized function and plays the role of a singular perturbation of a Laplace operator. Abstract theorem obtained earlier by the authors B.N. Biyarov and G.K. Abdrasheva applies to this. The main purpose of the study is the spectral issue. Singular perturbations for differential operators have been studied by many authors for the mathematical substantiation of solvable models of quantum mechanics, atomic physics, and solid state physics. In all these cases, the problems were self-adjoint. In this paper, we …
The dependence of singularities of scattering matrices of the abstract wave equation on the choice of asymptotically equivalent outgoing/incoming subspaces is studied. The obtained results are applied to the radial wave equation with nonlocal potential. In the latter case, the concept of associated inner function introduced in the Douglas-Shapiro-Shields work \cite{DSS} plays an essential role.
In this paper we give a description of the asymptotic behavior, as $ε\to 0$, of the $ε$-gradient flow in the finite dimensional case. Under very general assumptions we prove that it converges to an evolution obtained by connecting some smooth branches of solutions to the equilibrium equation (slow dynamics) through some heteroclinic solutions of the gradient flow (fast dynamics).
