Related papers: Approximations of Strongly Singular Evolution Equa…

We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state.…

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…

The reappearance of a sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be present in a recently introduced class of bilinear pseudodifferential operators…

The paper establishes conditions under which there are exact linear representations of nonlinear partial differential equations (Cauchy problems). By introducing a certain linear operator $A$, it is shown that under these conditions there…

We prove the well-posedness of non-autonomous linear evolution equations for generators $A(t): D(A(t)) \subset X \to X$ whose pairwise commutators are complex scalars and, in addition, we establish an explicit representation formula for the…

In this paper we consider evolution equations in the abstract Hilbert space under the special conditions imposed on the operator at the right-hand side of the equation. We establish the method that allows us to formulate the existence and…

We study two extension problems, and their interconnections: (i) extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; and (ii) (in case of Lie groups $G$) representations of the…

For Schr\"odinger operator $H=-\Delta+ V({\mathbf x})\cdot$, acting in the space $L_2(\mathbb R^d)\,(d\ge 3)$, necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum.are obtained without assumption that…

We consider one-dimensional Schroedinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity…

Given a complex, separable Hilbert space $\mathcal{H}$, we consider self-adjoint $L^2$-realizations of differential expressions $\tau = - (d^2/dx^2) I_{\mathcal{H}} + V(x)$, on half-lines and on the real line (assuming the limit-point…

We consider a class of partial differential equations with Carlitz derivatives over a local field of positive characteristic, for which an analog of the Cauchy problem is well-posed. Equations of such type correspond to quasi-holonomic…

We compute the deficiency spaces of operators of the form $H_A{\hat{\otimes}} I + I{\hat{\otimes}} H_B$, for symmetric $H_A$ and self-adjoint $H_B$. This enables us to construct self-adjoint extensions (if they exist) by means of von…

The diffusion operator $$ H_D=-\frac12\frac d{dx}a\frac d{dx}-b\frac d{dx}=-\frac12\exp(-2B)\frac d{dx}a\exp(2B)\frac d{dx}, $$ where $B(x)=\int_0^x\frac ba(y)dy$, defined either on $R^+=(0,\infty)$ with the Dirichlet boundary condition at…

We turn back to the well known problem of interpretation of the Schrodinger operator with the pseudopotential being the first derivative of the Dirac function. We show that the problem in its conventional formulation contains hidden…

We consider in a Hilbert space a self-adjoint operator H and a family Phi=(Phi_1,...,Phi_d) of mutually commuting self-adjoint operators. Under some regularity properties of H with respect to Phi, we propose two new formulae for a time…

Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form \[ \langle\ddot{z}(t),y \rangle + \mathfrak{d}[\dot{z} (t), y] + \mathfrak{a}_0 [z(t),y] = 0. \] Here…

We demonstrate criteria, purely based on finite subwords of the potential, to guarantee spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schr\"odinger operators on the discrete line or…

In this paper, we present a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential $\alpha x^{-2}$. Although the problem is quite old and well-studied, we believe that our…

Let $L$ be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations &(1) \quad \ddot{w}+ Lw=0, \quad w(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad…

We provide general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tool is the application of evolution semigroup methods, allowing the direct application of existing results on autonomous…