General Self-adjoint Operators Research Articles

The effect of inhomogeneous terms for asymptotic profiles of solutions to second-order abstract evolution equations with time-dependent dissipation

The inhomogeneous problem of second-order evolution equations with a time-dependent dissipation term of the form u″+Au+b(t)u′=g(t) in a real Hilbert space H is considered, where A is a general nonnegative selfadjoint operator in H. The result is the explicit description of asymptotic profile of solutions of such a problem when the dissipation b provides a diffusive structure with a suitable restriction. The proof is a basic energy method with a scope of well-behaved quantity for the diffusive structure.

Analyticity of resolvents of elliptic operators on quantum graphs with small edges

We consider an arbitrary metric graph, to which we glue another graph with edges of lengths proportional to ε, where ε is a small positive parameter. On such graph, we consider a general self-adjoint second order differential operator Hε with varying coefficients subject to general vertex conditions; all coefficients in differential expression and vertex conditions are supposed to be analytic in ε. We introduce a special operator on a certain graph obtained by rescaling the aforementioned small edges and assume that it has no embedded eigenvalues at the threshold of its essential spectrum. Under such assumption, we show that certain parts of the resolvent of Hε are analytic in ε. This allows us to represent the resolvent of Hε by a uniformly converging Taylor-like series and its partial sums can be used for approximating the resolvent up to an arbitrary power of ε. In particular, the zero-order approximation reproduces recent convergence results by G. Berkolaiko, Yu. Latushkin, S. Sukhtaiev and by C. Cacciapuoti, but we additionally show that next-to-leading terms in ε-expansions of the coefficients in the differential expression and vertex conditions can contribute to the limiting operator producing the Robin part at the vertices, to which small edges are incident. We also discuss possible generalizations of our model including both the cases of a more general geometry of the small parts of the graph and a non-analytic ε-dependence of the coefficients in the differential expression and vertex conditions.

An operational construction of the sum of two non-commuting observables in quantum theory and related constructions

The existence of a real linear space structure on the set of observables of a quantum system—i.e., the requirement that the linear combination of two generally non-commuting observables A, B is an observable as well—is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of a general observable of the form aA+bB (a,bin {{mathbb {R}}}) if such measuring instruments are given for the addends observables A and B when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of f(aA+bB) out of the spectral measures of A and B. We present such a construction with a formula which is valid for general unbounded self-adjoint operators A and B, whose spectral measures may not commute, and a wide class of functions f: {{mathbb {R}}}rightarrow {{mathbb {C}}}. In the bounded case, we prove that the Jordan product of A and B (and suitably symmetrized polynomials of A and B) can be constructed with the same procedure out of the spectral measures of A and B. The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman–Kac formula.

Selfadjoint operator Chebyshev–Grüss type inequalities

We present very general selfadjoint operator Chebyshev–Grüss type inequalities. We give applications.

Spectral analysis of discrete elliptic operators and applications in control theory

In this paper we propose an analysis of discrete spectral properties for a finite difference discretization of quite general 1D second-order self-adjoint elliptic operators. We particularly investigate some (uniform in the discretization parameter) qualitative behavior of eigenfunctions and eigenvalues. With those estimates we manage to obtain new results for the construction of bounded families of controls for semi-discrete parabolic PDEs, in particular for boundary controls of a coupled parabolic system with fewer controls than equations.

Correlation dimension Wonderland theorems

Existence of generic sets of self-adjoint operators, related to correlation dimensions of spectral measures, is investigated in separable Hilbert spaces. Typical results say that, given an orthonormal basis, the set of operators whose corresponding spectral measures are both 0-lower and 1-upper correlation dimensional is generic. The proofs rely on details of the relations among Fourier transform of spectral measures and Hausdorff and packing measures on the real line. Then such results are naturally combined with the Wonderland theorem. Applications are to classes of discrete one-dimensional Schrödinger operators and general (bounded) self-adjoint operators as well. Physical consequences include a proof of exotic dynamical behavior of singular continuous spectrum in some settings.

Fractional Laplacian on the torus

We study the fractional Laplacian [Formula: see text] on the [Formula: see text]-dimensional torus [Formula: see text], [Formula: see text]. First, we present a general extension problem that describes any fractional power [Formula: see text], [Formula: see text], where [Formula: see text] is a general nonnegative self-adjoint operator defined in an [Formula: see text]-space. This generalizes to all [Formula: see text] and to a large class of operators the previous known results by Caffarelli and Silvestre. In particular, it applies to the fractional Laplacian on the torus. The extension problem is used to prove interior and boundary Harnack’s inequalities for [Formula: see text], when [Formula: see text]. We deduce regularity estimates on Hölder, Lipschitz and Zygmund spaces. Finally, we obtain the pointwise integro-differential formula for the operator. Our method is based on the semigroup language approach.

The Weyl–von Neumann theorem and Borel complexity of unitary equivalence modulo compacts of self-adjoint operators

The Weyl–von Neumann theorem asserts that two bounded self-adjoint operators A, B on a Hilbert space H are unitarily equivalent modulo compacts, i.e.uAu* + K = B for some unitary u 𝜖 u(H) and compact self-adjoint operator K, if and only if A and B have the same essential spectrum: σess (A) = σess (B). We study, using methods from descriptive set theory, the problem of whether the above Weyl–von Neumann result can be extended to unbounded operators. We show that if H is separable infinite dimensional, the relation of unitary equivalence modulo compacts for bounded self-adjoint operators is smooth, while the same equivalence relation for general self-adjoint operators contains a dense Gδ-orbit but does not admit classification by countable structures. On the other hand, the apparently related equivalence relation A ~ B ⇔ ∃u 𝜖 U(H) [u(A-i)–1u* - (B-i)–1 is compact] is shown to be smooth.

On the convergence of the quadratic method

The convergence of the so-called quadratic method for computing eigenvalue enclosures of general self-adjoint operators is examined. Explicit asymptotic bounds for convergence to isolated eigenvalues are found. These bounds turn out to improve significantly upon those determined in previous investigations. The theory is illustrated by means of several numerical experiments performed on particularly simple benchmark models of one-dimensional Schrodinger operators.

Lieb–Thirring inequalities for generalized magnetic fields

Following an approach by Exner et al. (Commun Math Phys 26:531–541, 2014), we establish Lieb–Thirring inequalities for general self-adjoint and second-degree differential operators with matrix valued potentials acting in one space-dimension. These include and generalize the magnetic Schrödinger operator. Three different settings are considered, with functions defined on the whole real line, a semi-axis and an interval, respectively, leading to different types of bounds. An interpretation of the result in terms of Schrödinger operators acting on star graphs and graphs with two vertices is also given.

A sharp version of Ehrenfest’s theorem for general self-adjoint operators

We prove the Ehrenfest theorem of quantum mechanics under sharp assumptions on the operators involved.

Quantum-like microeconomics: Statistical model of distribution of investments and production

In this paper we demonstrate that the probabilistic quantum-like (QL) behavior–the Born’s rule, interference of probabilities, violation of Bell’s inequality, representation of variables by in general noncommutative self-adjoint operators, Schrödinger’s dynamics–can be exhibited not only by processes in the micro world, but also in economics. In our approach the QL-behavior is induced not by properties of systems. Here systems (commodities) are macroscopic. They could not be superpositions of two different states. In our approach the QL-behavior of economical statistics is a consequence of the organization of the process of production as well as investments. In particular, Hamiltonian (“financial energy”) is determined by rate of return.

A spectral approach to ill-posed problems for wave equations

In this article, we consider the problem of finding a solution to ill-posed problems for abstract wave equations in a Hilbert space, of the form $$\frac{{\rm d}^{2}u}{{\rm d}t^{2}}\left( t\right) + Au \left(t\right) = 0,\quad t \in \left(0, T\right),\quad u \left(0\right) = 0,\quad u \left(T\right) = u_{0},$$ when A is a general linear selfadjoint operator. We study issues like existence, uniqueness and continuance dependance of data and stability for this problem. Under precise constraint conditions on T, we make such problems well posed and in effect, generalize known results about these equations.

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.

Fractional telegraph equations

We investigate several aspects of the fractional telegraph equations, in an effort to better understand the anomalous diffusion processes observed in blood flow experiments. In the earlier work Eckstein et al. [Electron. J. Differential Equations Conf. 03 (1999) 39–50], the telegraph equation D 2 u+2 aDu+ Au=0 was used, where D= d/ dt, and it was shown that, as t tends to infinity, u is approximated by v, where 2 aDv+ Av=0; here A=− d 2/ dx 2 on L 2( R) , or A can be a more general nonnegative selfadjoint operator. In this paper the concern is with the fractional telegraph equation E 2 u+2 aEu+ Au=0, where E= D γ and 0< γ<1; after solving this equation it is shown that u is approximated by v, where 2 aEv+ Av=0.

Constrained dynamics for quantum mechanics. I. Restricting a particle to a surface

We analyze constrained quantum systems where the dynamics do not preserve the constraints. This is done, in particular, for the restriction of a quantum particle in Rn to a curved submanifold, and we propose a method of constraining and dynamics adjustment that produces the right Hamiltonian on the submanifold when tested on known examples. This method will be the germ of a “Dirac algorithm for quantum constraints.” We generalize it to the situation where the constraint is a general self-adjoint operator with some additional structures.

Spectral Enclosures and Complex Resonances for General Self-Adjoint Operators

AbstractThis paper considers a number of related problems concerning the computation of eigenvalues and complex resonances of a general self-adjoint operator H. The feature which ties the different sections together is that one restricts oneself to spectral properties of H which can be proved by using only vectors from a pre-assigned (possibly finite-dimensional) linear subspace L.

A mathematical interpretation of Dirac's formalism part b: generalized eigenfunctions in trajectory spaces

Starting with a Hilbert space L 2( R,μ) we introduce the dense subspace R( L 2( R,μ)) where R is a positive self-adjoint Hilbert–Schmidt operator on L 2( R ,μ). For the space R( L 2( R,μ)) a measure-theoretical Sobolev lemma is proved. The results for the spaces of type R( L 2( R,μ)) are applied to nuclear analyticity spaces S X, A = ⋃ t>0 e -t A (X) , where e −t A is a Hilbert–Schmidt operator on the Hilbert space X for each t>0. We solve the so-called generalized eigenvalue problem for a general self-adjoint operator P in X.