Some Self-adjoint Quantum Semimartingales
نویسنده
چکیده
Examples of symmetric quantum semimartingales are easy to find, but essentially self-adjoint quantum semimartingales have, in general, proved elusive. This is not unexpected since a quantum semimartingale is in some sense an indefinite integral of a family of unbounded operators, and strong conditions are required to ensure that the sum of even two unbounded self-adjoint operators is self-adjoint. The vacuum-adapted theory makes a striking contrast to this: the gauge integral preserves self-adjointness. More precisely, if H is a vacuum-adapted, self-adjoint process then
منابع مشابه
A note on $lambda$-Aluthge transforms of operators
Let $A=U|A|$ be the polar decomposition of an operator $A$ on a Hilbert space $mathscr{H}$ and $lambdain(0,1)$. The $lambda$-Aluthge transform of $A$ is defined by $tilde{A}_lambda:=|A|^lambda U|A|^{1-lambda}$. In this paper we show that emph{i}) when $mathscr{N}(|A|)=0$, $A$ is self-adjoint if and only if so is $tilde{A}_lambda$ for some $lambdaneq{1over2}$. Also $A$ is self adjoint if and onl...
متن کامل9 Self - adjoint extensions and spectral analysis in Calogero problem
In this paper, we present a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential αx−2 . Although the problem is quite old and well-studied, we believe that our consideration, based on a uniform approach to constructing a correct quantum-mechanical description for systems with singular potentials and/or boundaries, proposed in o...
متن کاملLimiting absorption principle for some long range perturbations of Dirac systems at threshold energies
We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to study a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non self-adjoint operators, which we develop in appendix. We also discuss so...
متن کاملError bounds in approximating n-time differentiable functions of self-adjoint operators in Hilbert spaces via a Taylor's type expansion
On utilizing the spectral representation of selfadjoint operators in Hilbert spaces, some error bounds in approximating $n$-time differentiable functions of selfadjoint operators in Hilbert Spaces via a Taylor's type expansion are given.
متن کاملThe Wave Equation in Non-classic Cases: Non-self Adjoint with Non-local and Non-periodic Boundary Conditions
In this paper has been studied the wave equation in some non-classic cases. In the rst case boundary conditions are non-local and non-periodic. At that case the associated spectral problem is a self-adjoint problem and consequently the eigenvalues are real. But the second case the associated spectral problem is non-self-adjoint and consequently the eigenvalues are complex numbers,in which two ...
متن کامل