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Register a new userComplete positivity and self-adjointness
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We specify the structure of completely positive operators and quantum Markov semigroup generators that are symmetric with respect to a family of inner products, also providing new information on the order strucure an extreme points in some previously studied cases.
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We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Greens function and when it gives a non-constant harmonic function which is square integrable.
We prove that the only entrywise transforms of rectangular matrices which preserve total positivity or total non-negativity are either constant or linear. This follows from an extended classification of preservers of these two properties for matrices of fixed dimension. We also prove that the same assertions hold upon working only with symmetric matrices; for total-positivity preservers our proofs proceed through solving two totally positive completion problems.
We consider deformations of unbounded operators by using the novel construction tool of warped convolutions. By using the Kato-Rellich theorem we show that unbounded self-adjoint deformed operators are self-adjoint if they satisfy a certain condition. This condition proves itself to be necessary for the oscillatory integral to be well-defined. Moreover, different proofs are given for self-adjointness of deformed unbounded operators in the context of quantum mechanics and quantum field theory.
In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe--Ritter to the sphere $mathbb{S}^n$. We determine the resulting Osterwalder--Schrader Hilbert space, a construction that can be viewed as the step from euclidean to relativistic quantum field theory. We show that this process gives rise to an irreducible unitary spherical representation of the orthochronous Lorentz group $G^c = mathrm{O}_{1,n}(mathbb{R})^{uparrow}$ and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key tool is a certain complex domain $Xi$, known as the crown of the hyperboloid, containing a half-sphere $mathbb{S}^n_+$ and the hyperboloid $mathbb{H}^n$ as totally real submanifolds. This domain provides a bridge between those two manifolds when we study unitary representations of $G^c$ in spaces of holomorphic functions on $Xi$. We connect this analysis with the boundary components which are the de Sitter space and the Lorentz cone of future pointing light like vectors.
We consider the problem of essential self-adjointness of the spatial part of the Klein-Gordon operator in stationary spacetimes. This operator is shown to be a Laplace-Beltrami type operator plus a potential. In globally hyperbolic spacetimes, essential selfadjointness is proven assuming smoothness of the metric components and semi-boundedness of the potential. This extends a recent result for static spacetimes to the stationary case. Furthermore, we generalize the results to certain non-globally hyperbolic spacetimes.
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