Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA characterization of singular packing subspaces with an application to limit-periodic operators
146
0
0.0
(
0
)
Added by
Silas Luiz Carvalho
Publication date
2016
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
A new characterization of the singular packing subspaces of general bounded self-adjoint operators is presented, which is used to show that the set of operators whose spectral measures have upper packing dimension equal to one is a $G_delta$ (in suitable metric spaces). As an application, it is proven that, generically (in space of continuous sampling functions), spectral measures of the limit-periodic Schrodinger operators have upper packing dimensions equal to one. Consequently, in a generic set, these operators are quasiballistic.
rate research
Read More
We study the one-dimensional Schrodinger operators $$ S(q)u:=-u+q(x)u,quad uin mathrm{Dom}left(S(q)right), $$ with $1$-periodic real-valued singular potentials $q(x)in H_{operatorname{per}}^{-1}(mathbb{R},mathbb{R})$ on the Hilbert space $L_{2}left(mathbb{R}right)$. We show equivalence of five basic definitions of the operators $S(q)$ and prove that they are self-adjoint. A new proof of continuity of the spectrum of the operators $S(q)$ is found. Endpoints of spectrum gaps are precisely described.
The asymptotic expansion of the heat-kernel for small values of its argument has been studied in many different cases and has been applied to 1-loop calculations in Quantum Field Theory. In this thesis we consider this asymptotic behavior for certain singular differential operators which can be related to quantum fields on manifolds with conical singularities. Our main result is that, due to the existence of this singularity and of infinitely many boundary conditions of physical relevance related to the admissible behavior of the fields on the singular point, the heat-kernel has an unusual asymptotic expansion. We describe examples where the heat-kernel admits an asymptotic expansion in powers of its argument whose exponents depend on external parameters. As far as we know, this kind of asymptotics had not been found and therefore its physical consequences are still unexplored.
In the present paper we continue our investigations of the representation theoretic side of reflection positivity by studying positive definite functions psi on the additive group (R,+) satisfying a suitably defined KMS condition. These functions take values in the space Bil(V) of bilinear forms on a real vector space V. As in quantum statistical mechanics, the KMS condition is defined in terms of an analytic continuation of psi to the strip { z in C: 0 leq Im z leq b} with a coupling condition psi (ib + t) = oline{psi (t)} on the boundary. Our first main result consists of a characterization of these functions in terms of modular objects (Delta, J) (J an antilinear involution and Delta > 0 selfadjoint with JDelta J = Delta^{-1}) and an integral representation. Our second main result is the existence of a Bil(V)-valued positive definite function f on the group R_tau = R rtimes {id_R,tau} with tau(t) = -t satisfying f(t,tau) = psi(it) for t in R. We thus obtain a 2b-periodic unitary one-parameter group on the GNS space H_f for which the one-parameter group on the GNS space H_psi is obtained by Osterwalder--Schrader quantization. Finally, we show that the building blocks of these representations arise from bundle-valued Sobolev spaces corresponding to the kernels 1/(lambda^2 - (d^2)/(dt^2}) on the circle R/bZ of length b.
We prove a limit theorem for quantum stochastic differential equations with unbounded coefficients which extends the Trotter-Kato theorem for contraction semigroups. From this theorem, general results on the convergence of approximations and singular perturbations are obtained. The results are illustrated in several examples of physical interest.
We characterize point transformations in quantum mechanics from the mathematical viewpoint. To conclude that the canonical variables given by each point transformation in quantum mechanics correctly describe the extended point transformation, we show that they are all selfadjoint operators in $L^2(mathbb{R}^n)$ and that the continuous spectrum of each coincides with $mathbb{R}$. They are also shown to satisfy the canonical commutation relations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
