273,993 research outputs found
Direct and inverse spectral theorems for a class of canonical systems with two singular endpoints
Part I of this paper deals with two-dimensional canonical systems
, , whose Hamiltonian is non-negative and
locally integrable, and where Weyl's limit point case takes place at both
endpoints and . We investigate a class of such systems defined by growth
restrictions on H towards a. For example, Hamiltonians on of the
form where
are included in this class. We develop a direct and inverse spectral theory
parallel to the theory of Weyl and de Branges for systems in the limit circle
case at . Our approach proceeds via - and is bound to - Pontryagin space
theory. It relies on spectral theory and operator models in such spaces, and on
the theory of de Branges Pontryagin spaces.
The main results concerning the direct problem are: (1) showing existence of
regularized boundary values at ; (2) construction of a singular Weyl
coefficient and a scalar spectral measure; (3) construction of a Fourier
transform and computation of its action and the action of its inverse as
integral transforms. The main results for the inverse problem are: (4)
characterization of the class of measures occurring above (positive Borel
measures with power growth at ); (5) a global uniqueness theorem (if
Weyl functions or spectral measures coincide, Hamiltonians essentially
coincide); (6) a local uniqueness theorem.
In Part II of the paper the results of Part I are applied to Sturm--Liouville
equations with singular coefficients. We investigate classes of equations
without potential (in particular, equations in impedance form) and
Schr\"odinger equations, where coefficients are assumed to be singular but
subject to growth restrictions. We obtain corresponding direct and inverse
spectral theorems
A statistical mechanics approach to mixing in stratified fluids
Predicting how much mixing occurs when a given amount of energy is injected
into a Boussinesq fluid is a longstanding problem in stratified turbulence. The
huge number of degrees of freedom involved in those processes renders extremely
difficult a deterministic approach to the problem. Here we present a
statistical mechanics approach yielding prediction for a cumulative, global
mixing efficiency as a function of a global Richardson number and the
background buoyancy profile.Comment: Accepted in Journal of Fluid Mechanic
State independence for tunneling processes through black hole horizons and Hawking radiation
Tunneling processes through black hole horizons have recently been
investigated in the framework of WKB theory discovering interesting interplay
with the Hawking radiation. In this paper we instead adopt the point of view
proper of QFT in curved spacetime, namely, we use a suitable scaling limit
technique to obtain the leading order of the correlation function related with
some tunneling process through a Killing horizon. The computation is done for
certain large class of reference quantum states for scalar fields. In the limit
of sharp localization either on the external side or on opposite sides of the
horizon, the quantum correlation functions appear to have thermal nature, where
in both cases the characteristic temperature is the Hawking one. Our approach
is valid for every stationary charged rotating non extremal black hole,
however, since the computation is completely local, it covers the case of a
Killing horizon which just temporarily exists in some finite region too. These
results give a strong support to the idea that the Hawking radiation, which is
detected at future infinity and needs some global structures to be defined, is
actually related to a local phenomenon taking place even for local geometric
structures (local Killing horizons) existing just for a while.Comment: 19 pages, one figure, some comments added, minor errors corrected,
accepted for publication in Communications in Mathematical Physic
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