Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpectral Functions of Singular Operators
178
0
0.0
(
0
)
Authors
Pablo Pisani
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
One of the many problems to which J.S. Dowker devoted his attention is the effect of a conical singularity in the base manifold on the behavior of the quantum fields. In particular, he studied the small-$t$ asymptotic expansion of the heat-kernel trace on a cone and its effects on physical quantities, as the Casimir energy. In this article we review some peculiar results found in the last decade, regarding the appearance of non-standard powers of $t$, and even negative integer powers of $log{t}$, in this asymptotic expansion for the selfadjoint extensions of some symmetric operators with singular coefficients. Similarly, we show that the $zeta$-function associated to these selfadjoint extensions presents an unusual analytic structure.
We consider the resolvent of a system of first order differential operators with a regular singularity, admitting a family of self-adjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents powers of $lambda$ which depend on the singularity, and can take even irrational values. The consequences for the pole structure of the corresponding $zeta$ and $eta$-functions are also discussed.
In this series of lectures, we (re)view the geometric method that reconstructs, from a geometric object: the spectral curve, an integrable system, and in particular its Tau function, Baker-Akhiezer functions and current amplitudes, all having an interpretation as CFT conformal blocks. The construction identifies Hamiltonians with cycles on the curve, and times with periods (integrals of forms over cycles). All the integrable structure is formulated in terms of homology of contours, the phase space is a space of cycles where the symplectic form is the intersection, the Hirota operator is a degree 2 second-kind cycle, a Sato shift is an addition of a 3rd kind cycle, the Hirota equations amount to saying that merging 3rd kind cycles (monopoles) yields a 2nd kind cycle (dipole). The lecture is divided into 3 parts: 1) classical case, perturbative: the spectral curve is a ramified cover of a base Riemann surface -- with some additional structure -- and the integrable system is defined as a formal power series of a small dispersion parameter $epsilon$. 2) dispersive classical case, non perturbative: the spectral curve is defined not as a ramified cover (which would be a bundle with discrete fiber), but as a vector bundle -- whose dispersionless limit consists in chosing a finite set of vectors in each fiber. 3) non-commutative case, and perturbative. The spectral curve is here a non-commutative surface, whose geometry will be defined in lecture III. 4) the full non-commutative dispersionless theory is under development is not presented in these lectures.
We use trace class scattering theory to exclude the possibility of absolutely continuous spectrum in a large class of self-adjoint operators with an underlying hierarchical structure and provide applications to certain random hierarchical operators and matrices. We proceed to contrast the localizing effect of the hierarchical structure in the deterministic setting with previous results and conjectures in the random setting. Furthermore, we survey stronger localization statements truly exploiting the disorder for the hierarchical Anderson model and report recent results concerning the spectral statistics of the ultrametric random matrix ensemble.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
