Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn eigenvalue problem related to the non-linear sigma-model: analytical and numerical results
132
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
An eigenvalue problem relevant for non-linear sigma model with singular metric is considered. We prove the existence of a non-degenerate pure point spectrum for all finite values of the size R of the system. In the infrared (IR) regime (large R) the eigenvalues admit a power series expansion around IR critical point Rtoinfty. We compute high order coefficients and prove that the series converges for all finite values of R. In the ultraviolet (UV) limit the spectrum condenses into a continuum spectrum with a set of residual bound states. The spectrum agrees nicely with the central charge computed by the Thermodynamic Bethe Ansatz method
rate research
Read More
We discuss the (right) eigenvalue equation for $mathbb{H}$, $mathbb{C}$ and $mathbb{R}$ linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows to translate the quaternionic problem into an {em equivalent} real or complex counterpart. Interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics.
We apply moment methods to obtaining an approximate analytical solution to Knudsen layers. Based on the hyperbolic regularized moment system for the Boltzmann equation with the Shakhov collision model, we derive a linearized hyperbolic moment system to model the scenario with the Knudsen layer vicinity to a solid wall with Maxwell boundary condition. We find that the reduced system is in an even-odd parity form that the reduced system proves to be well-posed under all accommodation coefficients. We show that the system may capture the temperature jump coefficient and the thermal Knudsen layer well with only a few moments. With the increasing number of moments used, qualitative convergence of the approximate solution is observed.
This work is divide in two cases. In the first case, we consider a spin manifold $M$ as the set of fixed points of an $S^{1}$-action on a spin manifold $X$, and in the second case we consider the spin manifold $M$ as the set of fixed points of an $S^{1}$-action on the loop space of $M$. For each case, we build on $M$ a vector bundle, a connection and a set of bundle endomorphisms. These objects are used to build global operators on $M$ which define an analytical index in each case. In the first case, the analytical index is equal to the topological equivariant Atiyah Singer index, and in the second case the analytical index is equal to a topological expression where the Witten genus appears.
This paper develops a method to carry out the large-$N$ asymptotic analysis of a class of $N$-dimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size $N$, but in the present problem, two scales $1/N^{alpha}$ and $1/N$ naturally occur. In our case, the equilibrium measure is $N^{alpha}$-dependent and characterised by means of the solution to a $2times 2$ Riemann--Hilbert problem, whose large-$N$ behavior is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-$N$ behavior of the free energy explicitly up to $o(1)$. The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales $1/N^{alpha}$ and $1/N$, thus waiving the analyticity assumptions often used in random matrix theory.
Prime numbers are the building blocks of our arithmetic, however, their distribution still poses fundamental questions. Bernhard Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the non-trivial zeros of the Riemann $zeta(s)$ function. According to the Hilbert-P{o}lya conjecture there exists a Hermitean operator of which the eigenvalues coincide with the real part of the non-trivial zeros of $zeta(s)$. This idea encourages physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Mar{v{c}}henko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the $zeta(s)$ function. We demonstrate the multifractal nature of these potentials by measuring the R{e}nyi dimension of their graphs. Our results offer hope for further analytical progress.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
