Related papers: Finite-zone PT-potentials
A particle moving on a circle in a purely imaginary one-step potential is studied in both the exact and broken $PT$-symmetric regime.
We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.
Assuming GRH, we prove an explicit upper bound for the number of zeros of a Dedekind zeta function having imaginary part in $[T-a,T+a]$. We also prove a bound for the multiplicity of the zeros.
In this paper, we give an approximate formula for the measure of extreme values for the logarithm of the Riemann zeta-function and its iterated integrals. The result recovers the unconditional best result for the $\Omega$-result of…
We give defining equations for function fields over finite fields with many rational places. They are obtained from composita of quadratic extensions of the rational function field.
We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…
We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let X be an arithmetic scheme (scheme of finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of its…
A Lefschetz formula is given that relates loops in a regular finite graph to traces of a certain representation. As an application the poles of the Ihara/Bass zeta function are expressed as dimensions of global section spaces of locally…
We discuss resent developments in the problem of description of finite rank Toeplitz operators in different Bergman spaces and give some applications in analysis and mathematical physics
Finite dimensional subspaces spanned by exponential functions in the space of square integrable functions on a finite interval of the real line are considered. Their limiting positions are studied and described in terms of expo-polynomials.
A Hermite type formula is introduced and used to study the zeta function over the real and complex n-projective space. This approach allows to compute the residua at the poles and the value at the origin as well as the value of the…
We review major appearances of the functional expression $\pm \Delta \rho ^{1/2}/ \rho ^{1/2}$ in the theory of diffusion-type processes and in quantum mechanically supported dynamical scenarios. Attention is paid to various manifestations…
We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta…
A review of some recent advances in zeta function techniques is given, in problems of pure mathematical nature but also as applied to the computation of quantum vacuum fluctuations in different field theories, and specially with a view to…
PT symmetric complex potential V(r) = - r^4 + i a r^3 + b r^2 + i c r + i d/r + e/r^2 is studied. Arbitrarily large multiplets of its closed bound-state solutions with real energies are shown obtainable quasi-exactly (i.e., with a certain…
By restricting the variables running over various (possibly different) subfields, we introduce the notion of a partial zeta function. We prove that the partial zeta function is rational in an interesting case, generalizing Dwork's well…
The purpose of this paper is to investigate coefficient matrices of functional equations of zeta functions associated with homogeneous cones, which are given explicitly in the previous paper, in detail. We prove that the coefficient matrix…
We present explicit formulas for the Faddeev eigenfunctions and related generalized scattering data for point (delta-type) potentials in two dimensions. In particular, we obtain the first explicit examples of such eigenfunctions with…
Within the framework of the recently proposed formalism using non-hermitean Hamiltonians constrained merely by their PT invariance we describe a new exactly solvable family of the harmonic-oscillator-like potentials with non-equidistant…
Using numerical, theoretical and general methods, we construct evaluation formulas for the Jacobi $\theta$ functions. Some of our results are conjectures, but are verified numerically.