Spectral Zeta Function Research Articles

Out-of-time-ordered-correlators for the pure inverted quartic oscillator: classical chaos meets quantum stability

Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of –2⟨x2⟩T ⟨p2⟩T at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.

Asymptotic Expansions for the Radii of Starlikeness of Normalized q-Bessel Functions

In this paper we study the asymptotic behavior of the radii of starlikeness of the normalized Jackson’s second and third q-Bessel functions, focusing on their large orders. To achieve this, we use the Rayleigh sums of positive zeros of both q-Bessel functions, and determine the coefficients of the asymptotic expansions. We derive complete asymptotic expansions for these radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums of positive zeros of q-Bessel functions, Kvitsinsky’s results on spectral zeta functions for q-Bessel functions and asymptotic inversion. Moreover, we derive bounds for the radius of starlikeness of q-Bessel functions by using potential polynomials.

A connection between discrete and regularized Laplacian determinants on fractals

The spectral zeta function of the Laplacian on self-similar fractal sets has been previously studied and shown to meromorphically extend to the complex plane. In this work we establish that under certain conditions a relationship exists between the logarithm of the determinant of the discrete graph Laplacian on the sequence of graphs approximating the fractal and the regularized Laplacian determinant on the fractal itself which is defined via help of the spectral zeta function. We then at the end present some concrete examples of this phenomenon.

Vacuum energy of scalar fields on spherical shells with general matching conditions

In this work we analyze the spectral zeta function for massless scalar fields propagating in a D-dimensional flat space under the influence of a shell potential. The static nature of the potential, and the spherical symmetry, allows us to focus on the spatial part of the field which satisfies a one-dimensional Schrodinger equation endowed with a point potential. The shell potential is defined in terms of the two-interval self-adjoint extensions of the one-dimensional Schrodinger equation that describes the radial part of the scalar field. After performing the necessary analytic continuation, we utilize the spectral zeta function of the system to compute the vacuum energy of the field.

Spectral asymptotics of elliptic operators on manifolds

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator [Formula: see text] directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function [Formula: see text] and the heat trace [Formula: see text]. The kernel [Formula: see text] of the heat semigroup [Formula: see text], called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as [Formula: see text], that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as [Formula: see text], that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods.

Quantum flux operators for Carrollian diffeomorphism in general dimensions

We construct Carrollian scalar field theories in general dimensions, mainly focusing on the boundaries of Minkowski and Rindler spacetime, whose quantum flux operators form a faithful representation of Carrollian diffeomorphism up to a central charge, respectively. At future/past null infinity, the fluxes are physically observable and encode rich information of the radiation. The central charge may be regularized to be finite by the spectral zeta function or heat kernel method on the unit sphere. For the theory at the Rindler horizon, the effective central charge is proportional to the area of the bifurcation surface after regularization. Moreover, the zero mode of supertranslation is identified as the modular Hamiltonian, linking Carrollian diffeomorphism to quantum information theory. Our results may hold for general null hypersurfaces and provide new insight in the study of the Carrollian field theory, asymptotic symmetry group and entanglement entropy.

Special values of spectral zeta functions of graphs and Dirichlet L-functions

In this paper, we establish relations between special values of Dirichlet L-functions and those of spectral zeta functions or L-functions of cycle graphs. In fact, they determine each other in a natural way. These two kinds of special values are bridged together by a combinatorial derivative formula obtained from studying spectral zeta functions of the first order self-adjoint differential operators on the unit circle.

The heat kernel of the asymmetric quantum Rabi model

In this paper we derive an explicit formula for the heat kernel of the asymmetric quantum Rabi model, a symmetry breaking generalization of the quantum Rabi model (QRM). The method described here is the extension of a recently developed method for the heat kernel of the QRM that uses the Trotter–Kato product formula instead of path integrals or stochastic methods. In addition to the heat kernel formula, we give applications including the explicit formula for the partition function and the Weyl law for the distribution of the eigenvalues, obtained from the corresponding spectral zeta function.

Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs

We consider a class of parabolic stochastic PDEs on bounded domains $D\subseteq\mathbb{R}^d$ that includes the stochastic heat equation, but with a fractional power $\gamma$ of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces $H_r$, with $r < \gamma - d/2$, we study its power variations in $H_r$ along regular partitions of the time-axis. As the mesh size tends to zero, we find a phase transition at $r=-d/2$: the solutions have a nontrivial quadratic variation when $r<-d/2$ and a nontrivial $p$th order variation for $p= 2\gamma/(\gamma-d/2-r)>2$ when $r>-d/2$. More generally, suitably normalized power variations of any order satisfy a genuine law of large numbers in the first case and a degenerate limit theorem in the second case. When $r<-d/2$, the quadratic variation is given explicitly via an expression that involves the spectral zeta function, which reduces to the Riemann zeta function when $d=1$ and $D$ is an interval.

On the spectral zeta function of second order semiregular non-commutative harmonic oscillators

In this paper we give a meromorphic continuation of the spectral zeta function for second order semiregular Non-Commutative Harmonic Oscillators (NCHO). By “semiregular systems” we mean systems with terms with degree of homogeneity scaling by 1 in their asymptotic expansion. As an application of our results, we first compute the meromorphic continuation of the Jaynes-Cummings (JC) model spectral zeta function. Then we compute the spectral zeta function of the JC generalization to a 3-level atom in a cavity. For both of them we show that it has only one pole in 1.

Casimir energy for spinor fields with δ-shell potentials

This work analyzes the Casimir energy of a massive spinor field propagating in flat space endowed with a spherically symmetric δ-function potential. By utilizing the spectral zeta function regularization method, the Casimir energy is evaluated after performing a suitable analytic continuation. Explicit numerical results are provided for specific cases in which the Casimir energy is unambiguously defined. The results described in this work represent a generalization of the MIT bag model for spinor fields.

Heat coefficients for magnetic Laplacians on the complex projective space P n(ℂ)

We denote by Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues β m , m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of Δ ν . Using a suitable polynomial decomposition of the multiplicity of each β m , we write down a trace formula for the heat operator associated with Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with Δ ν .

Exact sum rules for spectral zeta functions of homogeneous 1D quantum oscillators, revisited *

We survey sum rules for spectral zeta functions of homogeneous 1D Schrödinger operators, that mainly result from the exact Wentzel–Kramers–Brillouin method.

Mathcal{N} = 2* Schur indices

We find closed-form expressions for the Schur indices of 4d mathcal{N} = 2* super Yang-Mills theory with unitary gauge groups for arbitrary ranks via the Fermi-gas formulation. They can be written as a sum over the Young diagrams associated with spectral zeta functions of an ideal Fermi-gas system. These functions are expressed in terms of the twisted Weierstrass functions, generating functions for quasi-Jacobi forms. The indices lie in the polynomial ring generated by the Kronecker theta function and the Weierstrass functions which contains the polynomial ring of the quasi-Jacobi forms. The grand canonical ensemble allows for another simple exact form of the indices as infinite series. In addition, we find that the unflavored Schur indices and their limits can be expressed in terms of several generating functions for combinatorial objects, including sum of triangular numbers, generalized sums of divisors and overpartitions.

Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals

The purpose of the present paper is to study the number theoretic properties of the special values of the spectral zeta functions of the non-commutative harmonic oscillator (NcHO), especially in relation to modular forms and elliptic curves from the viewpoint of Fuchsian differential equations, and deepen the understanding of the spectrum of the NcHO. We study first the general expression of special values of the spectral zeta function $\zeta\_Q(s)$ of the NcHO at $s=n$ $(n=2,3,\dots)$ and then the generating and meta-generating functions for Apéry-like numbers defined through the analysis of special values $\zeta\_Q(n)$. Actually, we show that the generating function $w\_{2n}$ of such Apéry-like numbers appearing (as the “first anomaly”) in $\zeta\_Q(2n)$ for $n=2$ gives an example of automorphic integral with rational period functions in the sense of Knopp, but still a better explanation remains to be clarified explicitly for $n>2$. This is a generalization of our earlier result on showing that $w\_2$ is interpreted as a $\Gamma(2)$-modular form of weight $1$. Moreover, certain congruence relations over primes for “normalized” Apéry-like numbers are also proven. In order to describe $w\_{2n}$ in a similar manner as $w\_2$, we introduce a differential Eisenstein series by using analytic continuation of a classical generalized Eisenstein series due to Berndt. The differential Eisenstein series is actually a typical example of the automorphic integral of negative weight. We then have an explicit expression of $w\_4$ in terms of the differential Eisenstein series. We discuss also shortly the Hecke operators acting on such automorphic integrals and relating Eichler’s cohomology group.

Spectral zeta function on discrete tori and Epstein-Riemann Hypothesis

We consider the combinatorial Laplacian on a sequence of discrete tori which approximate the α-dimensional torus. In the special case α=1, Friedli and Karlsson derived an asymptotic expansion of the corresponding spectral zeta function in the critical strip, as the approximation parameter goes to infinity. There, the authors have also formulated a conjecture on this asymptotics, that is equivalent to the Riemann Hypothesis. In this paper, inspired by the work of Friedli and Karlsson, we prove that a similar asymptotic expansion holds for α=2. Similar argument applies to higher dimensions as well. A conjecture on this asymptotics gives an equivalent formulation of the Epstein-Riemann Hypothesis, if we replace the standard discrete Laplacian with the 9-point star discrete Laplacian.

Dynamical residues of Lorentzian spectral zeta functions

We define a dynamical residue which generalizes the Guillemin–Wodzicki residue density of pseudo-differential operators. More precisely, given a Schwartz kernel, the definition refers to Pollicott–Ruelle resonances for the dynamics of scaling towards the diagonal. We apply this formalism to complex powers of the wave operator and we prove that residues of Lorentzian spectral zeta functions are dynamical residues. The residues are shown to have local geometric content as expected from formal analogies with the Riemannian case.

Some domination inequalities for spectral zeta kernels on closed Riemannian manifolds

We first prove Kato's inequalities for the Laplacian and a Schr$\ddot{o}$dinger-type operator on smooth functions on closed Riemannian manifolds. We then apply the result to establish some new domination inequalities for spectral zeta functions and their related spectral zeta kernels on $n$-dimensional unit spheres using Kato's inequalities and majorisation techniques. Our results are the generalisations of Kato's comparison inequalities for Riemannian surfaces to $n$-dimensional closed Riemannian manifolds.

Casimir pistons with generalized boundary conditions: a step forward

In this work we study the Casimir effect for massless scalar fields propagating in a piston geometry of the type $I\times N$ where $I$ is an interval of the real line and $N$ is a smooth compact Riemannian manifold. Our analysis represents a generalization of previous results obtained for pistons configurations as we consider all possible boundary conditions that are allowed to be imposed on the scalar fields. We employ the spectral zeta function formalism in the framework of scattering theory in order to obtain an expression for the Casimir energy and the corresponding Casimir force on the piston. We provide explicit results for the Casimir force when the manifold $N$ is a $d$-dimensional sphere and a disk.

Heat kernel for the quantum Rabi model: II. Propagators and spectral determinants

The quantum Rabi model (QRM) is widely recognized as an important model in quantum systems, particularly in quantum optics. The Hamiltonian H Rabi is known to have a parity decomposition H Rabi = H + ⊕ H −. In this paper, we give the explicit formulas for the propagator of the Schrödinger equation (integral kernel of the time evolution operator) for the Hamiltonian H Rabi and H ± by the Wick rotation (meromorphic continuation) of the corresponding heat kernels. In addition, as in the case of the full Hamiltonian of the QRM, we show that for the Hamiltonians H ±, the spectral determinant is, up to a non-vanishing entire function, equal to the Braak G-function (for each parity) used to prove the integrability of the QRM. To do this, we show the meromorphic continuation of the spectral zeta function of the Hamiltonians H ± and give some of its basic properties.