Functional determinants of differential operators play a prominent role in many fields of theoretical and mathematical physics, ranging from condensed matter physics, to atomic, molecular and particle physics. They are, however, difficult to compute reliably in non-trivial cases. In one dimensional problems (i.e. functional determinants of ordinary differential operators), a classic result of Gel’fand and Yaglom greatly simplifies the computation of functional determinants. Here I report some recent progress in extending this approach to higher dimensions (i.e., functional determinants of partial differential operators), with applications in quantum field theory. 

A review is given of certain tridiagonal N-dimensional non-Hermitian J-parametric real-matrix quantum Hamiltonians H^(N). The domains D^(N) of reality of their spectra of energies are studied, with particular attention paid to their exceptional-point boundaries ∂D^(N). The strongest admissible couplings are specified in closed form for all N. 

Many combinatorial and arithmetical properties have been studied for infinite words ub associated with β-integers. Here, new results describing return words and recurrence function for a special case of ub will be presented. The methods used here can be applied to more general infinite words, but the description then becomes rather technical. 

In this talk I present the results from my paper Exact solvability of superintegrable Benenti systems, J. Math. Phys. 48 (2007), 052114. 

We study the selfmatching properties of Beatty sequences, in particular of the graph of the function jβ against j for every quadratic unit β ∈ (0, 1). We show that translation in the argument by an element G_i of a generalized Fibonacci sequence almost always causes the translation of the value of the function by G_i–1. More precisely, for fixed i ∈ N, we have β(j+G_i)=βj+G_i–1, where j ∈ U_i. We determine the set U_i of mismatches and show that it has a low frequency, namely βⁱ.

The adjoint representations of several small dimensional Lie algebras  on their universal enveloping algebras  are explicitly decomposed. It is shown that commutants of raising operators are generated as polynomials in several basic elements. The explicit form of these elements is given and the general method for obtaining  these elements is described. 

We have been studying the asymptotic energy distribution of the algebraic part of the spectrum of the one-dimensional sextic anharmonic oscillator. We review some (both old and recent) results on the multiparameter spectral problem and show that our problem ranks among the degenerate cases of Heine-Stieltjes spectral problem, and we derive the density of the corresponding probability measure. 

Pseudo-Hermitian quantum theories are those in which the Hamiltonian H satisfies H^†= ηHη^–1, where  η ≡ e^–Q is a positive-definite Hermitian operator, rather than the usual H^†= H. In the operator formulation of such theories the standard Hilbert-space metric must be modified by the inclusion of η in order to ensure their probabilistic interpretation. With possible generalizations to quantum field theory in mind, it is important to ask how the functional integral formalism for pseudo-Hermitian theories differs from that of standard theories. It turns out that here Q plays quite a different role, serving primarily to implement a canonical transformation of the variables. It does not appear explicitly in the expression for the vacuum generating functional. Instead, the relation to the Hermitian theory is encoded via the dependence of Z on the external source j(t). These points are illustrated and amplified in various versions of the Swanson model, a non-Hermitian transform of the simple harmonic oscillator. 

We present a general procedure by which solvable non-central potentials can be obtained in 2 and 3 dimensions by the separation of the angular and radial variables. The method is applied to generate solvable non-central PT-symmetric potentials in polar coordinates. General considerations are presented concerning the PT transformation properties of the eigenfunctions, their pseudo-norm and the nature of the energy eigenvalues. It is shown that within the present framework the spontaneous breakdown of PT symmetry can be implemented only in two dimensions. 

We briefly explain some simple arguments based on pseudo Hermiticity, supersymmetry and PT-symmetry which explain the reality of the spectrum of some non-Hermitian Hamiltonians. Subsequently we employ PT-symmetry as a guiding principle to construct deformations of some integrable systems, the Calogero-Moser-Sutherland model and the Korteweg deVries equation. Some properties of these models are discussed. 

Some particular properties of the parametric dependence of eigenvalues with emphasis on their complexification are discussed. The non-diagonalisability of PT-symmetric matrix Hamiltonians in exceptional points is compared with level-crossing prohibition of Hermitian systems. For non-matrix Hamiltonians, the different way of complexification between Klein-Gordon and Dirac Hamiltonians is demonstrated. 

We consider a quantum waveguide with a small PT-symmetric perturbation described by a potential. We study the spectrum of such a system and show that the perturbation can produce eigenvalues near the threshold of the continuous spectrum. 

The Classical Newton-Lagrange equations of motion represent the fundamental physical law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses a description of classical dynamics and presents some irresponsible speculations about its quantization by introducing a certain canonical two-form Ω. By its construction Ω embodies kinetic energy and forces acting within the system (not their potential). A new type of variational principle employing differential two-form Ω is introduced. Variation is performed over “umbilical surfaces“ instead of system histories. It provides correct Newton-Lagrange equations of motion. The quantization is inspired by the Feynman path integral approach. The quintessence is to rearrange it into an “umbilical world-sheet“ functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, the new approach reduces to the standard quantum mechanics. As an example, Quantum Mechanics with friction is analyzed in detail. 

A three by three integer matrix M is said to have the MEME Property if  MEM^T=±E, where E= . We characterize such matrices in terms of their spectra.

We present some basic features of pseudo-hermitian quantum mechanics and illustrate the use of pseudo-hermitian Hamiltonians in a description of physical systems. 

 brief overview is given of  recent results on the spectral properties of spherically symmetric MHD α²-dynamos. In particular, the spectra of sphere-confined fluid or plasma configurations with physically realistic boundary conditions (BCs) (surrounding vacuum) and with idealized BCs (super-conducting surrounding) are discussed. The subjects comprise third-order branch points of the spectrum, self-adjointness of the dynamo operator in a Krein space as well as the resonant unfolding of diabolical points. It is sketched how certain classes of dynamos with a strongly localized a-profile embedded in a conducting surrounding can be mode decoupled by a diagonalization of the dynamo operator matrix. A mapping of the dynamo eigenvalue problem to that of a quantum mechanical Hamiltonian with energy dependent potential is used to obtain qualitative information about the spectral behavior. Links to supersymmetric Quantum Mechanics and to the Dirac equation are indicated.