Compact Jacobi matrices: from Stieltjes to Krein and M(a,b)

In a note at the end of his paper {\it Recherches sur les fractions continues}, Stieltjes gave a necessary and sufficient condition when a continued fraction is represented by a meromorphic function. This result is related to the study of compact Jacobi matrices. We indicate how this notion was developped and used since Stieltjes, with special attention to the results by M. G. Krein. We also pay attention to the perturbation of a constant Jacobi matrix by a compact Jacobi matrix, work which basically started with Blumenthal in 1889 and which now is known as the theory for the class $M(a,b)$

Near-optimal perfectly matched layers for indefinite Helmholtz problems

A new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein's interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included.Comment: Accepted for publication in SIAM Review. To appear 201

The Jacobi matrices approach to Nevanlinna-Pick problems

A modification of the well-known step-by-step process for solving Nevanlinna-Pick problems in the class of \bR_0-functions gives rise to a linear pencil $H-\lambda J$, where $H$ and $J$ are Hermitian tridiagonal matrices. First, we show that $J$ is a positive operator. Then it is proved that the corresponding Nevanlinna-Pick problem has a unique solution iff the densely defined symmetric operator $J^{-1/2}HJ^{-1/2}$ is self-adjoint and some criteria for this operator to be self-adjoint are presented. Finally, by means of the operator technique, we obtain that multipoint diagonal Pad\'e approximants to a unique solution $\varphi$ of the Nevanlinna-Pick problem converge to $\varphi$ locally uniformly in \dC\setminus\dR. The proposed scheme extends the classical Jacobi matrix approach to moment problems and Pad\'e approximation for \bR_0-functions.Comment: 24 pages; Section 5 is modifed; some typos are correcte

Numerical Evidence that the Perturbation Expansion for a Non-Hermitian PT\mathcal{PT}-Symmetric Hamiltonian is Stieltjes

Recently, several studies of non-Hermitian Hamiltonians having $\mathcal{PT}$ symmetry have been conducted. Most striking about these complex Hamiltonians is how closely their properties resemble those of conventional Hermitian Hamiltonians. This paper presents further evidence of the similarity of these Hamiltonians to Hermitian Hamiltonians by examining the summation of the divergent weak-coupling perturbation series for the ground-state energy of the $\mathcal{PT}$-symmetric Hamiltonian $H=p^2+{1/4}x^2+i\lambda x^3$ recently studied by Bender and Dunne. For this purpose the first 193 (nonzero) coefficients of the Rayleigh-Schr\"odinger perturbation series in powers of $\lambda^2$ for the ground-state energy were calculated. Pad\'e-summation and Pad\'e-prediction techniques recently described by Weniger are applied to this perturbation series. The qualitative features of the results obtained in this way are indistinguishable from those obtained in the case of the perturbation series for the quartic anharmonic oscillator, which is known to be a Stieltjes series.Comment: 20 pages, 0 figure

Pseudo-factorials, elliptic functions, and continued fractions

This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a Dixonian and a Weierstrass function, which parametrize the Fermat cubic curve and are relative to a hexagonal lattice. A continued fraction expansion of the ordinary generating function of pseudo-factorials, first discovered empirically, is established here. This article also provides a characterization of the associated orthogonal polynomials, which appear to form a new family of "elliptic polynomials", as well as various other properties of pseudo-factorials, including a hexagonal lattice sum expression and elementary congruences.Comment: 24 pages; with correction of typos and minor revision. To appear in The Ramanujan Journa