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Register a new userUnique solvability of a coupling problem for entire functions
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Jonathan Eckhardt
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We establish the unique solvability of a coupling problem for entire functions which arises in inverse spectral theory for singular second order ordinary differential equations/two-dimensional first order systems and is also of relevance for the integration of certain nonlinear wave equations.
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To each nonzero sequence $s:= {s_{n}}_{n geq 0}$ of real numbers we associate the Hankel determinants $D_{n} = det mathcal{H}_{n}$ of the Hankel matrices $mathcal{H}_{n}:= (s_{i + j})_{i, j = 0}^{n}$, $n geq 0$, and the nonempty set $N_{s}:= {n geq 1 , | , D_{n-1} eq 0 }$. We also define the Hankel determinant polynomials $P_0:=1$, and $P_n$, $ngeq 1$ as the determinant of the Hankel matrix $mathcal H_n$ modified by replacing the last row by the monomials $1, x, ldots, x^n$. Clearly $P_n$ is a polynomial of degree at most $n$ and of degree $n$ if and only if $nin N_s $. Kronecker established in 1881 that if $N_s $ is finite then rank $mathcal{H}_{n} = r$ for each $n geq r-1$, where $r := max N_s $. By using an approach suggested by I.S.Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence ${t_n}_{ngeq 0}$ to be of the form $t_n=D_n$, $ngeq 0$ for a real sequence ${s_n}_{ngeq 0}$. This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial $ P_n $ satisfying deg$P_n = ngeq 1$ is preceded by a nonzero polynomial $P_{n-1}$ whose degree can be strictly less than $n-1$ and which has no common zeros with $ P_n $. As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that $D_0 > 0, ldots, D_{r-1} > 0 $ and $D_n=0$ for all $ngeq r$.
We show that the classical Hamburger moment problem can be included in the spectral theory of generalized indefinite strings. Namely, we introduce the class of Krein-Langer strings and show that there is a bijective correspondence between moment sequences and this class of generalized indefinite strings. This result can be viewed as a complement to the classical results of M. G. Krein on the connection between the Stieltjes moment problem and Krein-Stieltjes strings and I. S. Kac on the connection between the Hamburger moment problem and 2x2 canonical systems with Hamburger Hamiltonians.
We extend two theorems of Krein concerning entire functions of Cartwright class, and give applications for the Bernstein weighted approximation problem.
As a straightforward generalization and extension of our previous paper, J. Phys. A50 (2017) 215201 we study aspects of the quantum and classical dynamics of a $3$-body system with equal masses, each body with $d$ degrees of freedom, with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories (which are in the interaction plane) in the classical case are of this type. It corresponds to a three-dimensional quantum particle moving in a curved space with special $d$-dimension-independent metric in a certain $d$-dependent singular potential, while at $d=1$ it elegantly degenerates to a two-dimensional particle moving in flat space. It admits a description in terms of pure geometrical characteristics of the interaction triangle which is defined by the three relative distances. The kinetic energy of the system is $d$-independent, it has a hidden $sl(4,R)$ Lie (Poisson) algebra structure, alternatively, the hidden algebra $h^{(3)}$ typical for the $H_3$ Calogero model as in the $d=3$ case. We find an exactly-solvable three-body $S^3$-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential with singular terms. For both models an extra first order integral exists. It is shown that a straightforward generalization of the 3-body (rational) Calogero model to $d>1$ leads to two primitive quasi-exactly-solvable problems. The extension to the case of non-equal masses is straightforward and is briefly discussed.
If $f$ is an entire function and $a$ is a complex number, $a$ is said to be an asymptotic value of $f$ if there exists a path $gamma$ from $0$ to infinity such that $f(z) - a$ tends to $0$ as $z$ tends to infinity along $gamma$. The Denjoy--Carleman--Ahlfors Theorem asserts that if $f$ has $n$ distinct asymptotic values, then the rate of growth of $f$ is at least order $n/2$, mean type. A long-standing problem asks whether this conclusion holds for entire functions having $n$ distinct asymptotic (entire) functions, each of growth at most order $1/2$, minimal type. In this paper conditions on the function $f$ and associated asymptotic paths are obtained that are sufficient to guarantee that $f$ satisfies the conclusion of the Denjoy--Carleman--Ahlfors Theorem. In addition, for each positive integer $n$, an example is given of an entire function of order $n$ having $n$ distinct, prescribed asymptotic functions, each of order less than $1/2$.
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