2023-2024

Motivated by a recent study of Bessel operators in connection with a refinement of Hardy’s inequality involving 1/sin²(x) on the finite interval (0,π), we discuss domain properties of certain Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in L²((a,b);dx) associated with differential expressions of the form
and
ω_sa =-^d2⁄_dx²+^(s_a2-(1/4))⁄_(x-a)², s_a∈R, x∈(a,b),
τ_sa,sb=-^d2⁄_dx²+^(s_a2-(1/4))⁄_(x-a)²+^(s_b2-(1/4))⁄_(x-b)²+q(x), x∈(a,b),
s_a, s_b∈[0,∞), q∈L^∞((a,b);dx), q real-valued a.e. on (a,b),
where (a,b)⊂R is a bounded interval.
As an explicit illustration, we describe the bewildering variety of characterizations of the Friedrichs extension of the minimal operator corresponding to τ_sa,sb. Time permitting, we will illustrate the connection between the Sturm–Liouville problem τ_sa,sb,q=0u = zu and the confluent Heun differential equation.
This talk is based on joint work with Fritz Gesztesy and Michael M. H. Pang.

Rational approximations of functions offer a rich mathematical theory. Touching subjects such as orthogonal polynomials, potential theory and of course differential equations. In this talk we will discuss a specific type of rational approximant, factorial expansions. In recent work with O. Costin and R. Costin we have developed a theory of dyadic expansions which improve the domain and rate of convergence when compared to the classical methods found in the literature. These results provide a general method for producing rational approximations of Borel summable series with locally integrable branch points. Surprisingly, these expansions capture the asymptoticly important Stokes phenomena. Additionally, we find applications in operator theory on Hilbert spaces providing new representations for (bounded and unbounded) positive and self-adjoint operators in terms of the semigroups and unitary groups they generate. Finally, as an example of an important application we discuss representing the tritronquée solutions of Painlevé’s first equation.

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