On poles of the rational solution of the Toda equation of Painlevé-II type

On spectral asymptotic of quasi-exactly solvable quartic potential

“…Our strategy is to connect roots and coefficients of Yablonskii-Vorob'ev polynomials. Similar approach was used in [15]. Our results are sharper and we also derive for |z| > 3M.…”

Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour

“…Starting from the Painlevé-II equation (1-1), we would like to rescale p and y so that the zeros and poles of the rational solutions are (approximately) equally spaced. It is known that the maximum modulus of the zeros of P m (y) grows as m 2/3 [19]. This suggests the fact (which we will prove later) that the large-m asymptotic boundaries of the elliptic region T are fixed in the x-plane, where x = (m − 1 2 ) −2/3 y.…”

“…While c(x) has jump discontinuities across the three rays R − , R π/3 , and R −π/3 , Re(c(x)) extends to these rays from either side as a continuous harmonic function. To see this, one actually shows more by a direct calculation using (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20); namely for…”

“…• We characterize the boundary ∂T of the elliptic region T as explicitly as possible, expressing it as a level set of a function built from explicit elementary functions and the solution of a cubic equation (see (3)(4)(5)(6)(7), (3)(4)(5)(6)(7)(8), and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)).…”

“…The integration constant Π is determined as a function of x 0 in order that certain constant exponents that occur in the jump matrices of a model Riemann-Hilbert problem are purely imaginary, ensuring that the solution of the model problem is suitably bounded and that errors can be controlled. The precise conditions that determine Π we call Boutroux conditions, and they take the form (4-17), or equivalently, (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22). It turns out that Π is a complex-valued function of x 0 that is smooth (i.e., Re(Π) and Im(Π) are infinitely differentiable functions of Re(x 0 ) and Im(x 0 )) but nowhere analytic for x 0 ∈ T .…”