Subdominant Solutions Research Articles

Representation of solutions to delayed differential equations with a single delay by dominant and subdominant solutions

The representation and behavior is investigated of the solutions to an equation with a single constant delay ẋ(t)=−c(t)x(t−τ), t≥t0≥0 with c a positive function provided that it has a positive solution. A scheme is suggested for determining the representation of solutions defined by arbitrary initial function if dominant and subdominant solutions are known. Applications are given to equations with constant and perturbed coefficients with open problems included for future investigation.

ODE/IM correspondence for affine Lie algebras: a numerical approach

We study numerically the ODE/IM correspondence for untwisted affine Lie algebras associated with simple Lie algebras including exceptional type. We consider the linear problem obtained from the massless limit of that of the modified affine Toda field equation. We found that the Q-functions in integrable models are expressed as the inner product of the solution of the dual linear problem and the subdominant solution of the linear problem. Using Cheng’s algorithm to obtain the solution of the linear problem, we can determine efficiently the zeros of the Q-function, which is known to provide the solutions of the Bethe ansatz equations (BAEs). We calculate the zeros numerically, which are shown to agree with the results from the non-linear integral equations (NLIEs) for simply-laced affine Lie algebras including the exceptional type. By the folding procedure of the Dynkin diagrams of simply-laced Lie algebras, we also find the correspondence for the linear problem of the non-simply-laced affine Lie algebras.

Doubly infinite Jacobi matrices revisited: Resolvent and spectral measure

We study the resolvent and spectral measure of certain doubly infinite Jacobi matrices via asymptotic solutions of two-sided difference equations. By finding the minimal (or subdominant) solutions or calculating the continued fractions for the difference equations, we derive explicit formulas for the matrix entries of resolvent of doubly infinite Jacobi matrices corresponding to Lommel polynomials, associated ultraspherical polynomials, and Al-Salam–Ismail polynomials. The spectral measures are then obtained by inverting Stieltjes transformations.

Dominant and subdominant positive solutions to generalized Dickman equation

The paper considers a generalized Dickman equationtx˙(t)=−∑i=1saix(t−τi)for t → ∞ where s∈N,ai > 0, τi > 0, i=1,…,s and ∑i=1sai=1. It is proved that there are two mutually disjoint sets of positive decreasing solutions such that, for every two solutions from different sets, the limit of their ratio for t → ∞ equals 0 or ∞. The asymptotic behavior of such solutions is derived and a structure formula utilizing such solutions and describing all the solutions of a given equation is discussed. In addition, a criterion is proved giving sufficient conditions for initial functions to generate solutions falling into the first or the second set. Illustrative examples are given. Some open problems are suggested to be solved.

Positive solutions to generalized Dickman equation

The paper investigates large-time behaviour of positive solutions to a generalized Dickman equation. The asymptotic behaviour of dominant and subdominant positive solutions is analysed and a structure formula describing behaviour of all solutions is proved. A criterion is also given for sufficient conditions on initial functions to generate positive solutions with prescribed asymptotic behaviour with values of their weighted limits computed.

Exact asymptotics of positive solutions to Dickman equation

The paper considers the Dickman equation \begin{document}$\dot x (t)=-\frac{1}{t}\,x(t-1),$\end{document} for \begin{document} $t \to \infty $ \end{document} . The number theory uses what is called a Dickman (or Dickman -de Bruijn) function, which is the solution to this equation defined by an initial function \begin{document} $x(t)=1$ \end{document} if \begin{document} $0≤ t≤ 1$ \end{document} . The Dickman equation has two classes of asymptotically different positive solutions. The paper investigates their asymptotic behaviors in detail. A structure formula describing the asymptotic behavior of all solutions to the Dickman equation is given, an improvement of the well-known asymptotic behavior of the Dickman function, important in number theory, is derived and the problem of whether a given initial function defines dominant or subdominant solution is dealt with.

Splitting problem for WKB asymptotics in a nonresonant case and the reduction method for linear systems

As applied to the problem of asymptotic integration of linear systems of ordinary differential equations, we propose a reduction of order method that allows one to effectively construct solutions indistinguishable in the growth/decrease rate at infinity. In the case of a third-order equation, we use the developed approach to answer Bellman’s problem on splitting WKB asymptotics of subdominant solutions that decrease at the same rate. For a family of Wigner–von Neumann type potentials, the method allows one to formulate a selection rule for nonresonance values of the parameters (for which the corresponding second-order equation has a Jost solution).

Bethe Ansatz and the Spectral Theory of Affine Lie algebra–Valued Connections II: The Non Simply–Laced Case

We assess the ODE/IM correspondence for the quantum $\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\mathfrak{g}}^{(1)}$, and constructing the relevant $\Psi$-system among subdominant solutions. We then use the $\Psi$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.

Bethe Ansatz and the Spectral Theory of Affine Lie Algebra-Valued Connections I. The simply-laced Case

We study the ODE/IM correspondence for ODE associated to $\hat{\mathfrak g}$-valued connections, for a simply-laced Lie algebra $\mathfrak g$. We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called $\Psi$-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.

Pairing symmetry of the one-band Hubbard model in the paramagnetic weak-coupling limit: A numerical RPA study

We study the spin-fluctuation-mediated superconducting pairing gap in a weak-coupling approach to the Hubbard model for a two dimensional square lattice in the paramagnetic state. Performing a comprehensive theoretical study of the phase diagram as a function of filling, we find that the superconducting gap exhibits transitions from p-wave at very low electron fillings to d_{x^2-y^2}-wave symmetry close to half filling in agreement with previous reports. At intermediate filling levels, different gap symmetries appear as a consequence of the changes in the Fermi surface topology and the associated structure of the spin susceptibility. In particular, the vicinity of a van Hove singularity in the electronic structure close to the Fermi level has important consequences for the gap structure in favoring the otherwise sub-dominant triplet solution over the singlet d-wave solution. By solving the full gap equation, we find that the energetically favorable triplet solutions are chiral and break time reversal symmetry. Finally, we also calculate the detailed angular gap structure of the quasi-particle spectrum, and show how spin-fluctuation-mediated pairing leads to significant deviations from the first harmonics both in the singlet d_{x^2-y^2} gap as well as the chiral triplet gap solution.

On super-linear Emden–Fowler type differential equations

We study the second order Emden–Fowler type differential equation(a(t)|x′|αsgnx′)′+b(t)|x|βsgnx=0 in the super-linear case α<β. Using a Hölder-type inequality, we resolve the open problem on the possible coexistence on three possible types of nononscillatory solutions (subdominant, intermediate, and dominant solutions). Jointly with this, sufficient conditions for the existence of globally positive intermediate solutions are established. Some of our results are new also for the Emden–Fowler equation.

The semiclassical small-ℏ limit of loci of roots of subdominant solutions for polynomial potentials

In this paper, a description of the small-ℏ limit of loci of zeros of subdominant solutions for polynomial potentials is given called as fundamental solutions in our earlier papers. The considered potentials are those which provide us with the simple turning points only. Three types of Stokes graphs (SG’s) associated with the potentials are investigated – the general non-critical ones, the general critical ones but with only single internal Stokes line (SL), and the Stokes graphs corresponding to arbitrary multiple-well real even degree polynomial potentials with internal Stokes lines distributed on the real axis only. All these cases are considered in their both versions of the quantized and not quantized ℏ. In particular due to the fact that the small-ℏ limit is semiclassical it is shown that loci of roots of subdominant solutions in the cases considered are collected along Stokes lines. There are infinitely many roots of subdominant solutions on such lines escaping to infinity and a finite number of them on internal Stokes lines.

Stokes-multiplier expansion in an inhomogeneous differential equation with a small parameter

Accurate approximations to the solutions of a second-order inhomogeneous equation with a small parameter ϵ are derived using exponential asymptotics. The subdominant homogeneous solutions that are switched on by an inhomogeneous solution through a Stokes phenomenon are computed. The computation relies on a resurgence relation, and it provides the ϵ -dependent Stokes multiplier in the form of a power series. The ϵ -dependence of the Stokes multiplier is related to constants of integration that can be chosen arbitrarily in the WKB-type construction of the homogeneous solution. The equation under study governs the evolution of special solutions of the Boussinesq equations for rapidly rotating, strongly stratified fluids. In this context, the switching on of subdominant homogeneous solutions is interpreted as the generation of exponentially small inertia–gravity waves.

Subdominant positive solutions of the discrete equation Δu(k + n) = −p(k)u(k)

A delayed discrete equation Δu(k + n) = −p(k)u(k) with positive coefficient p is considered. Sufficient conditions with respect to p are formulated in order to guarantee the existence of positive solutions if k → ∞. As a tool of the proof of corresponding result, the method described in the author′s previous papers is used. Except for the fact of the existence of positive solutions, their upper estimation is given. The analysis shows that every positive solution of the indicated family of positive solutions tends to zero (if k → ∞) with the speednot smaller than the speed characterized by the function . A comparison with the known results is given and some open questions are discussed.

Positive Solutions of the Equation ẋ(t) = −c(t)x(t − τ) in the Critical Case

The equation ẋ(t)−c(t)x(t−τ) is considered in the critical case. For it, the asymptotic behavior of dominant and subdominant solutions is studied. A generalization is made and connections with known results are discussed.

Spectral Theory of Jacobi Matrices in $l^2 (\mathbb{Z})$ and the $su(1,1)$ Lie Algebra

The connection between orthogonal polynomials, continued fractions, difference equations, and self-adjoint Jacobi matrices acting in $l^2 (\mathbb{Z}^ + )$ and the extension of these connections to $l^2 (\mathbb{Z})$ are reviewed. This yields three different representations for the resolvent of the Jacobi matrix: an integral representation in terms of orthogonal polynomials, a representation in terms of continued fractions, and a representation in terms of the subdominant (or minimal) solution to the associated difference equation. This latter representation is given explicitly in terms of hypergeometric functions for the cases of associated Meixner, Meixner–Pollaczek, and Laguerre polynomials. It is also shown that it is precisely these cases that occur in the unitary irreducible representations of $su(1,1)$ for the resolvent of a real linear combination of the generators of the algebra.

Exceptional 𝑞-Askey-Wilson polynomials and continued fractions

Two linearly independent solutions of the three-term recurrence relation for the q q -Askey-Wilson polynomials are obtained for the special cases a b c d = q m , m = 1 , 2 , … abcd = {q^m},m = 1,2, \ldots .By obtaining the subdominant solution and employing Pincherle’s theorem, the associated continued fractions and properties of the corresponding weight functions are derived. The cases a b c d = q or q 2 abcd = q\;{\text {or}}\;{q^2} are exceptional. They differ from the cases considered by Askey and Wilson [1 ] and are limits of a family of associated cases considered by Ismail and Rahman [5].

Analytic treatment of the αx2+βx4 oscillator

We carry out an investigation of the Schrödinger equation with the αx2+βx4 interaction for 0&amp;lt;Re( x)&amp;lt;∞, and β=‖β‖ exp(i(π+λ)),−π&amp;lt;λ&amp;lt;π. In the sectors 0≤arg( x)&amp;lt;π/3−λ/6, 0≤λ&amp;lt;π, and −λ/6&amp;lt;arg( x)&amp;lt;π/3−λ/6, −π&amp;lt;λ&amp;lt;0, a subdominant solution is constructed essentially in terms of the Laplace transform of a function f (s), 0≤s&amp;lt;∞, which is expressible as a converging power series in s. The solution, y( x), thus obtained, is compared and contrasted with the one valid in the usual case where Re( x)≤0 is included and it is argued that y( x) may not be accessible to perturbative approaches.

Non-perturbative results for the ω2x2 + λx4 oscillator

For the ω 2 x 2 + λx 4, λ >0, oscillator we prove the existence of a basic set of solutions, the set consisting of a dominant and a subdominant solution. This result does not seem to be obtainable by means of some recently developed perturbation methods.

On the subdominant solution of a second-order differential equation with a complex parameter

On the subdominant solution of a second-order differential equation with a complex parameter