Yoshitsugu Takei

The Bender-Wu Analysis and the Voros Theory

In their pioneering work [BW], Bender and Wu presented the secular equation for anharmonic oscillators ([BW], (F.56)~(F.58) in p. 1257), which was supported by their ingenious WKB analysis. As we shall discuss in our forthcoming article, we can validate their conjecture by Voros’ epoch-making article [V]. At the same time several ideas contained in [BW] can be effectively employed to understand the Voros theory from the viewpoint of (micro)differential operators, and this is what we report here.

On the exact WKB analysis of operators admitting infinitely many phases

Abstract To analyze differential operators whose WKB solutions admit infinitely many phases, we introduce a class of differential operators of WKB type and analyze their exact WKB theoretic structure near their turning points. Our analysis makes full use of techniques and ideas in microlocal analysis; we use a quantized contact transformation to construct a WKB solution of a differential equation of WKB type, and we use a Spath-type division theorem for a differential operator of WKB type to study its structure near turning points. As an application, we show a connection formula for WKB solutions near a simple turning point.

On the exact steepest descent method: A new method for the description of Stokes curves

In order to determine the region where the Borel sum of a Wentzel–Kramers–Brillouin (WKB) solution ψ of an mth order (m⩾3) ordinary differential equation Pψ=0 with polynomial coefficients is well defined, we apply the steepest descent method to the Laplace integral of a WKB solution ψ of Pψ=0, with P denoting the Laplace transform of P. As a counterpart of the connection formula for ψ, we introduce a new notion of the exact steepest descent path, which is the union of bifurcated steepest descent paths. Both theoretical and computer-assisted experimental studies are given to show the importance of this notion.

Mathematical background for a method on quotient signal decomposition

The blind source separation problem is discussed in this article. Focusing on the assumption of independency of the sources in the time-frequency domain, we present a mathematical formulation for the estimation problem of the number of sources. The proposed method uses the quotient of complex valued time-frequency information of only two observed signals to detect the number of sources. No fewer number of observed signals than the detected number of sources is needed to separate sources. The assumption on sources is quite general independence in the time-frequency plane, which is different from that of independent component analysis. We propose algorithms with feedback and give numerical simulations to show the method works well even for noisy case.

On the WKB-theoretic structure of a Schrödinger operator with a merging pair of a simple pole and a simple turning point

The principal aim of this paper is to form a basis for the exact WKB analysis of a Schrödinger equation (0.1) d 2 dx 2 − η 2 Q(x, η) ψ = 0 (η : a large parameter) with one simple turning point and with one simple pole in the potential Q. As [Ko1] and [Ko2] emphasize, the Borel transform of a WKB solution of (0.1) displays, near the simple pole singularity, behavior similar to that near a simple turning point. Hence it is natural to expect that such an equation plays an important role in the exact WKB analysis in the large. Such an expectation has recently been enhanced by the discovery ([Ko4]) that the Voros coefficient of a WKB solution of (0.1) with (0.2) Q = 1 4 + α x + η −2 γ x 2 (α, γ : fixed complex numbers) can be explicitly written down with the help of the Bernoulli numbers. The potential Q given by (0.2) will play an important role in Section 2; the Schrödinger equation with the potential Q of the form (0.2), that is, the Whittaker equation with a large parameter η, gives us a WKB theoretic canonical form of a Schrödinger equation with one simple turning point and with one simple pole in its potential. We note that the parameter α contained in the Whittaker equation in Section 2 is an infinite series α(η) = k≥0 α k η −k (α k : a constant), and we call such an equation the ∞-Whittaker equation when we want to emphasize that α is not a genuine constant but an infinite series as above. In order to make a semi-global study of a Schrödinger equation with one simple turning point and with a simple pole in its potential, we let the simple pole singular point merge with the turning point and observe 2 what kind of equation appears. For example, what if we let α tend to 0 in (0.2) with γ being kept intact? Interestingly enough, the resulting equation is what we call a ghost equation ([Ko3]); we have been worrying where we should place the class of ghost equations in regard to the whole WKB analysis. A ghost equation has no turning point by its definition (cf. Remark 1.1 in Section 1); still a WKB solution of a ghost equation displays singularity similar to that which a WKB solution …

Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations

For a higher order linear ordinary differential operator P, its Stokes curve bifurcates in general when it hits another turning point of P. This phenomenon is most neatly understandable by taking into account Stokes curves emanating from virtual turning points, together with those from ordinary turning points. This understanding of the bifurcation of a Stokes curve plays an important role in resolving a paradox recently found in a system of linear differential equations associated with the fourth Painleve equation.

Virtual turning points

The discovery of a virtual turning point truly is a breakthrough in WKB analysis of higher order differential equations. This monograph expounds the core part of its theory together with its application to the analysis of higher order Painleve equations of the Noumi Yamada type and to the analysis of non-adiabatic transition probability problems in three levels. As M.V. Fedoryuk once lamented, global asymptotic analysis of higher order differential equations had been thought to be impossible to construct. In 1982, however, H.L. Berk, W.M. Nevins, and K.V. Roberts published a remarkable paper in the Journal of Mathematical Physics indicating that the traditional Stokes geometry cannot globally describe the Stokes phenomena of solutions of higher order equations; a new Stokes curve is necessary

On the Voros Coefficient for the Whittaker Equation with a Large Parameter \-- Some Progress around Sato's Conjecture in Exact WKB Analysis

Generalizing Sato’s conjecture for the Weber equation in exact WKB analysis, we explicitly determine the Voros coefficient of the Whittaker equation with a large parameter. By using our results we also compute alien derivatives of WKB solutions of the Whittaker equation at the so-called fixed singular points of their Borel transform. 2010 Mathematics Subject Classification: 34M30, 34M37, 34M55, 34M60.

On an exact WKB approach to Ablowitz-Segur's connection problem for the second Painlevé equation

We discuss Ablowitz-Segurs connection problem for the second Painleve equation from the viewpoint of WKB analysis of Painleve transcendents with a large parameter. The formula they first discovered is rederived from a suitable combination of connection formulas for the first Painleve equation.

Mathematical view of a blind source separation on a time frequency space

Abstract To treat the blind source separation problems, in many cases, either statistical independence or statistical orthogonality (uncorrelation) on the sources has been assumed. Napoletani–Berenstein–Krishnaprasad treated the problem under the linear independence of the windowed Fourier transforms of sources and the continuity of density functions defined statistically. In this paper, another independence of the windowed Fourier transforms of sources in a time–frequency domain is proposed without assuming any statistical conditions. This paper is a summary of the authors’ submitted papers.