Log-lightning computation of capacity and Green's function

See Video Abstract (click the "Video Abstract" button next to the "PDF" button) A basic measure of the size of a set E in the complex plane is the logarithmic capacity cap(E). Capacities are known analytically for a few simple shapes like ellipses, but in most cases they must be computed numerically. We explore their computation by the new "log-lightning'' method based on reciprocal-log approximations in the complex plane. For a sequence of 16 examples involving both connected and disconnected sets E, we compute capacities to 8–15 digits of accuracy at great speed in MATLAB. The convergence is almost-exponential with respect to the number of reciprocal-log poles employed, so it should be possible to compute many more digits if desired in Maple or another extended-precision environment. This is the first systematic exploration of applications of the log-lightning method, which opens up the possibility of solving Laplace problems with an efficiency not achievable by previous methods. The method computes not just the capacity, but also the Green's function and its harmonic conjugate. It also extends to "domains of negative measure" and other Riemann surfaces.

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On the spectra of SchroÌˆdinger and Jacobi operators with complex-valued quasi-periodic algebro-geometric coefficients

10.32469/10355/4136 ◽

2005 ◽

Author(s):

◽

Vladimir Batchenko

Keyword(s):

Green’S Function ◽

Green's Function ◽

Complex Plane ◽

Mean Value ◽

Qualitative Behavior ◽

Conditional Stability ◽

One Dimensional ◽

Jacobi Operators ◽

The Mean ◽

Complex Valued

In this thesis we characterize the spectrum of one-dimensional SchroÌˆdinger operators. H = -d2/dx2+V in L2(R; dx) with quasi-periodic complex-valued algebro geometric, potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green's function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well. These results extend to the Lp(R; dx)-setting for p 2 [1,1). In addition, we apply these techniques to the discrete case and characterize the spectrum of one-dimensional Jacobi operators H = aS+ + a-S- b in 2(Z) assuming a, b are complex-valued quasi-periodic algebro-geometric coefficients. In analogy to the case of SchroÌˆdinger operators, we prove that the spectrum of H coincides with the conditional stability set of H and can also explicitly be described in terms of the mean value of the Green's function of H. The qualitative behavior of the spectrum of H in the complex plane is similar to the SchroÌˆdinger case: the spectrum consists of finitely many bounded simple analytic arcs in the complex plane which may exhibit crossings as well as confluences.

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Some Theorems on Open Riemann Surfaces

Nagoya Mathematical Journal ◽

10.1017/s0027763000012277 ◽

1951 ◽

Vol 3 ◽

pp. 141-145 ◽

Cited By ~ 2

Author(s):

Masatsugu Tsuji

Keyword(s):

Riemann Surface ◽

Harmonic Function ◽

Green’S Function ◽

Green's Function ◽

Riemann Surfaces ◽

Dirichlet Integral ◽

Open Riemann Surface ◽

Surface Spread

Let F be an open Riemann surface spread over the z-plane. We say that F is of positive or null boundary, according as there exists a Green’s function on F or not, Let u(z) be a harmonic function on Fand be its Dirichlet integral As R. Nevanlinna proved, if F is of null boundary, there exists no one-valued non-constant harmonic function on F5 whose Dirichlet integral is finite, This Nevanlinna’s theorem was proved very simply by Kuroda.

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