By Batic D., Schmid H., Winklmeier M.
During this paper we learn for a given azimuthal quantum quantity ok the eigenvalues ofthe Chandrasekhar-Page angular equation with admire to the parameters mªamand nªav, the place a is the angular momentum consistent with unit mass of a black gap, m isthe leisure mass of the Dirac particle and v is the strength of the particle (as measuredat infinity). For this function, a self-adjoint holomorphic operator relatives Ask ;m ,ndassociated to this eigenvalue challenge is taken into account. initially we turn out that for fixedkPR\ s−12 , 12 d the spectrum of Ask ;m ,nd is discrete and that its eigenvalues dependanalytically on sm ,ndPC2. in addition, it will likely be proven that the eigenvalues satisfya first order partial differential equation with recognize to m and n, whose characteristicequations could be lowered to a Painlevé III equation. additionally, we derive apower sequence enlargement for the eigenvalues by way of n −m and n +m, and we givea recurrence relation for his or her coefficients. additional, will probably be proved that for fixedsm ,ndPC2 the eigenvalues of Ask ;m ,nd are the zeros of a holomorphic functionality Qwhich is outlined via a comparatively easy restrict formulation. eventually, we speak about the problemif there exists a closed expression for the eigenvalues of the Chandrasekhar-Page angular equation.
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Additional resources for On the eigenvalues of the Chandrasekhar-Page angular equation
46, 012504 (2005) The Chandrasekhar–Page angular equation TABLE I. The coefficients cm,n, 0 ഛ m + n ഛ 8, of the power series expansion (38) in the case = 2 and j = 1. 597 64 listed in Suffern et al. (1983, Table II). In order to test the reliability of our numerical result, we can use the statement of Lemma 3. That means, we approximate ⌰͑͒ defined in (25) by the second component ⌰n͑͒ of dn͑͒ for n = 8, and we compare ˜ ͒ and ⌰ ͑ˆ ͒ with the theoretical result ⌰͑ ͒ = 0. 511 64e − 04, our result seems to be more trustworthy.
Now, if we define v͑x͒ ª −P͑n␣,␤͒͑x͒, x ͑−1 , 1͒, then (A5) yields ±n u͑x͒ = ͑1 + x͒ = d ͑␣,␤͒ ␣+␤+n+1 ͑␣+1,␤+1͒ Pn + ␤ P͑n␣,␤͒ = + ␤ P͑n␣,␤͒ ͑1 + x͒Pn−1 2 dx ␣+␤+n+1 ͑␣+1,␤͒ ͓͑␤ + n͒Pn−1 + nP͑n␣+1,␤͔͒ + ␤ P͑n␣,␤͒ ␣ + ␤ + 2n + 1 = ͑␣ + ␤ + n + 1͒P͑n␣+1,␤͒ − ͑␣ + n + 1͒P͑n␣,␤͒ = ͑␤ + n͒P͑n␣+1,␤−1͒ = ͉±n ͉P͑n␣+1,␤−1͒ , where we applied the differentiation formulas and contiguous relations for Jacobi polynomials, see Magnus et al. (1966, Sec. 2). Hence, u͑x͒ = ± P͑n␣+1,␤−1͒͑x͒, x ͑−1 , 1͒, and since u satisfies the first condition in (A4), the numbers ±n are in fact eigenvalues of A͑ ; 0 , 0͒.
Anal. Geom. 3, 375–384 (2000). 210.
On the eigenvalues of the Chandrasekhar-Page angular equation by Batic D., Schmid H., Winklmeier M.