Calculating near-singular eigenvalues of the neutron transport operator with arbitrary order anisotropic scattering (Q991309)

A homogeneous one-speed neutron transport equation in plane geometry is considered \[ \mu\frac{\partial\psi(x,\mu)}{\partial x}+\Sigma_t\psi(x,\mu)= \frac{c\Sigma_t}{2}\sum_{l=0}^N(2l+1)f_lP_l(\mu) \int_{-1}^{+1}\psi(x,\mu')P_l(\mu')\,d\mu' \] where \(x\) is the spatial coordinate; \(\mu\) is the angular coordinate; \(\Sigma_t\) is the total macroscopic cross section; \(c\) is the number of secondary neutrons per collision; \(f_l\) are the scattering coefficients up to order \(N\) and \(P_l(\mu)\) is the Legandre polynomial of degree \(l\). The usual ansatz \[ \psi(x,\mu)= \exp\bigg(-\frac{x}{\nu}\bigg)\varphi_{\nu}(\mu) \] yields the corresponding characteristic equation \[ (\nu-\mu)\varphi_{\nu}(\mu)= \frac{c\Sigma_t}{2}\sum_{l=0}^N(2l+1)f_lP_l(\mu) \int_{-1}^{+1}\varphi_{\nu}(\mu')P_l(\mu')\,d\mu' \] where \(\nu\) is a parameter. In this paper is described the procedure to calculate discrete eigenvalues which are defined from the so-called dispersion relation \(\Lambda(\nu)=0\) with \[ \Lambda(\nu)=1-c\nu\sum_{l=0}^N(2l+1)f_lg_l(\nu)Q_l(\nu) \] where \(Q_l(\nu)\) denote the Legandre function of the second kind, \(g_l(\nu)\) denote the Chandrasekhar polynomials defined by the recurrence \[ (2l+1)\nu(1-cf_l)g_l(\nu)= (l+1)g_{l+1}(\nu)+lg_{l-1}(\nu), \quad l\geq 0 \] with \(g_0(\nu)=1.\) The authors found that there exist cases where there is a discrete eigenvalue located extremely close to the singular point at unity. The main part of this paper is devoted to the description how to calculate these so-called near-singular eigenvalues.