P.P. Summary

$ \def \EN {\quad \mbox{and} \quad} \def \OF {\quad \mbox{or} \quad} \def \slechts {\quad \Longleftrightarrow \quad} $ The coefficients of a tri-diagonal matrix, when associated with a 1-D uniform mesh, exhibit Persistent Properties. By definition, such P.P. are independent of the grid-spacing, while using a mesh refinement or coarsening procedure.
A (non exhaustive and sometimes redundant) list of Persistent Properties has been produced below. If appropriate, the coefficients associated with the coarser grid are indicated with a prime accent $'$. \begin{eqnarray*} & a = 0 \\ & b = 0 \\ & \\ & a > 0 \\ & b > 0 \\ & \\ & a = 0 \EN b = 0 \\ & a = 1/2 \EN b = 1/2 \\ & a = 0 \EN b = 1 \\ & a = 1 \EN b = 0 \\ & \\ & a' + b' < a + b < 1 \qquad (*) \\ & a' + b' = a + b = 1 \qquad (*) \\ & a' + b' > a + b > 1 \qquad (*) \\ & \\ & a < b \OF a'/b' < a/b < 1 \\ & a = b \OF a'/b' = a/b = 1 \\ & a > b \OF a'/b' > a/b > 1 \\ & \\ & a'.b' = a.b = 0 \\ & 1 - 4.a.b > 0 \OF a'.b' < a.b < 1/4 \\ & 1 - 4.a.b = 0 \OF a'.b' = a.b = 1/4 \\ & 1 - 4.a.b < 0 \OF a'.b' > a.b > 1/4 \\ & a'.b' = a.b = 1 \\ & \\ & a'.b' = \frac{3}{2} + \frac{\sqrt{5}}{2} \slechts a .b = \frac{3}{2} - \frac{\sqrt{5}}{2} \\ & a'.b' = \frac{3}{2} - \frac{\sqrt{5}}{2} \slechts a .b = \frac{3}{2} + \frac{\sqrt{5}}{2} \\ \end{eqnarray*} Note (*). We didn't actually carry out a complete proof for these statements. Here comes: $$ a' + b'= \frac{a^2 + b^2}{1 - 2.a.b} < a + b \slechts a^2 + b^2 - (a + b)(1 - 2.a.b) < 0 \slechts $$ $$ a^2 + b^2 + 2.a^2.b + 2.a.b^2 - a - b < 0 \slechts (a + b - 1)(a + b + 2.a.b) < 0 $$ $$ \slechts a + b < 1 \qquad \mbox{because} \quad a + b + 2.a.b > 0 $$ Then replace $<$ by $=$ and $>$ and repeat the sequence of arguments.