We show some interesting properties of tridiagonal and pentadiagonal matrices in the weak coupling limits. In the former case of this limit the ground state wave function amplitudes are identical to the Taylor expansion coefficients of the exponential function e (− v e ) . With regards to transition rates a dip in the pentadiagonal case which is not present in the tridiagonal case is explained. An intimate connection between energy denominators and exponential behavior is demonstrated.

In previous works we considered schematic Hamiltonians represented by simplified matrices. We defined 2 transition operators and calculated transition strengths from the ground state to all exited states.In many cases the strengths decreased nearly exponentially with excitation energy. Now we do the reverse We start with the highest energy state and calculate the cascade of transitions until the ground states is reached.On a log plot we show the average transition strength as a function of the number of energy intervals that were crossed. We give an analytic proof of exponential behavior for transition strength in the weak coupling limit for the T 2 transition operator.

We discuss L=0 vs L=2 couplings, symmetries of the pairing interaction for neutrons and protons, and the Jmax interaction. We show that certain schematic interactions yield exponentially decreasing transition strengths.We compare shell model calculations of B(E2)'s and quadrupole moments in the p-f shell with collective model results.

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