In a transition, if the density of final states (holeft(E_{n} ight)) is an increasing exponential, (holeft(E_{n} ight)=A

In a transition, if the density of final states \(ho\left(E_{n}\right)\) is an increasing exponential, \(ho\left(E_{n}\right)=A e^{\alpha E_{n}}\), \(\alpha>0\), what does the (physical) condition of a finite energy shift imply for the transition Hamiltonian matrix \(H_{1, i n}\), if \(E_{i}>0\) ?