= Solution
In the history basis, $M(s)=(1-s)D+sE$ is a Hermitian tridiagonal matrix. Its first diagonal entry is $s/2$, its interior entries are $1$, its last entry is $1-s/2$, and each off-diagonal entry is $-s/2$.
The <Gershgorin disc theorem> confines the first row's <eigenvalue> to $[0,s]$, the interior rows to $[1-s,1+s]$, and the last row to $[1-s,1]$. For $0\leq s\leq1/3$, the first disc is disjoint from all the others and therefore contains exactly one <eigenvalue>. Since $M(s)$ is <positive semidefinite>, that eigenvalue is the <ground state> energy. All other <eigenvalues> are at least $1-s$, while the lowest is at most $s$. Hence the <Gershgorin gap bound for an adiabatic history path> gives
$$
\boxed{\Delta(M(s))\geq1-2s\geq\frac13
\qquad(0\leq s\leq1/3).}
$$
This argument also covers $s=0$, where the first disc is the isolated point zero.
Back to article page
