L = 4;
H[t_] := SparseArray[
Flatten[
{
Table[{2 j - 1, 2 j} -> t, {j, 1, L}],
Table[{2 j, 2 j - 1} -> t, {j, 1, L}],
Table[{2 j, 2 j + 1} -> 1.0, {j, 1, L - 1}],
Table[{2 j + 1, 2 j} -> 1.0, {j, 1, L - 1}]
}
]
];
H[t] // MatrixForm
ev = Eigenvalues[H[t]]
Plot[ev, {t, -3.0, 3.0}]