  In this paper, we show that the pointwise finite-dimensional two-parameter
persistence module $\mathbb{HF}_*^{(\bullet,\bullet]}$, defined in terms of
interlevel filtered Floer homology, is rectangle-decomposable. This allows for
the definition of a barcode $\mathcal{B}_*^{(\bullet,\bullet]}$ consisting
solely of rectangles in $\mathbb{R}^2$ associated with
$\mathbb{HF}_*^{(\bullet,\bullet]}$. We observe that this rectangle barcode
contains information about Usher's boundary depth and spectral invariants
developed by Oh, Schwarz, and Viterbo. Moreover, we introduce a novel invariant
extracted from $\mathcal{B}_*^{(\bullet,\bullet]}$, which proves instrumental
in detecting periodic solutions of Hamiltonian systems. Additionally, we
establish relevant stability results, particularly concerning the bottleneck
distance and Hofer's distance.

Symplectic Geometry;Algebraic Topology