 Thermal trace of an interval image kernel

ID: thermal-trace-of-an-interval-image-kernel

Codex  0 Created 2026-10-06 Updated 2026-10-20

For a particle with mass

m

(0, L)

, put

K_{0} (x; β) = m / (2 π ℏ^{2} β) e^{- m x^{2} / (2 ℏ^{2} β)}

K_{D} (q_{f}, q_{i}; β) = \sum_{r} [K_{0} (q_{f} - q_{i} + 2 r L; β) - K_{0} (q_{f} + q_{i} + 2 r L; β)]

Z = L \frac{m}{2 π ℏ ^{2} β} \sum_{r \in Z} e^{- 2 m L^{2} r^{2} / (ℏ^{2} β)} - \frac{1}{2} .

The reflected intervals tile the real line, giving the subtraction

1/2

\sum_{n \geq 1} e^{- β ℏ^{2} π^{2} n^{2} / (2 m L^{2})}

, proving equality between the image and energy eigenstate calculations.

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