Abstract. The Witten Laplacian corresponding to a Morse function on the circle is studied using methods of complex WKB and resurgent analysis. It is shown that under certain assumptions the low-lying eigenvalues of the Witten Laplacian are resurgent.

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Abstract. The Witten Laplacian in one dimension is studied further by methods of resurgent analysis in order to approach Fukaya's conjectures relating WKB asymptotics and disc instantons. We carry out explicit computations of exponential asymptotic expansions of exponentially small (i.e. Oðe Àc=h Þ, c > 0, h ! 0þ) eigenvalues and of corresponding eigenfunctions of the Witten Laplacian; a general algorithm as well as two examples are discussed.