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Abstract. In this paper, we first provide an explicit description of all holomorphic discs ("disc instantons") attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute Fukaya-Oh-Ohta-Ono's (FOOO's) obstruction (co)chains and the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. In particular specializing to the formal parameter T 2π = e −1 , our computation verifies the folklore that FOOO's obstruction (co)chains correspond to the Landau-Ginzburg superpotentials under the mirror symmetry correspondence, and also proves the prediction made by K. Hori about the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. The latter states that the Floer cohomology (for the parameter value T 2π = e −1 ) of all the fibers vanish except at a finite number, the Euler characteristic of the toric manifold, of base points in the momentum polytope that are critical points of the superpotential of the Landau-Ginzburg mirror to the toric manifold. In the latter cases, we also prove that the Floer cohomology of the corresponding fiber is isomorphic to its singular cohomology.We also introduce a restricted version of the Floer cohomology of Lagrangian submanifolds, which is a priori more flexible to define in general, and which we call the adapted Floer cohomology. We then prove that the adapted Floer cohomology of any non-singular torus fiber of Fano toric manifolds is well-defined, invariant under the Hamiltonian isotopy and isomorphic to the Bott-Morse Floer cohomology of the fiber.

This is a continuation of part I in the series of the papers on Lagrangian Floer theory on toric manifolds. Using the deformations of Floer cohomology by the ambient cycles, which we call bulk deformations, we find a continuum of nondisplaceable Lagrangian fibers on some compact toric manifolds. We also provide a method of finding all fibers with non-vanishing Floer cohomology with bulk deformations in arbitrary compact toric manifolds, which we call bulk-balanced Lagrangian fibers.

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In this paper, we first construct multi-lump (nonlinear) bound states of the nonlinear Schrδdinger equation rot 2for sufficiently small h > 0, in which sense we call them "semiclassical bound states." We assume that 1 ^ /? < oo for n = 1, 2 and 1 ^ p < 1 + 4/(n -2) for n ^ 3, and that V is in the class (V) a in the sense of Kato for some a. For any finite collection {x l9 . . . , X N } of nondegenerate critical points of V, we construct a solution of the form e~i Et/h v(x) for E < α, where v is real and it is a small perturbation of a sum of one-lump solutions concentrated near x l9 ...,x N respectively. The concentration gets stronger as ft->0. And we also prove these solutions are positive, and unstable with respect to perturbations of initial conditions for possibly smaller h > 0. Indeed, for each such collection of critical points we construct 2 N ~ 1 distinct unstable bound states which may have nodes in general, and the above positive bound state is just one of them.