It is well known that the local decay estimates (2) are useful in studying nonlinear Schrödinger equations (see [8, §XI.13], [11]). On the other hand little seems to be known when one replaces the free operator HQ by more general Hamiltonians (4) H = -A + V(x), even when the potential V is in C^°(R). Obviously, one has to assume that H has no bound states for an estimate like (2) to it M hold f...

We are concerned with the almost automorphic solutions to the second-order elliptic differential equations of type ü(s) + 2Bu̇(s) + Au(s) = f(s) (∗), where A, B are densely defined closed linear operators acting in a Hilbert space H and f : R 7→ H is a vector-valued almost automorphic function. Using invariant subspaces, it will be shown that under appropriate assumptions; every solution to (∗) ...

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We show that an invariant surface allows to construct the Jacobi vector field along a geodesic and construct the formula for the normal component of the Jacobi field. If a geodesic is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic is hyperbolic), then we can construct a fundamental solution of the the Jacobi-Hill equation ü = −K(u)u....

We study the existence of homoclinic solutions for the second order Hamiltonian system ü+Vu(t, u) = f(t). Let V (t, u) = −K(t, u)+W (t, u) ∈ C1(R×Rn,R) be T -periodic in t, where K is a quadratic growth function and W may be asymptotically quadratic or super-quadratic at infinity. One homoclinic solution is obtained as a limit of solutions of a sequence of periodic second order differential equ...