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• Maslov index for composite power-law potentials depends on the intensity parameters. • For the first Exton potential the Maslov index varies in the interval (− 1 , 0). • The Maslov index reveals crucial reliance on the sign of a potential parameter. • There are two non-crossing branches of the curve describing the Maslov index. Using the exact solution of the one-dimensional Schrödinger equation, we demonstrate that the Maslov index involved in the Bohr-Sommerfeld semiclassical quantization rule for the first Exton potential depends on the intensity parameters of the potential. As a function of the intensity of the potential term proportional to x − 3 / 2 , it varies within a rather wide range, from − 1 to 0, adopting − 1 / 6 when this term and its conjugate term ∼ x − 1 , are absent, which is the case when the Exton potential reduces to the inverse-square-root potential. [ABSTRACT FROM AUTHOR]

We modify the construction of knot Floer homology to produce a one-parameter family of homologies tHFK for knots in S 3 . These invariants can be used to give homomorphisms from the smooth concordance group C to Z , giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms. [ABSTRACT FROM AUTHOR]

We discuss a definition of the Maslov index for Lagrangian pairs on R 2 n based on spectral flow, and develop many of its salient properties. We provide two applications to illustrate how our approach leads to a straightforward analysis of the relationship between the Maslov index and the Morse index for Schrödinger operators on [ 0 , 1 ] and R . [ABSTRACT FROM AUTHOR]

We prove a pumping lemma in order to characterize the infinite words determined by indexed languages. An infinite language L determines an infinite word α if every string in L is a prefix of α . If L is regular or context-free, it is known that α must be ultimately periodic. We show that if L is an indexed language, then α is a morphic word, i.e., α can be generated by iterating a morphism under a coding. Since the other direction, that every morphic word is determined by some indexed language, also holds, this implies that the infinite words determined by indexed languages are exactly the morphic words. The pumping lemma which we use to obtain this result generalizes one recently proved for the class ET0L. [ABSTRACT FROM AUTHOR]

Working with a general class of linear Hamiltonian systems on intervals with at least one singular endpoint which can be limit-point, limit-circle, or limit-intermediate, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of C 2 n. In the first part of the analysis we associate our linear Hamiltonian systems with families of well-defined self-adjoint operators, and in the latter part we employ the renormalized oscillation approach to count the number of eigenvalues these operators have on fixed intervals (λ 1 , λ 2) whose closures do not intersect the essential spectrum of the operators. We conclude the analysis with two illustrative examples, indicating how the theory can be implemented in practice. This extends previous work by the authors for regular linear Hamiltonian systems. [ABSTRACT FROM AUTHOR]

Assuming a symmetric potential and separated self-adjoint boundary conditions, we relate the Maslov and Morse indices for Schrödinger operators on [ 0 , 1 ] . We find that the Morse index can be computed in terms of the Maslov index and two associated matrix eigenvalue problems. This provides an efficient way to compute the Morse index for such operators. [ABSTRACT FROM AUTHOR]

Abstract: Let be a Lagrangian torus fibration with singularities such that the fibers are of Maslov index zero, and unobstructed. The paper constructs a rigid analytic space over the Novikov field which is a deformation of the semi-flat complex structure of the dual torus fibration over the smooth locus of π. Transition functions of are obtained via homomorphisms which capture the wall-crossing phenomenon of moduli spaces of holomorphic disks. [Copyright &y& Elsevier]

Abstract: The quantum mechanical Kowalevski top is studied by the direct diagonalization of the Hamiltonian. The spectra show different behaviors depending on the region divided by the bifurcation sets of the classical invariant tori. Some of these spectra are nearly degenerate due to the multiplicity of the invariant tori. The Kowalevski top has several symmetries and symmetry quantum numbers can be assigned to the eigenstates. We have also carried out the semiclassical quantization of the Kowalevski top by the EBK formulation. It is found that the semiclassical spectra are close to the exact values, thus the eigenstates can be also labeled by the integer quantum numbers. The symmetries of the system are shown to have close relations with the semiclassical quantum numbers and the near-degeneracy of the spectra. [Copyright &y& Elsevier]

Abstract: Morse-type index theorems for self-adjoint elliptic second order boundary value problems arise as the second variation of an energy functional corresponding to some variational problem. The celebrated Morse index theorem establishes a precise relation between the Morse index of a geodesic (as critical point of the geodesic action functional) and the number of conjugate points along the curve. Generalization of this theorem to linear elliptic boundary value problems appeared since seventies. (See, for instance, Smale (1965) , Uhlenbeck (1973) and Simons (1968) among others.) The aim of this paper is to prove a Morse–Smale index theorem for a second order self-adjoint elliptic boundary value problem in divergence form on a star-shaped domain of the N-dimensional Euclidean space with Dirichlet and Neumann boundary conditions. This result will be achieved by generalizing a recent new idea introduced by authors in Deng and Jones (2011) , based on the idea of shrinking the boundary. [Copyright &y& Elsevier]

Abstract: In lines 8–11 of Lu (2009) we wrote: “For integer , if M is -smooth and -smooth satisfies the assumptions (L1)–(L3), then the functional is -smooth, bounded below, satisfies the Palais–Smale condition, and all critical points of it have finite Morse indexes and nullities (see and )”. However, as proved in Abbondandolo and Schwarz (2009) the claim that is -smooth is true if and only if for every the function is a polynomial of degree at most 2. So the arguments in Lu (2009) are only valid for the physical Hamiltonian in (1.2) and corresponding Lagrangian therein. In this note we shall correct our arguments in Lu (2009) with a new splitting lemma obtained in Lu (2011) . [Copyright &y& Elsevier]

Abstract: In this paper, we define a relative Morse index for two continuous symmetric matrices paths in satisfying condition (B1) and study its relation with the Maslov-type indices under brake orbit boundary value of these two symmetric matrices paths. As applications, using this relation we obtain a multiple existence of periodic brake orbit solutions of asymptotically linear Hamiltonian system in the presence of symmetries. [Copyright &y& Elsevier]

Abstract: In this paper we study bifurcation from simple eigenvalues for systems of differential equations; we prove the existence of global bifurcating branches of solutions on which the Maslov index of suitable associated linear systems is preserved. [Copyright &y& Elsevier]

Abstract: Motivated by the generalization of the Maslov index to tube domains and by numerous applications of related index function in infinite-dimensional situations, we describe in this paper a topologically oriented approach to an index function generalizing the Maslov index for bounded symmetric domains of tube type to a variety of infinite-dimensional situations containing in particular the class of all bounded symmetric domains of tube type in Banach spaces. The framework is that of 3-graded Banach–Lie groups and corresponding Jordan triple systems. [Copyright &y& Elsevier]

Abstract: We obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution. We study electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct.Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point. [Copyright &y& Elsevier]

We study the relationship between the twist condition in the Poincare´–Birkhoff fixed point theorem and the assumptions on the Maslov index for asymptotically linear planar Hamiltonian systems. For this aim, we develop a variant of the Poincare´–Birkhoff theorem which, together with its classical version, allows to obtain a lower bound for the number of nontrivial periodic solutions in terms of the gap between the Maslov indexes associated to the linearizations of the planar system at zero and at infinity. [Copyright &y& Elsevier]

We consider a Morse–Sturm system in Rn whose coefficient matrix is symmetric with respect to a (not necessarily positive definite) nondegenerate symmetric bilinear form on Rn. The main motivation for studying such systems comes from semi-Riemannian geometry, where the Morse–Sturm system is obtained from the Jacobi equation along a geodesic by writing the equation in terms of a parallelly transported basis of the tangent bundle along the geodesic. Two integer numbers are naturally associated to such systems: the Maslov index, which gives a sort of algebraic count of the conjugate instants, and the spectral index, which gives an algebraic count of the negative eigenvalues of the corresponding second-order differential operator. In this paper we prove that these two integer numbers are equal; in the case of Riemannian geometry, this equality is precisely the Morse Index Theorem. Such equality is already known to hold under a suitable nondegeneracy assumption on the eigenvalues of the Jacobi operator; we give a proof of the equality in the degenerate case using a perturbation argument. [Copyright &y& Elsevier]

We develop an index theory for parabolic and collision solutions to the classical n -body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of trajectories ending with a total collapse or expanding with vanishing limiting velocities. Both problems suffer from a lack of compactness and can be brought in a similar form of a Lagrangian System on the half time line by a regularising change of coordinates which preserve the Lagrangian structure. We then introduce a Maslov-type index which is suitable to capture the asymptotic nature of these trajectories as half-clinic orbits: by taking into account the underlying Hamiltonian structure we define the appropriate notion of geometric index for this class of solutions and we develop the relative index theory. [ABSTRACT FROM AUTHOR]