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We study the eigenvalue problem for a self-adjoint 1D Dirac operator. It is known that, near an energy level where the square of the potential makes a simple well, the eigenvalues are approximated by a Bohr–Sommerfeld type quantization rule. A remarkable difference from the Schrödinger case appears in the Maslov correction term. This fact was recently found by Hirota [Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov–Shabat operator, J. Math. Phys. 58 (2017) 102–108] under the analyticity condition of the potential using a complex WKB method. In this paper, we approach this problem with a microlocal technique focusing on the asymptotic behavior of the eigenfunction along the characteristic set to generalize the result to C∞ potentials. [ABSTRACT FROM AUTHOR]

Working with a general class of regular linear Hamiltonian systems, 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 . We verify that our applicability class includes Dirac and Sturm–Liouville systems, as well as a system arising from differential-algebraic equations for which the spectral parameter appears nonlinearly. [ABSTRACT FROM AUTHOR]

In this paper, we study several closely related invariants associated to Dirac operators on odd-dimensional manifolds with boundary with an action of the compact group H of isometries. In particular, the equality between equivariant winding numbers, equivariant spectral flow and equivariant Maslov indices is established. We also study equivariant η -invariants which play a fundamental role in the equivariant analog of Getzler's spectral flow formula. As a consequence, we establish a relation between equivariant η -invariants and equivariant Maslov triple indices in the splitting of manifolds. [ABSTRACT FROM AUTHOR]

In this paper we present the theory of oscillation numbers and dual oscillation numbers for continuous Lagrangian paths in R 2 n . Our main results include a connection of the oscillation numbers of the given Lagrangian path with the Lidskii angles of a special symplectic orthogonal matrix. We also present Sturmian type comparison and separation theorems for the difference of the oscillation numbers of two continuous Lagrangian paths. These results, as well as the definition of the oscillation number itself, are based on the comparative index theory (Elyseeva, 2009). The applications of these results are directed to the theory of Maslov index of two continuous Lagrangian paths. We derive a formula for the Maslov index via the Lidskii angles of a special symplectic orthogonal matrix, and hence we express the Maslov index as the oscillation number of a certain transformed Lagrangian path. The results and methods are based on a generalization of the recently introduced oscillation numbers and dual oscillation numbers for conjoined bases of linear Hamiltonian systems (Elyseeva, 2019 and 2020) and on the connection between the comparative index and Lidskii angles of symplectic matrices (Šepitka and Šimon Hilscher, 2021). [ABSTRACT FROM AUTHOR]

We use the Maslov index to study the spectrum of a class of linear Hamiltonian differential operators. We provide a lower bound on the number of positive real eigenvalues, which includes a contribution to the Maslov index from a nonregular crossing. A close study of the eigenvalue curves, which represent the evolution of the eigenvalues as the domain is shrunk or expanded, yields formulas for their concavity at the nonregular crossing in terms of the corresponding Jordan chains. This enables the computation of the Maslov index at such a crossing via a homotopy argument. We apply our theory to study the spectral (in)stability of standing waves in the nonlinear Schr\"odinger equation on a compact interval. We derive stability results in the spirit of the Jones--Grillakis instability theorem and the Vakhitov--Kolokolov criterion, both originally formulated on the real line. A fundamental difference on passing from the real line to the compact interval is the loss of translational invariance, in which case the zero eigenvalue of the linearized operator is (typically) geometrically simple. Consequently, the stability results differ depending on the boundary conditions satisfied by the wave. We compare our lower bound to existing results involving constrained eigenvalue counts, finding a direct relationship between the correction factors found therein and the objects of our analysis, including the second-order Maslov crossing form. [ABSTRACT FROM AUTHOR]

The geometric properties of the Maslov index on symplectic manifolds are discussed. The Maslov index is constructed as a homological invariant on a Lagrangian submanifold of a symplectic manifold. In the simplest case, a Lagrangian submanifold is a submanifold in the symplectic space , in which the symplectic structure is given by the nondegenerate form and is a submanifold, , on which the form is trivial. In the general case, a symplectic manifold and the bundle of Lagrangian Grassmannians is considered. The question under study is as follows: when is the Maslov index, given on an individual Lagrangian manifold as a one-dimensional cohomology class, the image of a one-dimensional cohomology class of the total space of bundles of Lagrangian Grassmannians? An answer is given for various classes of bundles of Lagrangian Grassmannians. [ABSTRACT FROM AUTHOR]

In the previous papers (Hu et al. in J Differ Equ 266(7):4106–4136, 2019; Kong et al. in J Differ Equ 156(2):328–354, 1999), the jump phenomena of the j-th eigenvalue were completely characterized for Sturm–Liouville problems. In this paper, we show that the jump number of these eigenvalue branches is exactly the Maslov index for the path of corresponding boundary conditions. Then we determine the sharp range of the jth eigenvalue on each layer of the space of boundary conditions. Finally, we prove that the graph of monodromy matrix tends to the Dirichlet boundary condition as the spectral parameter goes to - ∞ . [ABSTRACT FROM AUTHOR]

The spectral flow is a well‐known quantity in spectral theory that measures the variation of spectra about 0 along paths of selfadjoint Fredholm operators. The aim of this work is twofold. Firstly, we consider homotopy invariance properties of the spectral flow and establish a simple formula which comprises its classical homotopy invariance and yields a comparison theorem for the spectral flow under compact perturbations. We apply our result to the existence of non‐trivial solutions of boundary value problems of Hamiltonian systems. Secondly, the spectral flow was axiomatically characterised by Lesch, and by Ciriza, Fitzpatrick and Pejsachowicz under the assumption that the endpoints of the paths of selfadjoint Fredholm operators are invertible. We propose a different approach to the uniqueness of spectral flow which lifts this additional assumption. As application of the latter result, we discuss the relation between the spectral flow and the Maslov index in symplectic Hilbert spaces. [ABSTRACT FROM AUTHOR]

We extend the definition of the Maslov index to a broad class of non-Hamiltonian dynamical systems. To do this, we introduce a family of topological spaces--which we call Maslov--Arnold spaces--that share key topological features with the Lagrangian Grassmannian and hence admit a similar index theory. This family contains the Lagrangian Grassmannian and much more. We construct a family of examples, called hyperplane Maslov--Arnold spaces, that are dense in the Grassmannian and hence are much larger than the Lagrangian Grassmannian (which is a submanifold of positive codimension). The resulting index is then used to study eigenvalue problems for nonsymmetric reaction-diffusion systems. A highlight of our analysis is a topological interpretation of the Turing instability: the bifurcation that occurs as one increases the ratio of diffusion coefficients corresponds to a change in the generalized Maslov index. [ABSTRACT FROM AUTHOR]

Classical results from Sturm--Liouville theory state that the number of unstable eigenvalues of a scalar, second-order linear operator is equal to the number of associated conjugate points. Recent work has extended these results to a much more general setting, thus allowing for spectral stability of nonlinear waves in a variety of contexts to be determined by counting conjugate points. However, in practice, it is not yet clear whether it is easier to compute conjugate points than to just directly count unstable eigenvalues. We address this issue by developing a framework for the computation of conjugate points using validated numerics. Moreover, we apply our method to a parameter-dependent system of bistable equations and show that there exist both stable and unstable standing fronts. This application can be seen as complementary to the classical result via Sturm--Liouville theory that in scalar reaction-diffusion equations pulses are unstable whereas fronts are stable and to the more recent result of Beck et al. [Philos. Trans. Roy. Soc. A, 376 (2018), 20170187] that symmetric pulses in reaction-diffusion systems with gradient nonlinearity are also necessarily unstable. [ABSTRACT FROM AUTHOR]

A unified geometric approach for the stability analysis of traveling pulse solutions for reaction diffusion equations with skew-gradient structure has been established in a previous paper (see Paul Cornwell [Indiana Univ. Math. J. 68 (2019), pp. 1801-1832]), but essentially no results have been found in the case of traveling front solutions. In this work, we will bridge this gap. For such cases, a Maslov index of the traveling wave is well-defined, and we will show how it can be used to provide the spectral information of the waves. As an application, we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of FitzHugh-Nagumo equations. [ABSTRACT FROM AUTHOR]

• 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 provide a short and self-contained proof of an equality of Maslov indices in a linear symplectic reduction and apply it to obtain an equality of Maslov focal indices after reducing by symmetries an electromagnetic Lagrangian system. [ABSTRACT FROM AUTHOR]

Motivated by the recent discoveries on the stability properties of symmetric periodic solutions of Hamiltonian systems, we establish a Bott-type iteration formula for dihedraly equivariant Hamiltonian systems. We apply the abstract theory for computing the Morse indices of the celebrated Chenciner and Montgomery figure-eight orbit for the planar three body problem in different equivariant spaces. Finally we provide a hyperbolicity criterion for reversible Lagrangian systems. [ABSTRACT FROM AUTHOR]

In this paper, we prove Morse index theorem of Lagrangian systems with self-adjoint boundary conditions. Based on it, we give some nontrivial estimates on the difference of Morse indices. As an application, we get a new criterion for the stability problem of brake periodic orbit. [ABSTRACT FROM AUTHOR]

The mechanism of energy quantization is studied for classical dynamics on a highly anharmonic potential, ranging from integrable, mixed, and chaotic motions. The quantum eigenstates (standing waves) are created by the phase factors (the action integrals and the Maslov index) irrespective of the integrability, when the amplitude factors are relatively slowly varying. Indeed we show numerically that the time Fourier transform of an approximate semiclassical correlation function in which the amplitude factors are totally removed reproduces the spectral positions (energy eigenvalues) accurately in chaotic regime. Quantization with the phase information alone brings about dramatic simplification to molecular science, since the amplitude factors in the lowest order semiclassical approximation diverge exponentially in a chaotic domain. [ABSTRACT FROM AUTHOR]

In view of the rapid progress in experiments of the tunneling dynamics in the time domain, we develop a quasisemiclassical method that is aimed at a study of the proton-transfer dynamics in a large system such as tropolone and its interesting derivatives, to which not only full quantum mechanics, but even a standard semiclassical theory is never easy to apply. In our very tractable method for multidimensional systems, the tunneling paths are generated in terms of the generalized classical mechanics [K. Takatsuka and H. Ushiyama, Phys. Rev. A 51, 4353 (1995)], but the quantum phases arising from the action integral, the Maslov index, and the semicalssical amplitude factor as well in the semiclassical kernels are entirely neglected. This approach is called the quasisemiclassical method. One of the technical issues involved in the general semiclassical scheme is how to locate points from which a tunneling path emanates. Hence the studies of such tunneling points and the quasisemiclassical method should be examined collectively. We test several ways of determining the tunneling point, including those already proposed in the literature and a newly proposed one. It is shown numerically that the quasisemiclassical method with an appropriate choice of tunneling points reproduces the full quantum mechanical tunneling probability reasonably well. This case study indicates that the present conventional approach is promising to the study of large systems. The role of tunneling points in the initial process of tunneling is also discussed. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Abstract We provide an integral formula for the Maslov index of a pair (E , F) over a surface Σ , where E → Σ is a complex vector bundle and F ⊂ E | ∂ Σ is a totally real subbundle. As in Chern–Weil theory, this formula is written in terms of the curvature of E plus a boundary contribution. When (E , F) is obtained via an immersion of (Σ , ∂ Σ) into a pair (M , L) where M is Kähler and L is totally real, the formula allows us to control the Maslov index in terms of the geometry of (M , L). We exhibit natural conditions on (M , L) which lead to bounds and monotonicity results. [ABSTRACT FROM AUTHOR]

Working with general linear Hamiltonian systems on [0, 1], and with a wide range of self-adjoint boundary conditions, including both separated and coupled, we develop a general framework for relating the Maslov index to spectral counts. Our approach is illustrated with applications to Schrödinger systems on R with periodic coefficients, and to Euler-Bernoulli systems in the same context. [ABSTRACT FROM AUTHOR]

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher-KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure. [ABSTRACT FROM AUTHOR]

We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over Q , Z and Z 2 , the Kervaire invariant, the f-invariant, and the ν-invariant. This requires either a framing or a corner structure. The canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies. We characterize the flux quantization condition on the C-field and the topological action of the M5-brane via the Hopf invariant, and the dual of the C-field as (a refinement of) an element of Hopf invariant two. In the signature formulation, the contribution to the M-brane effective action is given by the Maslov index of the corner. The Kervaire invariant implies that the effective action of the M5-brane is quadratic. Our study leads to viewing the self-dual string, which is the boundary of the M2-brane on the M5-brane worldvolume, as a string theory in the sense of cobordism of manifolds with corners. We show that the dynamics of the C-field and its dual are encoded in a unified way in the 4-sphere, which suggests the corresponding spectrum as the generalized cohomology theory describing the fields. The effective action of the corner is captured by the f-invariant, which is an invariant at chromatic level two. Finally, considering M-theory on manifolds with G2 holonomy, we show that the canonical G2 structure minimizes the topological part of the M5-brane action. This is done via the ν-invariant and a variant that we introduce related to the one-loop polynomial. [ABSTRACT FROM AUTHOR]

We calculate the Hörmander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle. [ABSTRACT FROM AUTHOR]

In this work we apply the asymptotic method suggested by Maslov [1] to obtain the Hugoniot-Maslov chain for shock type solutions of conservation laws systems with quadratic flux. Additionally to the ODE infinite system that make up the chain, it was obtained an algebraic compatibility condition that must be satisfied by some of the coefficients of the asymptotic expansion of the shock solution. We give a new geometrical interpretation for this compatibility condition by means of certain singular surface whose projections represent time-dependent Hugoniot locus through the left limit state of the Shock. [ABSTRACT FROM AUTHOR]

Taking the signature of the closure of a braid defines a map from the braid group to the integers. In 2005, Gambaudo and Ghys expressed the homomorphism defect of this map in terms of the Meyer cocycle and the Burau representation. In the present paper, we simultaneously extend this result in two directions, considering the multivariable signature of the closure of a coloured tangle. The corresponding defect is expressed in terms of the Maslov index and of the Lagrangian functor defined by Turaev and the first-named author. [ABSTRACT FROM AUTHOR]

We address in this paper the notion of relative phase shift for mixed quantum systems. We study the Pancharatnam–Sjöqvist phase shift φ ( t ) = Arg Tr ( U ^ t ρ ^ ) for metaplectic isotopies acting on Gaussian mixed states. We complete and generalize the previous results obtained by one of us, while giving rigorous proofs. The key actor in this study is the theory of the Conley–Zehnder index which is an intersection index related to the Maslov index. [ABSTRACT FROM AUTHOR]

Periodic and localized travelling waves such as wave trains, pulses, fronts and patterns of more complex structure often occur in natural and experimentally built systems. In mathematics, these objects are realized as solutions of nonlinear partial differential equations. The existence, dynamic properties and bifurcations of those solutions are of interest. In particular, their stability is important for applications, as the waves that are observable are usually stable. When the waves are unstable, further investigation is warranted of the way the instability is exhibited, i.e. the nature of the instability, and also coherent structures that appear as a result of an instability of travelling waves. A variety of analytical, numerical and hybrid techniques are used to study travelling waves and their properties. [ABSTRACT FROM AUTHOR]

In a scalar reaction-diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state's critical points. In particular, this implies that pulse solutions are unstable. We extend this picture to pulses in reaction-diffusion systems with gradient nonlinearity. In particular, we associate a Maslov index to any asymptotically constant state, generalizing existing definitions of the Maslov index for homoclinic orbits. It is shown that this index equals the number of unstable eigenvalues for the linearized evolution equation. Finally, we use a symmetry argument to show that any pulse solution must have non-zero Maslov index, and hence be unstable. [ABSTRACT FROM AUTHOR]

The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space. Then a direct proof of the iteration formulae for the Maslov-type indices of symplectic paths is given. [ABSTRACT FROM AUTHOR]

The FitzHugh-Nagumo equation, which was deri ved as a simplification of the Hodgkin-Huxley model for nerve impulse propagation, has been extensively studied as a paradigmatic activator-inhibitor system. We consider the version of this system in which two agents diffuse at an equal rate. Using geometric singular perturbation theory, we prove the existence and stability of fast traveling pulses. The stability proof makes use of the Maslov index--an invariant of symplectic geometry--to count unstable eigenvalues for the linearization about the wave. The calculation of the Maslov index is carried out by tracking the evolution of the uns table manifold of the rest state using the timescale separation. This entails a careful consideration of how the transition from fast to slow dynamics occurs in the tangent bundle over the wave. Finally, we observe in the calculation that the Maslov index lacks monotonicity in the spatial paramete r, which distinguishes this application of the Maslov index from similar analyses of Hamiltonian systems. [ABSTRACT FROM AUTHOR]

We study the Maslov index as a tool to analyze stability of steady state solutions to a reaction-diffusion equation in one spatial dimension. We show that the path of unstable subspaces associated to this equation is governed by a matrix Riccati equation whose solution S develops singularities when changes in the Maslov index occur. Our main result proves that at these singularities the change in Maslov index equals the number of eigenvalues of S that increase to $$+\infty $$ minus the number of eigenvalues that decrease to $$-\infty $$ . [ABSTRACT FROM AUTHOR]

In this paper, we use the central configuration coordinate decomposition to study the linearized Hamiltonian system near the 3-body elliptic Euler solutions. Then using the Maslov-type $${\omega}$$ -index theory of symplectic paths and the theory of linear operators we compute the $${\omega}$$ -indices and obtain certain properties of linear stability of the Euler elliptic solutions of the classical three-body problem. [ABSTRACT FROM AUTHOR]

In this paper, a new notion of limiting problem for nonsmooth nonoscillating solutions to linear parabolic equations of Maslov tunnel type is introduced. This notion allows one to reduce the construction of asymptotic solutions to the construction of (nonsmooth) solutions to the Hamilton-- Jacobi and continuity equations. [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]

The problem of constructing the asymptotics of the Green function for the Helmholtz operator h Δ + n ( x), x ∈ R , with a small positive parameter h and smooth n ( x) has been studied by many authors; see, e.g., [1, 2, 4]. In the case of variable coefficients, the asymptotics was constructed by matching the asymptotics of the Green function for the equation with frozen coefficients and a WKB-type asymptotics or, in a more general situation, the Maslov canonical operator. The paper presents a different method for evaluating the Green function, which does not suppose the knowledge of the exact Green function for the operator with frozen variables. This approach applies to a larger class of operators, even when the right-hand side is a smooth localized function rather than a δ-function. In particular, the method works for the linearized water wave equations. [ABSTRACT FROM AUTHOR]

We define Maslov-type indices associated to contact and symplectic transformation groups. There are two such families of indices. The first class of indices are maps from the homotopy groups of the contactomorphism or symplectomorphism group to a quotient of Z . These are based on a generalization of the Maslov index. The second class of indices are maps from the homotopy groups of the space of contact structures or the space of cohomologous symplectic forms to the homotopy groups of a simple homogeneous space. We provide a detailed construction and describe some properties of these indices and their applications. [ABSTRACT FROM AUTHOR]

We examine symplectic topological features of a certain family of monotone Lagrangian submanifolds in CPn. First we give cohomological constraints on a Lagrangian submanifold in CPn whose first integral homology is p-torsion. In the case where (n, p) = (5, 3), (8, 3), we prove that the cohomologies with coefficients in Z2 of such Lagrangian submanifolds are isomorphic to that of SU(3)/(SO(3)3) and SU(3)/3, respectively. Then we calculate the Floer cohomology with coefficients in Z2 of a monotone Lagrangian submanifold SU(p)/Zp in CPp2-1. [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]

The Maslov index in the semiclassical Bohr-Sommerfeld quantization rule is calculated for one-dimensional power-law potentials V ( x) = − V / x , x > 0, 0 < s < 2 The result for the potential V(x)=- V / x is compared with the recently reported exact solution. The case of a spherically symmetric power-law potential is also considered. [ABSTRACT FROM AUTHOR]

We study the spectrum of the Schr¨odinger operators with n x n matrix valued potentials on a finite interval subject to θ-periodic boundary conditions. For two such operators, corresponding to different values of θ, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. In addition we derive a formula for the derivatives of the eigenvalues with respect to θ in terms of the Maslov crossing form. Finally, we give a new shorter proof of a recent result relating the Morse and Maslov indices of the Schr¨odinger operator for a fixed θ. [ABSTRACT FROM AUTHOR]

We obtain an alternative class of Lagrangians in the so-called the multiplicative form for a system with one degree of freedom in the nonrelativistic and the relativistic cases. This new form of the Lagrangian can be regarded as a one-parameter class with the parameter λ obtained using an extension of the standard additive form of the Lagrangian because both forms yield the same equation of motion. We note that the multiplicative form of the Lagrangian can be regarded as a generating function for obtaining an infinite hierarchy of Lagrangians that yield the same equation of motion. This nontrivial set of Lagrangians confirms that the Lagrange function is in fact nonunique. [ABSTRACT FROM AUTHOR]

We study the Schrödinger operator L = -Δ+V on a star-shaped domain O in Rd with Lipschitz boundary ∂Ω. The operator is equipped with quite general Dirichlet- or Robin-type boundary conditions induced by operators between H1/2(∂Ω) and H-1/2(∂Ω), and the potential takes values in the set of symmetric N × N matrices. By shrinking the domain and rescaling the operator we obtain a path in the Fredholm-Lagrangian Grassmannian of the subspace of H1/2(∂Ω)H-1/2(∂Ω) corresponding to the given boundary condition. The path is formed by computing the Dirichlet and Neumann traces of weak solutions to the rescaled eigenvalue equation. We prove a formula relating the number of negative eigenvalues of L (the Morse index), the signed crossings of the path (the Maslov index), the number of negative eigenvalues of the potential matrix evaluated at the center of the domain, and the number of negative eigenvalues of a bilinear form related to the boundary operator. [ABSTRACT FROM AUTHOR]

In this paper we show how to combinatorially compute the rotation class of a large family of embedded Legendrian tori in R 5 with the standard contact form. In particular, we give a formula to compute the Maslov index for any loop on the torus and compute the Maslov number of the Legendrian torus. These formulas are a necessary component in computing contact homology. Our methods use a new way to represent knotted Legendrian tori called Lagrangian hypercube diagrams. [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]

Transition-state theories describing barrierless chemical reactions, or more general activated problems, are often hampered by the lack of a saddle around which the dividing surface can be constructed. For example, the time-dependent transition-state trajectory uncovering the nonrecrossing dividing surface in thermal reactions in the framework of the Langevin equation has relied on perturbative approaches in the vicinity of the saddle. We recently obtained an alternative approach using Lagrangian descriptors to construct time-dependent and recrossing-free dividing surfaces. This is a nonperturbative approach making no reference to a putative saddle. Here we show how the Lagrangian descriptor can be used to obtain the transition-state geometry of a dissipated and thermalized reaction across barrierless potentials. We illustrate the method in the case of a 1D Brownian motion for both barrierless and step potentials; however, the method is not restricted and can be directly applied to different kinds of potentials and higher dimensional systems. [ABSTRACT FROM AUTHOR]

We study the semiclassical mechanics of systems with second-class constraints. We assume that the quantum mechanics of such a system is constructed by the Batalin-Fradkin-Tyutin method, where some additional coordinates and momenta are introduced and second-class constraints are converted into firstclass constraints. We also assume that the algebraic quantization method is used to quantize the extended system. To construct the semiclassical approximation, we use the Maslov complex germ theory. We study the semiclassical states of a system with second-class constraints, their scalar product, and the action of semiclassical observables in the first order of the semiclassical expansion. We consider the transformation of semiclassical states in the course of evolution. [ABSTRACT FROM AUTHOR]

High frequency field expression at the focal region of a spherical reflector antenna backed by isotropic and homogeneous chiral substrate is derived using Maslov's method. As Geometrical Optics (GO) solution fails at the focal point, so Maslov's method is used to overcome this problem. Field patterns are calculated numerically and the results are plotted. Analysis is done to show the effects of thickness ( d ) of chiral substrate, chirality parameter ( β ) of the chiral substrate and permittivity ( ɛ ) of the medium. [ABSTRACT FROM AUTHOR]