Oscillation Numbers for Continuous Lagrangian Paths and Maslov Index

Kód výsledku v IS VaVaI

RIV/00216224:14310/23:00134002 - isvavai.cz

Výsledek na webu

https://link.springer.com/article/10.1007/s10884-022-10140-7

DOI - Digital Object Identifier

10.1007/s10884-022-10140-7

Jazyk výsledku

angličtina

Název v původním jazyce

Oscillation Numbers for Continuous Lagrangian Paths and Maslov Index

Popis výsledku v původním jazyce

In this paper we present the theory of oscillation numbers and dual oscillation numbers for continuous Lagrangian paths in R-2n. 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).

Název v anglickém jazyce

Oscillation Numbers for Continuous Lagrangian Paths and Maslov Index

Popis výsledku anglicky

In this paper we present the theory of oscillation numbers and dual oscillation numbers for continuous Lagrangian paths in R-2n. 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).

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

GA19-01246S: Nová oscilační teorie pro lineární hamiltonovské a symplektické systémy

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Dynamics and Differential Equations

ISSN

1040-7294

e-ISSN

1572-9222

Svazek periodika

35

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

32

Strana od-do

2589-2620

Kód UT WoS článku

000763193100001

EID výsledku v databázi Scopus

2-s2.0-85125526670