Related papers: Aspects of the ODE/IM correspondence
We briefly explain some simple arguments based on pseudo Hermiticity, supersymmetry and PT-symmetry which explain the reality of the spectrum of some non-Hermitian Hamiltonians. Subsequently we employ PT-symmetry as a guiding principle to…
We describe differential invariants of infinite-dimensional algebras being equivalence algebras of some classes of PDE and study structure of these algebras.
We exhibit a relationship between the massless $a_2^{(2)}$ integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the…
Inverse spectral problems are studied for the second order integro-differential operators on a finite interval. Properties of spectral characteristic are established, and the uniqueness theorem is proved for this class of inverse problems.
A master equation expressing the classical integrability of two-dimensional non-linear sigma models is found. The geometrical properties of this equation are outlined. In particular, a closer connection between integrability and T-duality…
A discussion is presented, within a simple unifying scheme, about different types of symmetry of PDE's, with the introduction and a precise characterization of the notions of "standard" and "weak" conditional symmetries, together with their…
In the correspondence between spectral problems and topological strings, it is natural to consider complex values for the string theory moduli. In the spectral theory side, this corresponds to non-Hermitian quantum curves with complex…
In this paper, we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions. For this purpose, we introduce two types of nonlocal transformations which relate specific classes of nonlinear…
In this work, a connection between some spectral properties of direct integral of operators in the direct integral of Hilbert spaces and their coordinate operators has been investigated.
The theory of Lie systems has recently been applied to Quantum Mechanics and additionally some integrability conditions for Lie systems of differential equations have also recently been analysed from a geometric perspective. In this paper…
Symmetries concerning the ordinary coordinate spacetime and internal spacetime are discussed. A possible unification model of electroweak, strong and gravitational interactions is briefly described.
The simplified models of interaction of charged matter with resonance modes of radiation generalizing the well-known Jaynes-Cummings and Dicke models are considered. It is found that these new models are integrable for arbitrary numbers of…
It is well established numerically that spectral statistics of pseudo-integrable models differs considerably from the reference statistics of integrable and chaotic systems. In [PRL,93 (2004) 254102] statistical properties of a certain…
Isotropic oscillator and Coulomb problems are known to have interesting correspondence. We focus on 2D quantum problems and present complete treatment on the correspondence including the Schroedinger equation, eigenfunctions and…
This is a brief review of several algebraic constructions related to generalized fermionic spectra, of the type which appear in integrable quantum spin chains and integrable quantum field theories. We discuss the connection between…
The Weyl-Sims classification for a second-order ordinary differential equation with general complex coefficients is investigated. Connections are then established between the associated m-function and the spectral properties of…
The mergings of energy levels associated with the breaking of PT symmetry in the model of Bender and Boettcher, and in its generalisation to incorporate a centrifugal term, are analysed in detail. Even though conventional WKB techniques…
Symmetry methods are by now recognized as one of the main tools to attack deterministic differential equations (both ODEs and PDEs); the situation is quite different for what concerns stochastic differential equations: here, symmetry…
The role of singular solutions in some simple quantum mechanical models is studied. The space of the states of two-dimensional quantum harmonic oscillator is shown to be separated into sets of states with different properties.
In this article we consider a class of integrable operators and investigate its connections with the following theories:the spectral theory of non-self-adjoint operators, the Riemann-Hilbert problem, the canonical differential systems and…