Condensing Functions and Approximate Endpoint Criterion for the Existence Analysis of Quantum Integro-Difference FBVPs

A nonlinear quantum boundary value problem (q-FBVP) formulated in the sense of quantum Caputo derivative, with fractional q-integro-difference conditions along with its fractional quantum-difference inclusion q-BVP are investigated in this research. To prove the solutions’ existence for these quantum systems, we rely on the notions such as the condensing functions and approximate endpoint criterion (AEPC). Two numerical examples are provided to apply and validate our main results in this research work.

A general quantum Laplace transform

In this paper, we introduce a general quantum Laplace transform $\mathcal{L}_{\beta }$ L β and some of its properties associated with the general quantum difference operator ${D}_{\beta }f(t)= ({f(\beta (t))-f(t)} )/ ({ \beta (t)-t} )$ D β f ( t ) = ( f ( β ( t ) ) − f ( t ) ) / ( β ( t ) − t ) , β is a strictly increasing continuous function. In addition, we compute the β-Laplace transform of some fundamental functions. As application we solve some β-difference equations using the β-Laplace transform. Finally, we present the inverse β-Laplace transform $\mathcal{L}_{\beta }^{-1}$ L β − 1 .

On Sevostyanov’s Construction of Quantum Difference Toda Lattices

We propose a natural generalization of the construction of the quantum difference Toda lattice [6, 22] associated with a simple Lie algebra $\mathfrak{g}$. Our construction depends on two orientations of the Dynkin diagram of $\mathfrak{g}$ and some other data (which we refer to as a pair of Sevostyanov triples). In types $A$ and $C$, we provide an alternative construction via Lax matrix formalism, cf. [15]. We also show that the generating function of the pairing of Whittaker vectors in the Verma modules is an eigenfunction of the corresponding modified quantum difference Toda system and derive fermionic formulas for the former in spirit of [7]. We give a geometric interpretation of all Whittaker vectors in type $A$ via line bundles on the Laumon moduli spaces and obtain an edge-weight path model for them, generalizing the construction of [4].