Related papers: On $b$-Whittaker functions
It is well known that there is an integral theorem for quaternion-valued functions analogous to Cauchys Theorem for complex-valued functions, namely Fueters Theorem. The class of quaternionic functions for which this applies are generally…
We use nonstandard analysis to formulate quantum mechanics in hyperfinite-dimensional spaces. Self-adjoint operators on hyperfinite-dimensional spaces have complete eigensets, and bound states and continuum states of a Hamiltonian can thus…
We define a tau function for a generic Riemann-Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed…
This is an extended (factor 2.5) version of arXiv:math/0601371 and arXiv:0808.3486. We present new results in the theory of the classical $\theta$-functions of Jacobi: series expansions and defining ordinary differential equations (\odes).…
Motivated by $T\bar T$, we introduce and study a wide class of solvable deformations of quantum-mechanical theories. These deformations map the Hamiltonian to a function of itself. We solve these theories by computing all finite-temperature…
We identify Whittaker vectors for $\mathcal{W}_k(\mathfrak{g})$-modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable…
The quasiclassical solution to the extended Toda chain hierarchy, corresponding to the deformation of the simplest Seiberg-Witten theory by all descendants of the dual topological string model, is constructed explicitly in terms of the…
The extended flow equations of the multi-component Toda hierarchy are constructed. We give the Hirota bilinear equations and tau function of this new extended multi-component Toda hierarchy(EMTH). Because of logarithmic terms, some extended…
Whittaker functions on non-archimedean fields were first introduced in the work of Jacquet, and they were characterized explicitly by Shintani. We obtain an explicit inversion formula and a Plancherel formula for the $p$-adic Whittaker…
We give a proof of Awata and Yamada's conjecture for the explicit formula of Whittaker vector of the deformed Virasoro algebra realized in the Fock space. The formula is expressed as a summation over Macdonald symmetric functions with…
We review new aspects of integrable systems discovered recently in N=2 supersymmetric gauge theories and their topologically twisted versions. The main topics are (i) an explicit construction of Whitham deformations of the Seiberg-Witten…
The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of…
The Inozemtsev Hamiltonian is an elliptic generalization of the differential operator defining the BC_N trigonometric quantum Calogero-Sutherland model, and its eigenvalue equation is a natural many-variable generalization of the Heun…
Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and output space ${\bf C}$ and state space $H$. The function $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$ on $L^2((0, \infty…
In this paper, we extend results connecting quantum groups to spherical Whittaker functions on metaplectic covers of $GL_r(F)$, for $F$ a nonarchimedean local field. Brubaker, Buciumas, and Bump showed that for a certain metaplectic…
In this paper, we begin by applying the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. By utilizing the solution to the Pythagorean equation $a^2 + b^2 = c^2$, these…
We show that the Hamiltonians of the open relativistic Toda system are elements of the generic basis of a cluster algebra, and in particular are cluster characters of nonrigid representations of a quiver with potential. Using cluster…
In the present paper we describe the procedure of the Q-operators construction for the q-deformed model, described by the Lax operator, which is important to formulate the Bethe ansatz for the Sin-Gordon model. This Lax operator can also be…
This paper is devoted to revealing the relationship between matrix-valued $\theta$-deformed bi-orthogonal polynomials and non-commutative Toda-type hierarchies. In this procedure, Wronski quasi-determinants are widely used and play the role…
The main object considered in this paper is the trace function, defined as a suitably normalized trace of a product of intertwining operators for the Drinfeld-Jimbo quantum group, multiplied by the exponential of an element of the Cartan…