We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold when the diffusion constant $ε$ goes to zero. We assume that the first order term is given by a vector field $b$, whose recurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the normalized eigenfunctions concentrate on the recurrent sets of maximal dimension where the topological pressure \cite{Kifer90} is attained. On the cycles and torii, the limit measures are absolutely continuous with respect to the invariant probability measure on these sets. We have determined these limit measures, using a blow-up analysis.
We construct the orthogonal eigenbasis for a discrete elliptic Ruijsenaars type quantum particle Hamiltonian with hyperoctahedral symmetry. In the trigonometric limit the eigenfunctions in question recover a previously studied $q$-Racah type reduction of the Koornwinder-Macdonald polynomials. When the inter-particle interaction degenerates to that of impenetrable bosons, the orthogonal eigenbasis simplifies in terms of generalized Schur polynomials on the spectrum associated with recently found elliptic Racah polynomials.
Let $(\otimes_{j=1}^nV_j)[0]$ be the zero weight subspace of a tensor product of finite-dimensional irreducible $\frak{sl}_2$-modules. The dynamical elliptic Bethe algebra is a commutative algebra of differential operators acting on $(\otimes_{j=1}^nV_j)[0]$-valued functions on the Cartan subalgebra of $\frak{sl}_2$. The algebra is generated by values of the coefficient $S_2(x)$ of a certain differential operator $D=(d/dx)^2+S_2(x)$, defined to V. Rubtsov, A. Silantyev, D. Talalaev in 2009. We express $S_2(x)$ in terms of the KZB operators introduced by G. Felder and C. Wieszerkowski in 1994. …
We demonstrate that the symmetric elliptic polynomials $E_λ(x)$ originally discovered in the study of generalized Noumi-Shiraishi functions are eigenfunctions of the elliptic Ruijsenaars-Schneider (eRS) Hamiltonians that act on the mother function variable $y_i$ (substitute of the Young-diagram variable $λ$). This means they are eigenfunctions of the dual eRS system. At the same time, their orthogonal complements in the Schur scalar product, $P_R(x)$ are eigenfunctions of the elliptic reduction of the Koroteev-Shakirov (KS) Hamiltonians. This means that these latter are related to the dual eRS…
This work is concerned with operators of the type A = --c$Δ$ acting in domains $Ω$ := $Ω$ ' x (0, H) $\subseteq$ R^d x R ^+. The diffusion coefficient c > 0 depends on one coordinate y $\in$ (0, H) and is bounded but may be discontinuous. This corresponds to the physical model of ''layered media'', appearing in acoustics, elasticity, optical fibers... Dirichlet boundary conditions are assumed. In general, for each $ε$ > 0, the set of eigenfunctions is divided into a disjoint union of three subsets : Fng (non-guided), Fg (guided) and Fres (residual). The residual set shrinks as $ε$ $\rightarrow…
We consider mass concentration properties of Laplace eigenfunctions $φ_λ$, that is, smooth functions satisfying the equation $-Δφ_λ= λφ_λ$, on a smooth closed Riemannian manifold. Using a heat diffusion technique, we first discuss mass concentration/localization properties of eigenfunctions around their nodal sets. Second, we discuss the problem of avoided crossings and (non)existence of nodal domains which continue to be thin over relatively long distances. Further, using the above techniques, we discuss the decay of Laplace eigenfunctions on Euclidean domains which have a central "thick" par…
We present how explicit eigenfunctions of the principal Hamiltonian for the $BC_{m}$ relativistic Calogero-Moser-Sutherland model, due to van Diejen, can be constructed using gauge and integral transformations. In particular, we find that certain $BC$-type elliptic hypergeometric integrals, including elliptic Selberg integrals, of both Type I and Type II arise as eigenfunctions of the van Diejen model under some parameter restrictions. Among these are also joint eigenfunctions of so-called modular pairs of van Diejen operators. Furthermore, these transformations are related to reflections of t…
In this article we develop new techniques for studying concentration of Laplace eigenfunctions $ϕ_λ$ as their frequency, $λ$, grows. The method consists of controlling $ϕ_λ(x)$ by decomposing $ϕ_λ$ into a superposition of geodesic beams that run through the point $x$. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than $λ^{-\frac{1}{2}}$. We control $ϕ_λ(x)$ by the $L^2$-mass of $ϕ_λ$ on each geodesic tube and derive a purely dynamical statement through which $ϕ_λ(x)$ can be studied. In particular, we obtain estimates on $ϕ_λ(x)$…
Consider the Dirichlet-Laplacian in $Ω:= (0,L)\times (0,H)$ and choose another open set $ω\subset Ω$. The estimate $0<C_ω\leq R_ω(u):=\frac{\Vert u\Vert^{2}_{L^{2}(ω)}}{\Vert u\Vert^{2}_{L^{2}(Ω)}}\leq \frac{vol(ω)}{vol(ω)}$, for all the eigenfunctions, is well known. This is no longer true for an inhomogeneous elliptic selfadjoint operator $A$. In this work we create a partition among the set of eigenfunctions: $\forall ω$, the eigenfunctions satisfy $R_{omega}>C_ω>0,\exists ω, ω\not=\emptyset$, such that $\inf R_ω(u)=0$,and we wish to characterize these two sets. For two patterns we give a s…
We prove sharp uniform $L^p$-bounds for low-lying eigenfunctions of non-self-adjoint semiclassical pseudodifferential operators $P$ on $\mathbb{R}^{n}$ whose principal symbols are doubly-characteristic at the origin of $\mathbb{R}^{2n}$. Our bounds hold under two main assumptions on $P$: (1) the total symbol of $P$ extends holomorphically to a neighborhood of $\mathbb{R}^{2n}$ in $\mathbb{C}^{2n}$, and (2) the quadratic approximation to the principal symbol of $P$ at the origin is elliptic along its singular space. Most notably, our assumptions on the quadratic approximation are less restricti…
By means of a truncation condition on the parameters, the elliptic Ruijsenaars difference operators are restricted onto a finite lattice of points encoded by bounded partitions. A corresponding orthogonal basis of joint eigenfunctions is constructed in terms of polynomials on the joint spectrum. In the trigonometric limit, this recovers the diagonalization of the truncated Macdonald difference operators by a finite-dimensional basis of Macdonald polynomials.
We analyze the 1D cubic nonlinear stationary Schrödinger equation on a ring with a defect for both focusing and defocusing nonlinearity. All possible $δ$ and $δ'$ boundary conditions are considered at the defect, computing for each of them the real eigenfunctions, written as Jacobi elliptic functions, and eigenvalues for the ground state and first few excited energy levels. All six independent Jacobi elliptic functions are found to be solutions of some boundary condition. We also provide a way to map all eigenfunctions satisfying $δ$/$δ'$ conditions to any other general boundary condition or p…
In this paper the discrete eigenvalues of elliptic second order differential operators in $L^2(\mathbb{R}^n)$, $n \in \mathbb{N}$, with singular $δ$- and $δ'$-interactions are studied. We show the self-adjointness of these operators and derive equivalent formulations for the eigenvalue problems involving boundary integral operators. These formulations are suitable for the numerical computations of the discrete eigenvalues and the corresponding eigenfunctions by boundary element methods. We provide convergence results and show numerical examples.
