We prove an explicit formula for the spectral expansions in $L^2(\R)$ generated by selfadjoint differential operators $$ (-1)^n\frac{d^{2n}}{dx^{2n}}+\sum\limits_{j=0}^{n-1}\frac{d^{j}}{dx^{j}} p_j(x)\frac{d^{j}}{dx^{j}},\quad p_j(x+π)=p_j(x),\quad x\in\R. $$
Analogs of generalized resolvents and eigenfunction expansions of relations generated by pair of differential operator expressions one of which depends on spectral parameter in nonlinear manner
We consider an arbitrary selfadjoint operator on a separable Hilbert space. To this operator we construct an expansion in generalized eigenfunctions in which the original Hilbert space is decomposed as a direct integral of Hilbert spaces consisting of general eigenfunctions. This automatically gives a Plancherel type formula. For suitable operators on metric measure spaces we discuss some growth restrictions on the generalized eigenfunctions. For Laplacians on locally finite graphs the generalized eigenfunctions are exactly the solutions of the corresponding difference equation.
In this paper we analyse the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds, continuing the research in our previous paper. We prove that such spaces of Fourier coefficients are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on spaces of Fourier coefficients and characterise their adjoint mappings. In particular, the considered classes include spaces of analytic and Gevrey functions, as well as spaces of ultradistributions, yielding tensor representations for linear mapp…
In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.
We consider the Laplacian on a periodic metric graph and obtain its decomposition into a direct fiber integral in terms of the corresponding discrete Laplacian. Eigenfunctions and eigenvalues of the fiber metric Laplacian are expressed explicitly in terms of eigenfunctions and eigenvalues of the corresponding fiber discrete Laplacian and eigenfunctions of the Dirichlet problem on the unit interval. We show that all these eigenfunctions are uniformly bounded. We apply these results to the periodic metric Laplacian perturbed by real integrable potentials. We prove the following: a) the wave oper…
Uniform convergence of the expansion of an absolutely continuous function for eigenfunctions of the Sturm-Liouville problem $-y" + q \left( x \right) y = μy,$ $y \left(0\right)=0,$ $y\left( π\right)\cos β+ y'\left( π\right)\sin β= 0,$ $β\in \left( 0, π\right)$ with summable potential $q \in L_{\mathbb{R}}^1 \left[0, π\right]$ is proved. This result is used to obtain more precise asymptotic formulae for eigenvalues and norming constants of this problem.
We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called ``pockets''. We prove there are only finitely many immersed periodic tubes missing the pockets and moreover establish a new quantitative estimate for the lengths of such tubes. This extends well-known results in dimension $2$. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results on irrational polyh…
Following a number of recent studies of resolvent and spectral convergence of non-uniformly elliptic families of differential operators describing the behaviour of periodic composite media with high contrast, we study the corresponding one-dimensional version that includes a "defect": an inclusion of fixed size with a given set of material parameters. It is known that the spectrum of the purely periodic case without the defect and its limit, as the period $\varepsilon$ goes to zero, has a band-gap structure. We consider a sequence of eigenvalues $λ_\varepsilon$ that are induced by the defect a…
The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. For a high contrast small $\ve$-size periodicity and a finite size defect we consider the critical $\ve^2$-scaling for the contrast. We employ (high contrast) two-scale homogenization for deriving asymptotically explicit limit equations for the localised modes (exponentially decaying eigenfunctions) and associated eigenvalues. Those are expressed in terms of the eigenvalues and eigenfunctions of a perturbed version of a two-scale limit operator introduced by V.V. Zhikov with an emergent explicit n…
We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic equation with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the $j$th eigenfunction can be approximated by a rescaled function that is constructed in terms of the $j$th eigenfunction of fourth or second order order effective opera…
We consider the one-dimensional Schrödinger equation $-f"+q_κf = Ef$ on the positive half-axis with the potential $q_κ(r)=(κ^2-1/4)r^{-2}$. For each complex number $\vartheta$, we construct a solution $u^κ_\vartheta(E)$ of this equation that is analytic in $κ$ in a complex neighborhood of the interval $(-1,1)$ and, in particular, at the "singular" point $κ= 0$. For $-1<κ<1$ and real $\vartheta$, the solutions $u^κ_\vartheta(E)$ determine a unitary eigenfunction expansion operator $U_{κ,\vartheta}\colon L_2(0,\infty)\to L_2(\mathbb R,\mathcal V_{κ,\vartheta})$, where $\mathcal V_{κ,\vartheta}$ …
We study the spectral expansion of the semigroup of a general stable process killed on the first exit from the positive half-line. Starting with the Wiener-Hopf factorization we obtain the q-resolvent density for the killed process, from which we derive the spectral expansion of the semigroup via the inverse Laplace transform. The eigenfunctions and co-eigenfunctions are given rather explicitly in terms of the double sine function and they give rise to a pair of integral transforms which generalize the classical Fourier sine transform. Our results provide the first explicit example of a spectr…
We consider the Schrödinger operator on a combinatorial graph consisting of a finite graph and a finite number of discrete half-lines, all jointed together, and compute an asymptotic expansion of its resolvent around the threshold $0$. Precise expressions are obtained for the first few coefficients of the expansion in terms of the generalized eigenfunctions. This result justifies the classification of threshold types solely by growth properties of the generalized eigenfunctions. By choosing an appropriate free operator a priori possessing no zero eigenvalue or zero resonance we can simplify th…
We consider a Schrödinger operator $H=-Δ+V(\vec x)$ in dimension two with a quasi-periodic potential $V(\vec x)$. We prove that the absolutely continuous spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i\langle \vec \varkappa,\vec x\rangle}$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\vec \varkappa$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor t…
We analyze a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well potential and $\varepsilon$ is a small parameter. We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of $H_\varepsilon$. These are obtained from eigenfunctions or quasimodes for the operator $H_\varepsilon$, acting on $L^2(\mathbb{R}^d)$, via restriction to the lattice $\varepsilon\mathbb{Z}^d$.
Scattering problems in periodic waveguides are interesting but also challenging topics in mathematics, both theoretically and numerically. Due to the existence of eigenvalues, the unique solvability of these problems is not always guaranteed. To obtain a unique solution that is "physically meaningful", the limiting absorption principle (LAP) is a commonly used method. LAP assumes that the limit of a family of solutions with absorbing media converges, as the absorption parameter tends to 0, and the limit is the "physically meaningful solution". It is also called the LAP solution in this paper. …
We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blo…
In this paper we investigate the spectral expansion for the asymptotically spectral differential operators generated in all real line by ordinary differential expression of arbitrary order with periodic matrix-valued coefficients
