Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPerturbations of periodic Sturm--Liouville operators
216
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study perturbations of the self-adjoint periodic Sturm--Liouville operator [ A_0 = frac{1}{r_0}left(-frac{mathrm d}{mathrm dx} p_0 frac{mathrm d}{mathrm dx} + q_0right) ] and conclude under $L^1$-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.
rate research
Read More
We extend the classical boundary values begin{align*} & g(a) = - W(u_{a}(lambda_0,.), g)(a) = lim_{x downarrow a} frac{g(x)}{hat u_{a}(lambda_0,x)}, &g^{[1]}(a) = (p g)(a) = W(hat u_{a}(lambda_0,.), g)(a) = lim_{x downarrow a} frac{g(x) - g(a) hat u_{a}(lambda_0,x)}{u_{a}(lambda_0,x)} end{align*} for regular Sturm-Liouville operators associated with differential expressions of the type $tau = r(x)^{-1}[-(d/dx)p(x)(d/dx) + q(x)]$ for a.e. $xin[a,b] subset mathbb{R}$, to the case where $tau$ is singular on $(a,b) subseteq mathbb{R}$ and the associated minimal operator $T_{min}$ is bounded from below. Here $u_a(lambda_0, cdot)$ and $hat u_a(lambda_0, cdot)$ denote suitably normalized principal and nonprincipal solutions of $tau u = lambda_0 u$ for appropriate $lambda_0 in mathbb{R}$, respectively. We briefly discuss the singular Weyl-Titchmarsh-Kodaira $m$-function and finally illustrate the theory in some detail with the examples of the Bessel, Legendre, and Kummer (resp., Laguerre) operators.
The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $zeta$-functions to efficiently compute values of spectral $zeta$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $tau$. Depending on the underlying boundary conditions, we express the $zeta$-function values in terms of a fundamental system of solutions of $tau y = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $zeta$-function through a Liouville transformation and provide an explicit expression for the $zeta$-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schr{o}dinger operators with zero, piecewise constant, and a linear potential on a compact interval.
Let $dot A$ be a densely defined, closed, symmetric operator in the complex, separable Hilbert space $mathcal{H}$ with equal deficiency indices and denote by $mathcal{N}_i = ker big(big(dot Abig)^* - i I_{mathcal{H}}big)$, $dim , (mathcal{N}_i)=kin mathbb{N} cup {infty}$, the associated deficiency subspace of $dot A$ . If $A$ denotes a self-adjoint extension of $dot A$ in $mathcal{H}$, the Donoghue $m$-operator $M_{A,mathcal{N}_i}^{Do} (, cdot ,)$ in $mathcal{N}_i$ associated with the pair $(A,mathcal{N}_i)$ is given by [ M_{A,mathcal{N}_i}^{Do}(z)=zI_{mathcal{N}_i} + (z^2+1) P_{mathcal{N}_i} (A - z I_{mathcal{H}})^{-1} P_{mathcal{N}_i} bigvert_{mathcal{N}_i},, quad zin mathbb{C} backslash mathbb{R}, ] with $I_{mathcal{N}_i}$ the identity operator in $mathcal{N}_i$, and $P_{mathcal{N}_i}$ the orthogonal projection in $mathcal{H}$ onto $mathcal{N}_i$. Assuming the standard local integrability hypotheses on the coefficients $p, q,r$, we study all self-adjoint realizations corresponding to the differential expression [ tau=frac{1}{r(x)}left[-frac{d}{dx}p(x)frac{d}{dx} + q(x)right] , text{ for a.e. $xin(a,b) subseteq mathbb{R}$,} ] in $L^2((a,b); rdx)$, and, as the principal aim of this paper, systematically construct the associated Donoghue $m$-functions (resp., $2 times 2$ matrices) in all cases where $tau$ is in the limit circle case at least at one interval endpoint $a$ or $b$.
We examine the spectrum of a family of Sturm--Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described in a previor result of the last two authors with Berkolaiko, where it was used to study the nodal deficiency of Laplacian eigenfunctions. Here we consider the eigenfunctions of these operators. In particular, we give explicit formulas for the limiting eigenfunctions, and also characterize the eigenfunctions and eigenvalues for all values for the spectral flow parameter (not just in the limit). We also develop spectrally accurate numerical tools for comparison and visualization.
We derive explicit Krein resolvent identities for generally singular Sturm-Liouville operators in terms of boundary condition bases and the Lagrange bracket. As an application of the resolvent identities obtained, we compute the trace of the resolvent difference of a pair of self-adjoint realizations of the Bessel expression $-d^2/dx^2+( u^2-(1/4))x^{-2}$ on $(0,infty)$ for values of the parameter $ uin[0,1)$ and use the resulting trace formula to explicitly determine the spectral shift function for the pair.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
