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Register a new userOn self-adjoint boundary conditions for singular Sturm-Liouville operators bounded from below
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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.
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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.
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 $b$.
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.
Considering singular Sturm--Liouville differential expressions of the type [ tau_{alpha} = -(d/dx)x^{alpha}(d/dx) + q(x), quad x in (0,b), ; alpha in mathbb{R}, ] we employ some Sturm comparison-type results in the spirit of Kurss to derive criteria for $tau_{alpha}$ to be in the limit point and limit circle case at $x=0$. More precisely, if $alpha in mathbb{R}$ and for $0 < x$ sufficiently small, [ q(x) geq [(3/4)-(alpha/2)]x^{alpha-2}, ] or, if $alphain (-infty,2)$ and there exist $Ninmathbb{N}$, and $varepsilon>0$ such that for $0<x$ sufficiently small, begin{align*} &q(x)geq[(3/4)-(alpha/2)]x^{alpha-2} - (1/2) (2 - alpha) x^{alpha-2} sum_{j=1}^{N}prod_{ell=1}^{j}[ln_{ell}(x)]^{-1} &quadquadquad +[(3/4)+varepsilon] x^{alpha-2}[ln_{1}(x)]^{-2}. end{align*} then $tau_{alpha}$ is nonoscillatory and in the limit point case at $x=0$. Here iterated logarithms for $0 < x$ sufficiently small are of the form, [ ln_1(x) = |ln(x)| = ln(1/x), quad ln_{j+1}(x) = ln(ln_j(x)), quad j in mathbb{N}. ] Analogous results are derived for $tau_{alpha}$ to be in the limit circle case at $x=0$. We also discuss a multi-dimensional application to partial differential expressions of the type [ - Div |x|^{alpha} abla + q(|x|), quad alpha in mathbb{R}, ; x in B_n(0;R) backslash{0}, ] with $B_n(0;R)$ the open ball in $mathbb{R}^n$, $nin mathbb{N}$, $n geq 2$, centered at $x=0$ of radius $R in (0, infty)$.
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.
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