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Register a new userExplicit Krein Resolvent Identities for Singular Sturm-Liouville Operators with Applications to Bessel Operators
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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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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 spectrum of the singular indefinite Sturm-Liouville operator $$A=text{rm sgn}(cdot)bigl(-tfrac{d^2}{dx^2}+qbigr)$$ with a real potential $qin L^1(mathbb R)$ covers the whole real line and, in addition, non-real eigenvalues may appear if the potential $q$ assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction were obtained. In this paper the bound $$|lambda|leq |q|_{L^1}^2$$ on the absolute values of the non-real eigenvalues $lambda$ of $A$ is obtained. Furthermore, separate bounds on the imaginary parts and absolute values of these eigenvalues are proved in terms of the $L^1$-norm of the negative part of $q$.
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.
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.
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.
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