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We provide an elementary derivation of the Bessel analog of the celebrated Riesz composition formula and use the former to effortlessly derive the latter.

We consider the generalized eigenvalue problem for the classical Euler differential equation and demonstrate its intimate connection with Meijer’s G-functions. In the course of deriving the solution of the generalized Euler eigenvalue equation we review some of the basics of generalized hypergeometric functions and Meijer’s G-functions and some of its special cases where the underlying Mellin-type integrand exhibits higher-order poles.

The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy–Rellich inequality on n-dimensional balls, valid for the largest variety of underlying parameters and for all dimensions n∈N, n≥2.

Using the variational characterization of the principal (i.e., smallest) eigenvalue below the essential spectrum of a lower semibounded self-adjoint operator, we prove strict domain monotonicity (with respect to changing the finite interval length) of the principal eigenvalue of the Friedrichs extension TFof the minimal operator for regular four-coefficient Sturm–Liouville differential expressions. In the more general singular context, these four-coefficient differential expressions act according to τf=1r(-(f[1])′+sf[1]+qf)withf[1]=p[f′+sf]on(a,b)⊆R,where the coefficients p, q, r, s are real-valued and Lebesgue measurable on (a, b), with p > 0, r > 0 a.e. on (a, b), and p-1,q,r,s∈Lloc1((a,b);dx), and fis supposed to satisfy f∈ACloc((a,b)),p[f′+sf]∈ACloc((a,b)).This setup is sufficiently general so that τpermits certain distributional potential coefficients q, including potentials in Hloc-1((a,b)).

The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension Tof a symmetric operator Sin a Hilbert space $$\mathfrak {H}$$ H , employing the technique of quasi boundary triples for T. The general results are illustrated with couplings of Schrödinger operators on Lipschitz domains on smooth, boundaryless, compact Riemannian manifolds.

Assuming T0to be an m‐accretive operator in the complex Hilbert space H, we use a resolvent method due to Kato to appropriately define the additive perturbation T=T0+Wand prove stability of square root domains, that is, dom((T0+W)1/2)=dom(T01/2).Moreover, assuming in addition that dom(T01/2)=dom((T0*)1/2), we prove stability of square root domains in the form dom((T0+W)1/2)=dom(T01/2)=dom((T0*)1/2)=dom(((T0+W)*)1/2),which is most suitable for partial differential equation applications. We apply this approach to elliptic second‐order partial differential operators of the form −div(a∇⋅)+(B→1⋅∇⋅)+div(B→2⋅)+Vin L2(Ω)on certain open sets Ω⊆Rn, n∈N, with Dirichlet, Neumann, and mixed boundary conditions on ∂Ω, under general hypotheses on the (typically, non‐smooth, unbounded) coefficients and on ∂Ω.

Assuming $T_{0}$to be an m-accretive operator in the complex Hilbert space ${\mathcal{H}}$, we use a resolvent method due to Kato to appropriately define the additive perturbation $T=T_{0}+W$and prove stability of square root domains, that is, $$\begin{eqnarray}\text{dom}((T_{0}+W)^{1/2})=\text{dom}(T_{0}^{1/2}).\end{eqnarray}$$Moreover, assuming in addition that $\text{dom}(T_{0}^{1/2})=\text{dom}((T_{0}^{\ast })^{1/2})$, we prove stability of square root domains in the form $$\begin{eqnarray}\text{dom}((T_{0}+W)^{1/2})=\text{dom}(T_{0}^{1/2})=\text{dom}((T_{0}^{\ast })^{1/2})=\text{dom}(((T_{0}+W)^{\ast })^{1/2}),\end{eqnarray}$$which is most suitable for partial differential equation applications. We apply this approach to elliptic second-order partial differential operators of the form $$\begin{eqnarray}-\text{div}(a{\rm\nabla}\,\cdot )+(\vec{B}_{1}\cdot {\rm\nabla}\,\cdot )+\text{div}(\vec{B}_{2}\,\cdot )+V\end{eqnarray}$$in $L^{2}({\rm\Omega})$on certain open sets ${\rm\Omega}\subseteq \mathbb{R}^{n}$, $n\in \mathbb{N}$, with Dirichlet, Neumann, and mixed boundary conditions on $\partial {\rm\Omega}$, under general hypotheses on the (typically, non-smooth, unbounded) coefficients and on $\partial {\rm\Omega}$.

A Borg-type uniqueness theorem for matrix-valued Schrödinger operators is proved. More precisely, assuming a reflectionless potential matrix and its spectrum a half-line [0, ∞), we derive the triviality of the potential matrix. Our approach is based on trace formulas and matrix-valued Herglotz representation theorems. As a by-product of our techniques, we obtain an extension of Borg's classical result from the class of periodic scalar potentials to the class of reflectionless matrix-valued potentials.

Abstract:: We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl–Titchmarsh m-functions, m j (z), of two Schr�dinger operators , j≡ 1,2 in L 2((0,R)), 01≡q 2 a.e. on [0,a]. The result applies to any boundary conditions at x≡ 0 and x≡R and should be considered a local version of the celebrated Borg–Marchenko uniqueness result (which is quickly recovered as a corollary to our proof). Moreover, we extend the local uniqueness result to matrix-valued Schr�dinger operators.

The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and ζ-functions to efficiently compute values of spectral ζ-functions at positive integers associated with regular (three-coefficient) self-adjoint Sturm–Liouville differential expressions τ. Depending on the underlying boundary conditions, we express the ζ-function values in terms of a fundamental system of solutions of τy=zyand their expansions about the spectral point z=0. Furthermore, we give the full analytic continuation of the ζ-function through a Liouville transformation and provide an explicit expression for the ζ-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 Schrödinger operators with zero, piecewise constant, and a linear potential on a compact interval.

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We provide additional results in connection with Krein's formula, which describes the resolvent difference of two self-adjoint extensionsA1andA2of a densely defined closed symmetric linear operatorȦwith deficiency indices (n,n),n∈N∪{∞}. In particular, we explicitly derive the linear fractional transformation relating the operator-valued Weyl–TitchmarshM-functionsM1(z) andM2(z) corresponding toA1andA2.