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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., Comment: 21 pages

We study convergence of the spectral shift function for the finite interval restrictions of a pair of full-line Schr\"odinger operators to an interval of the form $(-\ell,\ell)$ with coupled boundary conditions at the endpoints as $\ell\to \infty$ in the case when the finite interval restrictions are relatively prime to those with Dirichlet boundary conditions. Using a Krein-type resolvent identity we show that the spectral shift function for the finite interval restrictions converges weakly to that for the pair of full-line Schr\"odinger operators as the length of the interval tends to infinity., Comment: 23 pages

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 $T_F$ of 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 \[ \tau f = \frac{1}{r} \left( - \big(f^{[1]}\big)' + s f^{[1]} + qf\right)\,\text{ with $f^{[1]} = p [f' + s f]$ on $(a,b) \subseteq \mathbb{R}$}, \] 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 $p^{-1}$, $q$, $r$, $s \in L^1_{loc}((a,b); dx)$, and $f$ is supposed to satisfy \[ f \in AC_{loc}((a,b)), \; p[f' + s f] \in AC_{loc}((a,b)). \] This setup is sufficiently general so that $\tau$ permits certain distributional potential coefficients $q$, including potentials in $H^{-1}_{loc}((a,b))$. As a consequence of the strict domain monotonicity of the principal eigenvalue of the Friedrichs extension in the regular case, and on the basis of oscillation theory in the singular context, in our main result, we characterize all lower bounds of $T_F$ as those $\lambda\in \mathbb{R}$ for which the differential equation $\tau u = \lambda u$ has a strictly positive solution $u > 0$ on $(a,b)$., Comment: 26 pages, some references added. arXiv admin note: text overlap with arXiv:1208.4677

We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms of a specially chosen basis for the kernel of the maximal operator and employs a description of the Friedrichs extension due to M\"oller and Zettl.

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 $\mathcal{N}_i = \ker \big(\big(\dot A\big)^* - i I_{\mathcal{H}}\big)$, $\dim \, (\mathcal{N}_i)=k\in \mathbb{N} \cup \{\infty\}$, 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 \[ M_{A,\mathcal{N}_i}^{Do}(z)=zI_{\mathcal{N}_i} + (z^2+1) P_{\mathcal{N}_i} (A - z I_{\mathcal{H}})^{-1} P_{\mathcal{N}_i} \big\vert_{\mathcal{N}_i}\,, \quad z\in \mathbb{C} \backslash \mathbb{R}, \] 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 \[ \tau=\frac{1}{r(x)}\left[-\frac{d}{dx}p(x)\frac{d}{dx} + q(x)\right] \, \text{ for a.e. $x\in(a,b) \subseteq \mathbb{R}$,} \] 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 $b$., Comment: 35 pages. arXiv admin note: text overlap with arXiv:1910.13117, arXiv:2102.00685

Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrodinger operators with sufficiently short-range interactions in general space dimensions. More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions $n \geq 3$, the absence of resonances at $\pm m$ for massive Dirac operators (with mass $m > 0$) in dimensions $n \geq 5$, and recall the well-known case of absence of zero-energy resonances for Schr\"odinger operators in dimension $n \geq 5$., Comment: 34 pages. arXiv admin note: substantial text overlap with arXiv:2105.03024

We derive a limiting absorption principle on any compact interval in $\mathbb{R} \backslash \{0\}$ for the free massless Dirac operator, $H_0 = \alpha \cdot (-i \nabla)$ in $[L^2(\mathbb{R}^n)]^N$, $n \geq 2$, $N=2^{\lfloor(n+1)/2\rfloor}$, and then prove the absence of singular continuous spectrum of interacting massless Dirac operators $H = H_0 +V$, where $V$ decays like $O(|x|^{-1 - \varepsilon})$. Expressing the spectral shift function $\xi(\,\cdot\,; H,H_0)$ as normal boundary values of regularized Fredholm determinants, we prove that for sufficiently decaying $V$, $\xi(\,\cdot\,;H,H_0) \in C((-\infty,0) \cup (0,\infty))$, and that the left and right limits at zero, $\xi(0_{\pm}; H,H_0)$, exist. Introducing the non-Fredholm operator $\boldsymbol{D}_{\boldsymbol{A}} = \frac{d}{dt} + \boldsymbol{A}$ in $L^2\big(\mathbb{R};[L^2(\mathbb{R}^n)]^N\big)$, where $\boldsymbol{A} = \boldsymbol{A_-} + \boldsymbol{B}$, $\boldsymbol{A_-}$, and $\boldsymbol{B}$ are generated in terms of $H, H_0$ and $V$, via $A(t) = A_- + B(t)$, $A_- = H_0$, $B(t)=b(t) V$, $t \in \mathbb{R}$, assuming $b$ is smooth, $b(-\infty) = 0$, $b(+\infty) = 1$, and introducing $\boldsymbol{H_1} = \boldsymbol{D}_{\boldsymbol{A}}^{*} \boldsymbol{D}_{\boldsymbol{A}}$, $\boldsymbol{H_2} = \boldsymbol{D}_{\boldsymbol{A}} \boldsymbol{D}_{\boldsymbol{A}}^{*}$, one of the principal results in this manuscript expresses the $k$th resolvent regularized Witten index $W_{k,r}(\boldsymbol{D}_{\boldsymbol{A}})$ ($k \in \mathbb{N}$, $k \geq \lceil n/2 \rceil$) in terms of spectral shift functions as \[ W_{k,r}(\boldsymbol{D}_{\boldsymbol{A}}) = \xi(0_+; \boldsymbol{H_2}, \boldsymbol{H_1}) = [\xi(0_+;H,H_0) + \xi(0_-;H,H_0)]/2. \] Here $L^2(\mathbb{R};\mathcal{H}) = \int_{\mathbb{R}}^{\oplus} dt \, \mathcal{H}$ and $\boldsymbol{T} = \int_{\mathbb{R}}^{\oplus} dt \, T(t)$ abbreviate direct integrals., Comment: 158 pages

We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint., Comment: 26 pages. arXiv admin note: text overlap with arXiv:1910.13117

We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type \[ \|A f\|_{\mathcal{X}}^2 \leq C \|f\|_{\mathcal{X}} \big\|A^2 f\big\|_{\mathcal{X}}, \quad f \in dom\big(A^2\big), \] and recall that under exceedingly stronger hypotheses on the operator $A$ and/or the Banach space $\mathcal{X}$, the optimal constant $C$ in these inequalities diminishes from $4$ (e.g., when $A$ is the generator of a $C_0$ contraction semigroup on a Banach space $\mathcal{X}$) all the way down to $1$ (e.g., when $A$ is a symmetric operator on a Hilbert space $\mathcal{H}$). We also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt., Comment: 28 pages, some updates added

For trace class operators $A, B \in \mathcal{B}_1(\mathcal{H})$ ($\mathcal{H}$ a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form \[ {\det}_{\mathcal{H}} ((I_{\mathcal{H}} - A) (I_{\mathcal{H}} - B)) = {\det}_{\mathcal{H}} (I_{\mathcal{H}} - A) {\det}_{\mathcal{H}} (I_{\mathcal{H}} - B). \] When trace class operators are replaced by Hilbert--Schmidt operators $A, B \in \mathcal{B}_2(\mathcal{H})$ and the Fredholm determinant ${\det}_{\mathcal{H}}(I_{\mathcal{H}} - A)$, $A \in \mathcal{B}_1(\mathcal{H})$, by the 2nd regularized Fredholm determinant ${\det}_{\mathcal{H},2}(I_{\mathcal{H}} - A) = {\det}_{\mathcal{H}} ((I_{\mathcal{H}} - A) \exp(A))$, $A \in \mathcal{B}_2(\mathcal{H})$, the product formula must be replaced by \[ {\det}_{\mathcal{H},2} ((I_{\mathcal{H}} - A) (I_{\mathcal{H}} - B)) = {\det}_{\mathcal{H},2} (I_{\mathcal{H}} - A) {\det}_{\mathcal{H},2} (I_{\mathcal{H}} - B) \exp(- {\rm tr}(AB)). \] The product formula for the case of higher regularized Fredholm determinants ${\det}_{\mathcal{H},k}(I_{\mathcal{H}} - A)$, $A \in \mathcal{B}_k(\mathcal{H})$, $k \in \mathbb{N}$, $k \geq 2$, does not seem to be easily accessible and hence this note aims at filling this gap in the literature., Comment: 10 pages, reference added, extended proofs in Sect. 2 a bit

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. $x\in[a,b] \subset \mathbb{R}$, 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., Comment: 38 pages

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 $-d^2/dx^2+(\nu^2-(1/4))x^{-2}$ on $(0,\infty)$ for values of the parameter $\nu\in[0,1)$ and use the resulting trace formula to explicitly determine the spectral shift function for the pair., Comment: 50 pages

We prove a global limiting absorption principle on the entire real line for free, massless Dirac operators $H_0 = \alpha \cdot (-i \nabla)$ for all space dimensions $n \in \mathbb{N}$, $n \geq 2$. This is a new result for all dimensions other than three, in particular, it applies to the two-dimensional case which is known to be of some relevance in applications to graphene. We also prove an essential self-adjointness result for first-order matrix-valued differential operators with Lipschitz coefficients., Comment: 22 pages

We introduce a generalized index for certain meromorphic, unbounded, operator-valued functions. The class of functions is chosen such that energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schr\"odinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl-Titchmarsh $M$-functions and Donoghue-type $M$-functions corresponding to closed extensions of symmetric operators belong to it. The principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of Robin realizations of non-self-adjoint Schr\"{o}dinger operators, and more abstract pairs of closed operators in Hilbert spaces with the generalized index of the corresponding energy dependent Dirichlet-to-Neumann maps and abstract Weyl-Titchmarsh $M$-functions, respectively., Comment: A natural continuation of our arXiv:1512.06962 submission

We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces. Applications to abstract perturbation theory and associated Birman-Schwinger-type operators and to the operator-valued Weyl-Titchmarsh functions associated to closed extensions of dual pairs of closed operators are provided., Comment: 24 pages, some corrections and updates were made. This paper revisits and extends some parts of arXiv:1405.4910

We derive two main results: First, assume that $A$, $B$, $A_n$, $B_n$ are self-adjoint operators in the Hilbert space $\mathcal{H}$, and suppose that $A_n$ converges to $A$ and $B_n$ to $B$ in strong resolvent sense as $n \to \infty$. Fix $m \in \mathbb{N}$, $m$ odd, $p \in [1,\infty)$, and assume that $T:= \big[( A + iI_{\mathcal{H}})^{-m} - ( B + iI_{\mathcal{H}})^{-m}\big] \in \mathcal{B}_p(\mathcal{H})$, $T_n := \big[( A_n + iI_{\mathcal{H}})^{-m} - ( B_n + iI_{\mathcal{H}})^{-m}\big] \in \mathcal{B}_p(\mathcal{H})$, and $\lim_{n \rightarrow \infty} \|T_n - T\|_{\mathcal{B}_p(\mathcal{H})} =0$. Then for any function $f$ in the class $\mathfrak F_{k}(\mathbb{R}) \supset C_0^{\infty}(\mathbb{R})$ (cf. (1.1)), $$ \lim_{n \rightarrow \infty} \big\| [f(A_n) - f(B_n)] - [f(A)- f(B)]\big\|_{\mathcal{B}_p(\mathcal{H})}=0. $$ Our second result concerns the continuity of spectral shift functions $\xi(\cdot; B,B_0)$ with respect to the operator parameter $B$. For $T$ self-adjoint in $\mathcal{H}$ we denote by $\Gamma_m(T)$, $m \in \mathbb{N}$ odd, the set of all self-adjoint operators $S$ in $\mathcal{H}$ satisfying $\big[(S - z I_{\mathcal{H}})^{-m} - (T - z I_{\mathcal{H}})^{-m}\big] \in \mathcal{B}_1(\mathcal{H})$, $z \in \mathbb{C}\backslash \mathbb{R}$. Employing a suitable topology on $\Gamma_m(T)$ (cf. (1.9), we prove the following: Suppose that $B_1\in \Gamma_m(B_0)$ and let $\{B_{\tau}\}_{\tau\in [0,1]}\subset \Gamma_m(B_0)$ denote a path from $B_0$ to $B_1$ in $\Gamma_m(B_0)$ depending continuously on $\tau\in [0,1]$ with respect to the topology on $\Gamma_m(B_0)$. If $f \in L^{\infty}(\mathbb{R})$, then $$ \lim_{\tau\to 0^+} \|\xi(\, \cdot \, ; B_{\tau}, A_0) f - \xi(\, \cdot \, ; B_0, A_0) f\|_{L^1(\mathbb{R}; (|\nu|^{m+1} + 1)^{-1}d\nu)} = 0. $$, Comment: 23 pages; we fixed an error in the original Theorem 2.12 which lead to considerable changes throughout this manuscript

Using the Davies-Helffer-Sj\"ostrand functional calculus based on almost analytic extensions, we address the following problem: Given self-adjoint operators $S_j$, $j=1,2$, in $\mathcal{H}$, and functions $f$ in an appropriate class, for instance, $f \in C_0^{\infty}(\mathbb{R})$, how to control the norm $\|f(S_2) - f(S_1)\|_{\mathcal{B}(\mathcal{H})}$ in terms of the norm of the difference of resolvents, $\|(S_2 - z_0 I_{\mathcal{H}})^{-1} - (S_2 - z_0 I_{\mathcal{H}})^{-1}\|_{\mathcal{B}(\mathcal{H})}$, for some $z_0 \in \mathbb{C}\backslash\mathbb{R}$. We are particularly interested in the case where $\mathcal{B}(\mathcal{H})$ is replaced by a trace ideal, $\mathcal{B}_p(\mathcal{H})$, $p \in [1,\infty)$., Comment: 20 pages, references and Remark 3.3 got updated

We discuss stability of square root domains for uniformly elliptic partial differential operators $L_{a,\Omega,\Gamma} = -\nabla\cdot a \nabla$ in $L^2(\Omega)$, with mixed boundary conditions on $\partial \Omega$, with respect to additive perturbations. We consider open, bounded, and connected sets $\Omega \in \mathbb{R}^n$, $n \in \mathbb{N} \backslash\{1\}$, that satisfy the interior corkscrew condition and prove stability of square root domains of the operator $L_{a,\Omega,\Gamma}$ with respect to additive potential perturbations $V \in L^p(\Omega) + L^{\infty}(\Omega)$, $p>n/2$. Special emphasis is put on the case of uniformly elliptic systems with mixed boundary conditions., Comment: 13 pages, to appear in Journal of Differential Equations

We consider additive perturbations of the type $K_t=K_0+tW$, $t\in [0,1]$, where $K_0$ and $W$ are self-adjoint operators in a separable Hilbert space $\mathcal{H}$ and $W$ is bounded. In addition, we assume that the range of $W$ is a generating (i.e., cyclic) subspace for $K_0$. If $\lambda_0$ is an eigenvalue of $K_0$, then under the additional assumption that $W$ is nonnegative, the Lebesgue measure of the set of all $t\in [0,1]$ for which $\lambda_0$ is an eigenvalue of $K_t$ is known to be zero. We recall this result with its proof and show by explicit counterexample that the nonnegativity assumption $W\geq 0$ cannot be removed., Comment: 10 pages; added Lemma 2.4, typos removed; to appear in J. Math. Anal. Appl

We study several natural multiplicity questions that arise in the context of the Birman-Schwinger principle applied to non-self-adjoint operators. In particular, we re-prove (and extend) a recent result by Latushkin and Sukhtyaev by employing a different technique based on factorizations of analytic operator-valued functions due to Howland. Factorizations of analytic operator-valued functions are of particular interest in themselves and again we re-derive Howland's results and subsequently extend them. Considering algebraic multiplicities of finitely meromorphic operator-valued functions, we recall the notion of the index of a finitely meromorphic operator-valued function and use that to prove an analog of the well-known Weinstein-Aronszajn formula relating algebraic multiplicities of the underlying unperturbed and perturbed operators. Finally, we consider pairs of projections for which the difference belongs to the trace class and relate their Fredholm index to the index of the naturally underlying Birman-Schwinger operator., Comment: 31 pages, corrected and extended the 1st paragraph in the proof of Theorem 4.5 and Theorem 5.5

We study the analog of semi-separable integral kernels in $\mathcal{H}$ of the type $$ K(x,x')=\begin{cases} F_1(x)G_1(x'), & a

The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schr\"odinger-type operators on a half-line from the underlying Weyl-Titchmarsh function., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1206.4966

The main objective of this paper is to identify principal solutions associated with Sturm-Liouville operators on arbitrary open intervals $(a,b) \subseteq \mathbb{R}$, as introduced by Leighton and Morse in the scalar context in 1936 and by Hartman in the matrix-valued situation in 1957, with Weyl-Titchmarsh solutions, as long as the underlying Sturm-Liouville differential expression is nonoscillatory (resp., disconjugate or bounded from below near an endpoint) and in the limit point case at the endpoint in question. In addition, we derive an explicit formula for Weyl-Titchmarsh functions in this case (the latter appears to be new in the matrix-valued context)., Comment: 27 pages, expanded Sect. 2, added references

The principal aim of this short note is to extend a recent result on Gaussian heat kernel bounds for self-adjoint $L^2(\Om; d^n x)$-realizations, $n\in\bbN$, $n\geq 2$, of divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$, where $$ Lu = - \sum_{j,k=1}^n\partial_j a_{j,k}\partial_k u. $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ \nu\cdot A\nabla u + \Theta\big[u\big|_{\partial\Om}\big]=0 \, \text{on} \, \partial\Omega, $$ where $\Theta$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Om$ of $\Om$, and $\nu$ denotes the outward pointing normal unit vector on $\partial\Om$., Comment: 14 pages

We determine square root domains for non-self-adjoint Sturm--Liouville operators of the type $$ L_{p,q,r,s} = - \frac{d}{dx}p\frac{d}{dx}+r\frac{d}{dx}-\frac{d}{dx}s+q $$ in $L^2((c,d);dx)$, where either $(c,d)$ coincides with the real line $\mathbb{R}$, the half-line $(a,\infty)$, $a \in \mathbb{R}$, or with the bounded interval $(a,b) \subset \mathbb{R}$, under very general conditions on the coefficients $q, r, s$. We treat Dirichlet and Neumann boundary conditions at $a$ in the half-line case, and Dirichlet and/or Neumann boundary conditions at $a,b$ in the final interval context. (In the particular case $p=1$ a.e.\ on $(a,b)$, we treat all separated boundary conditions at $a, b$.), Comment: 36 pages

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, $$ dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big). $$ Moreover, assuming in addition that $dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big)$, we prove stability of square root domains in the form $$dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big) = dom\big(((T_0 + W)^*)^{1/2}\big), $$ which is most suitable for PDE applications. We apply this approach to elliptic second-order partial differential operators of the form $$ - div(a\nabla \, \cdot \,) + \big(\mathbf{B}_1\cdot \nabla \cdot \big) + div \big(\mathbf{B}_2 \cdot \big) + V $$ in $L^2(\Omega)$ on certain open sets $\Omega \subseteq \mathbb{R}^n$, $n \in \mathbb{N}$, with Dirichlet, Neumann, and mixed boundary conditions on $\partial \Omega$, under general hypotheses on the (typically, nonsmooth, unbounded) coefficients and on $\partial\Omega$., Comment: 61 pages, to appear in Mathematika

We discuss inverse spectral theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[\tau f = \frac{1}{r} \left(- \big(p[f' + s f]\big)' + s p[f' + s f] + qf\right), \] where the coefficients $p$, $q$, $r$, $s$ are Lebesgue measurable on $(a,b)$ with $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$ and real-valued with $p\not=0$ and $r>0$ a.e.\ on $(a,b)$. In particular, we explicitly permit certain distributional potential coefficients. The inverse spectral theory results derived in this paper include those implied by the spectral measure, by two-spectra and three-spectra, as well as local Borg-Marchenko-type inverse spectral results. The special cases of Schr\"odinger operators with distributional potentials and Sturm--Liouville operators in impedance form are isolated, in particular., Comment: 29 pages

One of the principal topics of this paper concerns the realization of self-adjoint operators $L_{\Theta, \Om}$ in $L^2(\Om; d^n x)^m$, $m, n \in \bbN$, associated with divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$. In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions $L$ which act as $$ Lu = - \biggl(\sum_{j,k=1}^n\partial_j\bigg(\sum_{\beta = 1}^m a^{\alpha,\beta}_{j,k}\partial_k u_\beta\bigg) \bigg)_{1\leq\alpha\leq m}, \quad u=(u_1,...,u_m). $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ \nu \cdot A D u + \Theta \big[u\big|_{\partial \Om}\big] = 0 \, \text{on $\partial \Om$}, $$ where $\Theta$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Om$ of $\Om$, $\nu$ denotes the outward pointing normal unit vector on $\partial\Om$, and $Du:=\bigl(\partial_j u_\alpha\bigr)_{\substack{1\leq\alpha\leq m 1\leq j\leq n}}$. Assuming $\Theta \geq 0$ in the scalar case $m=1$, we prove Gaussian heat kernel bounds for $L_{\Theta, \Om}$ by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on $\partial \Om$. We also discuss additional zero-order potential coefficients $V$ and hence operators corresponding to the form sum $L_{\Theta, \Om} + V$., Comment: 45 pages; small corrections are made in this version