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We derive an optimal power-weighted Hardy-type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the latter inequality differs from the former. Moreover, by iterating these inequalities we derive the sequence of power-weighted Birman-Hardy-Rellich-type inequalities in integral form on finite intervals and then also prove the analogous sequence of inequalities in differential form. We use the one-dimensional Hardy-type result in differential form to derive an optimal multi-dimensional version of the power-weighted Hardy inequality in differential form on annuli (i.e., spherical shell domains), and once more employ an iteration procedure to derive the Birman-Hardy-Rellich-type sequence of power-weighted higher-order Hardy-type inequalities for annuli. In the limit as the annulus approaches $\mathbb{R}^n$, we recover well-known prior results on Rellich-type inequalities on $\mathbb{R}^n$., Comment: 19 pages

We use a model operator approach and the spectral theorem for self-adjoint operators in a Hilbert space to derive the basic results of abstract left-definite theory in a straightforward manner. The theory is amply illustrated with a variety of concrete examples employing scales of Hilbert spaces, fractional Sobolev spaces, and domains of (strictly) positive fractional powers of operators, employing interpolation theory. In particular, we explicitly describe the domains of positive powers of the harmonic oscillator operator in $L^2(\mathbb{R})$ $\big($and hence that of the Hermite operator in $L^2\big(\mathbb{R}; e^{-x^2}dx)\big)\big)$ in terms of fractional Sobolev spaces, certain commutation techniques, and positive powers of (the absolute value of) the operator of multiplication by the independent variable in $L^2(\mathbb{R})$., Comment: 50 pages

We revisit and extend a variety of inequalities related to power weighted Rellich and Hardy--Rellich inequalities, including an inequality due to Schmincke., Comment: 37 pages, corrections made and slight reorganization of appendix

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 $n \in \mathbb{N}$, $n\geq 2$., Comment: 9 pages

We prove an abstract criterion on spectral instability of nonnegative selfadjoint extensions of a symmetric operator and apply this to self-adjoint Neumann Laplacians on bounded Lipschitz domains, intervals, and graphs. Our results can be viewed as variants of the classical weak coupling phenomenon for Schr\"odinger operators in $L^2(\mathbb R^n)$ for $n=1,2$., Comment: to appear in Proc. Amer. Math. Soc

Let $n\in\mathbb{N}, n\geq 2$. We prove that the strongly singular differential operator \[\left(\Delta^2 +c|x|^{-4}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \backslash \{0\})}, \quad c \in \mathbb{R}, \] is essentially self-adjoint in $L^2(\mathbb{R}^n; d^n x)$ if and only if \[c\geq \begin{cases}3(n+2)(6-n)&\mbox{for $2\leq n\leq 5$};\\[5pt] {\displaystyle -\frac{n(n+4)(n-4)(n-8)}{16}}&\mbox{for $n\geq 6$}.\end{cases}\] Via separation of variables, our proof reduces to studying the essential self-adjointness on the space $C_0^{\infty}((0,\infty))$ of fourth-order Euler-type differential operators of the form \[ \frac{d^4}{dr^4}+c_1\left(\frac{1}{r^2}\frac{d^2}{dr^2}+\frac{d^2}{dr^2}\frac{1}{r^2}\right)+\frac{c_2}{r^4},\quad r\in(0,\infty),\quad(c_1,c_2)\in \mathbb{R}^2,\] in $L^2((0,\infty);dr)$. Our methods generalize to differential operators related to higher-order powers of the Laplacian, however, there are some nontrivial subtleties that arise. For example, the natural expectation that for $m,n\in\mathbb{N}$, $n \geq 2$, there exist $c_{m,n}\in\mathbb{R}$ such that $\left(\Delta^m+c|x|^{-2m}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \backslash \{0\})}$ is essentially self-adjoint in $L^2(\mathbb{R}^n; d^n x)$ if and only if $c \geq c_{m,n}$, turns out to be false. Indeed, for $n=20$, we prove that the differential operator \[ \left((-\Delta)^5+c|x|^{-10}\right)\big|_{C_0^{\infty}(\mathbb{R}^{20} \backslash \{0\})}, \quad c \in \mathbb{R},\] is essentially self-adjoint in $L^2\big( \mathbb{R}^{20}; d^{20} x\big)$ if and only if $c\in [0,\beta]\cup [\gamma,\infty)$, where $\beta\approx 1.0436\times 10^{10}$, and $\gamma\approx 1.8324\times 10^{10}$ are the two real roots of the quartic equation \begin{align*}&3125z^4-83914629120000z^3+429438995162964368031744 z^2\\&\quad+1045471534388841527438982355353600z\\&\quad +629847004905001626921946285352115240960000=0.\end{align*}, Comment: 24 pages

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., Comment: 21 pages

We consider essential self-adjointness on the space $C_0^{\infty}((0,\infty))$ of even order, strongly singular, homogeneous differential operators associated with differential expressions of the type \[ \tau_{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}}, \quad x > 0, \; n \in \mathbb{N}, \; c \in \mathbb{R}, \] in $L^2((0,\infty);dx)$. While the special case $n=1$ is classical and it is well-known that $\tau_2(c)\big|_{C_0^\infty((0,\infty))}$ is essentially self-adjoint if and only if $c \geq 3/4$, the case $n \in \mathbb{N}$, $n \geq 2$, is far from obvious. In particular, it is not at all clear from the outset that \[ \text{ there exists } c_n \in \mathbb{R}, \, n \in \mathbb{N}, \text{ such that } \tau_{2n}(c)\big|_{C_0^\infty((0,\infty))} \, \text{ is essentially self-adjoint if and only if } c \geq c_n. \tag{*}\label{0.1} \] As one of the principal results of this paper we indeed establish the existence of $c_n$, satisfying $c_n \geq (4n-1)!!\big/2^{2n}$, such that property \eqref{0.1} holds. In sharp contrast to the analogous lower semiboundedness question, \[ \text{ for which values of } c \, \text{\it is } \tau_{2n}(c)\big|_{C_0^{\infty}((0,\infty))} \, \text{ bounded from below?}, \] which permits the sharp (and explicit) answer $c \geq [(2n -1)!!]^{2}\big/2^{2n}$, $n \in \mathbb{N}$, the answer for \eqref{0.1} is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly, \[ c_1 = 3/4, \quad c_2= 45, \quad c_3 = 2240 \big(214+7 \sqrt{1009}\,\big)\big/27, \] and remark that $c_n$ is the root of a polynomial of degree $n-1$. We demonstrate that for $n=6,7$, $c_n$ are algebraic numbers not expressible as radicals over $\mathbb{Q}$ (and conjecture this is in fact true for general $n \geq 6$)., Comment: 36 pages

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 offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha} (1+x)^{-\beta}(d/dx) \big((1-x)^{\alpha+1}(1+x)^{\beta+1}\big) (d/dx),& \\ \alpha, \beta \in \mathbb{R}, \; x \in (-1,1),& \end{align*} in $L^2\big((-1,1); (1-x)^{\alpha} (1+x)^{\beta} dx\big)$, $\alpha, \beta \in \mathbb{R}$. In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general $\eta$-periodic and Krein--von Neumann extensions. In particular, we treat all underlying Weyl-Titchmarsh-Kodaira and Green's function induced $m$-functions and revisit their Nevanlinna-Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev., Comment: 59 pages. arXiv admin note: text overlap with arXiv:2102.00685, arXiv:2110.15913, arXiv:1910.13117

In this note we provide estimates for the lower bound of the self-adjoint operator associated with the three-coefficient Sturm-Liouville differential expression $$ \frac{1}{r} \left(-\frac{\mathrm d}{\mathrm dx} p \frac{\mathrm d}{\mathrm dx} + q\right) $$ in the weighted $L^2$-Hilbert space $L^2(\mathbb R; rdx)$.

We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows "forbidden" endpoints and permits the consideration of functions exhibiting very limited regularity. This is done at the (necessary) expense of stipulating an additional regularity condition involving the action of the Laplacian on the functions in question which, nonetheless, works perfectly with the Dirichlet and Neumann realizations of the Schr\"odinger differential expression $-\Delta+V$. In turn, this boundary trace theory serves as a platform for developing a spectral theory for Schr\"odinger operators on bounded Lipschitz domains, along with their associated Weyl-Titchmarsh operators. Overall, this pushes the present state of knowledge a significant step further. For example, we succeed in extending the Dirichlet and Neumann trace operators in such a way that all self-adjoint extensions of a Schr\"odinger operator on a bounded Lipschitz domain may be described with explicit boundary conditions, thus providing a final answer to a problem that has been investigated for more than 60 years in the mathematical literature. Along the way, a number of other open problems are solved. The most general geometric and analytic setting in which the theory developed here yields satisfactory results is that of Lipschitz subdomains of Riemannian manifolds and for the corresponding Laplace-Beltrami operator (in place of the standard flat-space Laplacian). In particular, such an extension yields results for variable coefficient Schr\"odinger operators on bounded Lipschitz domains., Comment: 189 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

In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2\big((-1,1); (1-x)^{\alpha} (1+x)^{\beta} dx\big)$, associated to the differential expression \begin{align*} \begin{split} \tau_{\alpha,\beta} = - (1-x)^{-\alpha} (1+x)^{-\beta}(d/dx) \big((1-x)^{\alpha + 1}(1+x)^{\beta + 1}\big) (d/dx),& \\ x \in (-1,1), \; \alpha, \beta \in \mathbb{R}, \end{split} \end{align*} whenever at least one endpoint, $x=\pm 1$, is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's $m$-functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case, a topic not covered in the literature., Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:2107.09832; text overlap with arXiv:2102.00685, arXiv:1910.13117

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

Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin^2(x)$ on the finite interval $(0,\pi)$, we now take a closer look at the underlying Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in $L^2((a,b); dx)$ associated with differential expressions of the form \[ \omega_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in \mathbb{R}, \; x \in (a,b), \] and \begin{align*} \tau_{s_a,s_b} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2} + \frac{s_b^2 - (1/4)}{(x-b)^2} + q(x), \quad x \in (a,b),& \\ s_a, s_b \in [0,\infty), \; q \in L^{\infty}((a,b); dx), \; q \text{ real-valued~a.e.~on $(a,b)$,}& \end{align*} where $(a,b) \subset \mathbb{R}$ is a bounded interval. As an explicit illustration we describe the Krein-von Neumann extension of the minimal operator corresponding $\omega_{s_a}$ and $\tau_{s_a,s_b}$., Comment: 37 pages, references updated

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 $\alpha\in (-\infty,2)$ and there exist $N\in\mathbb{N}$, and $\varepsilon>0$ such that for $0

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

The principal aim of this paper is to employ Bessel-type operators in proving the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f \equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schr\"{o}dinger operator associated with the differential expression \begin{align*} \tau_s=-\dfrac{d^2}{dx^2}+\dfrac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), \; x \in (0,\pi). \end{align*} The new inequality represents a refinement of Hardy's classical inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} it also improves upon one of its well-known extensions in the form \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{d_{(0,\pi)}(x)^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} where $d_{(0,\pi)}(x)$ represents the distance from $x \in (0,\pi)$ to the boundary $\{0,\pi\}$ of $(0,\pi)$., Comment: 23 pages, updated references

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., Comment: 44 pages

We reconsider generalizations of Hardy's inequality corresponding to the case of (point) dipole potentials $V_{\gamma}(x) = \gamma (u, x) |x|^{-3}$, $x \in \mathbb{R}^n \backslash \{0\}$, $\gamma \in [0,\infty)$, $u \in \mathbb{R}^n$, $|u|=1$, $n \in \mathbb{N}$, $n \geq 3$. More precisely, for $n \geq 3$, we provide an alternative proof of the existence of a critical dipole coupling constant $\gamma_{c,n} > 0$, such that \begin{align*} &\text{for all $\gamma \in [0,\gamma_{c,n}]$, and all $u \in \mathbb{R}^n$, $|u|=1$,} \\ &\quad \int_{\mathbb{R}^n} d^n x \, |(\nabla f)(x)|^2 \geq \pm \gamma \int_{\mathbb{R}^n} d^n x \, (u, x) |x|^{-3} |f(x)|^2, \quad f \in D^1(\mathbb{R}^n). \end{align*} with $D^1(\mathbb{R}^n)$ denoting the completion of $C_0^{\infty}(\mathbb{R}^n)$ with respect to the norm induced by the gradient. Here $\gamma_{c,n}$ is sharp, that is, the largest possible such constant, and we discuss a numerical scheme for its computation. Moreover, we discuss upper and lower bounds for $\gamma_{c,n} > 0$. We also consider the case of multicenter dipole interactions with dipoles centered on an infinite discrete set., Comment: 37 pages, 1 figure, 1 table, introduction and references updated, some typos removed

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

The principal aim of this paper is to extend Birman's sequence of integral inequalities originally obtained in 1961, and containing Hardy's and Rellich's inequality as special cases, to a sequence of inequalities that incorporates power weights on either side and logarithmic refinements on the right-hand side of the inequality as well. Our new technique of proof for this sequence of inequalities relies on a combination of transforms originally due to Hartman and M\"uller-Pfeiffer. The results obtained considerably improve on prior results in the literature., Comment: 32 pages, typos removed

We revisit an archive submission by P. B. Denton, S. J. Parke, T. Tao, and X. Zhang, arXiv:1908.03795, on $n \times n$ self-adjoint matrices from the point of view of self-adjoint Dirichlet Schr\"odinger operators on a compact interval., Comment: 11 pages, new Remark 2.2 introduced

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 revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof., Comment: 20 pages

The principal objective in this paper is a new inverse approach to general Dirac-type systems modeled after B. Simon's 1999 inverse approach to half-line Schr\"odinger operators. In particular, we derive the so-called A-equation associated to Dirac-type systems and, given a fundamental positivity condition, we prove that this integro-differential equation for A is uniquely solvable. We show how to recover the matrix-valued potential coefficient from A. This approach is also applicable to matrix-valued Schr\"odinger operators on a half-line., Comment: 30 pages

We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers., Comment: 19 pages, 1 figure

After recalling a fundamental identity relating traces and modified Fredholm determinants, we apply it to a class of half-line Schr\"odinger operators $(- d^2/dx^2) + q$ on $(0,\infty)$ with purely discrete spectra. Roughly speaking, the class considered is generated by potentials $q$ that, for some fixed $C_0 > 0$, $\varepsilon > 0$, $x_0 \in (0, \infty)$, diverge at infinity in the manner that $q(x) \geq C_0 x^{(2/3) + \varepsilon_0}$ for all $x \geq x_0$. We treat all self-adjoint boundary conditions at the left endpoint $0$., Comment: 16 pages. Slight improvements were made in the file. This is a follow up to the submission arXiv:1712.00928

This note deals with the operator $T^*T$, where $T$ is a densely defined operator on a complex Hilbert space. We reprove a recent result of Z. Sebesty\'en and Zs. Tarcsay [13]: If $T^*T$ and $TT^*$ are self-adjoint, then $T$ is closed. In addition, we describe the Friedrichs extension of $S^2$, where $S$ is a symmetric operator, recovering results due to Yu. Arlinski\u{i} and Yu. Kovalev [1], [2]., Comment: 6 pages

We characterize positivity preserving, translation invariant, linear operators in $L^p(\mathbb{R}^n)^m$, $p \in [1,\infty)$, $m,n \in \mathbb{N}$., Comment: 14 pages

The principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents (and resolvent differences), Fredholm determinants, $\zeta$-functions, and $\zeta$-function regularized determinants associated with linear operators in a Hilbert space. In particular, we detail the connection between Fredholm and $\zeta$-function regularized determinants. Concrete applications of our formalism to general (i.e., three-coefficient) regular Sturm-Liouville operators on compact intervals with various (separated and coupled) boundary conditions, and Schr\"odinger operators on a half-line, are provided and further illustrated with an array of examples., Comment: 35 pages, corrected Hypothesis 3.9, fixed some typos

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

In 1961, Birman proved a sequence of inequalities $\{I_{n}\},$ for $n\in\mathbb{N},$ valid for functions in $C_0^{n}((0,\infty))\subset L^{2}((0,\infty)).$ In particular, $I_{1}$ is the classical (integral) Hardy inequality and $I_{2}$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_{n}([0,\infty))$ of functions defined on $[0,\infty).$ Moreover, $f\in H_{n}([0,\infty))$ implies $f^{\prime}\in H_{n-1}([0,\infty));$ as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0,$ these inequalities hold on the standard Sobolev space $H_0^{n}((0,b))$. Furthermore, in all cases, the Birman constants $[(2n-1)!!]^{2}/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^{2}((0,\infty))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Ces\`aro averaging operator whose spectral properties we determine in detail., Comment: 33 pages, 2 figures

For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps. Here $\delta_{\cal{C}}$ denotes a singular $\delta$-potential which is supported on a smooth compact hypersurface $\mathcal{C}\subset\mathbb{R}^n$ and $\alpha$ is a real-valued function on $\mathcal{C}$., Comment: 22 pages, this was originally part of arXiv:1609.08292, the latter will soon be resubmitted to the archive in a shortened form that is to appear in Math. Annalen