arXiv subjects
Publications and source records attributed to Fritz Gesztesy.
The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension $T$ of a symmetric operator $S$ in a Hilbert space $\mathfrak H$, 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 Riemannian manifolds.
For an arbitrary open, nonempty, bounded set $Ω\subset \mathbb{R}^n$, $n \in \mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we consider the closed, strictly positive, higher-order differential operator $A_{Ω, 2m} (a,b,q)$ in $L^2(Ω)$ defined on $W_0^{2m,2}(Ω)$, associated with the higher-order differential expression $$ τ_{2m} (a,b,q) := \bigg(\sum_{j,k=1}^{n} (-i \partial_j - b_j) a_{j,k} (-i \partial_k - b_k)+q\bigg)^m, \quad m \in \mathbb{N}, $$ and its Krein--von Neumann extension $A_{K, Ω, 2m} (a,b,q)$ in $L^2(Ω)$. Denoting by $N(λ; A_{K, Ω, 2m} (a,b,q))$, $λ> 0$, the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K, Ω, 2m} (a,b,q)$, we derive the bound $$ N(λ; A_{K, Ω, 2m} (a,b,q)) \leq C v_n (2π)^{-n} \bigg(1+\frac{2m}{2m+n}\bigg)^{n/(2m)} λ^{n/(2m)} , \quad λ> 0, $$ where $C = C(a,b,q,Ω)>0$ (with $C(I_n,0,0,Ω) = |Ω|$) is connected to the eigenfunction expansion of the self-adjoint operator $\widetilde A_{2m} (a,b,q)$ in $L^2(\mathbb{R}^n)$ defined on $W^{2m,2}(\mathbb{R}^n)$, corresponding to $τ_{2m} (a,b,q)$. Here $v_n := π^{n/2}/Γ((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in $\mathbb{R}^n$. Our method of proof relies on variational considerations exploiting the fundamental link between the Krein--von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of $\widetilde A_{2} (a,b,q)$ in $L^2(\mathbb{R}^n)$. We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension $A_{F,Ω, 2m} (a,b,q)$ in $L^2(Ω)$ of $A_{Ω, 2m} (a,b,q)$. No assumptions on the boundary $\partial Ω$ of $Ω$ are made.
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 Schrödinger 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 Schrödinger 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.
Using the Davies-Helffer-Sjöstrand 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, $\mathcal{B}_p(\mathcal{H})$, $p \in [1,\infty)$.
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 $ξ(\cdot; B,B_0)$ with respect to the operator parameter $B$. For $T$ self-adjoint in $\mathcal{H}$ we denote by $Γ_m(T)$, $m \in \mathbb{N}$ 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 $Γ_m(T)$ (cf. (1.9), we prove the following: Suppose that $B_1\in Γ_m(B_0)$ and let $\{B_τ\}_{τ\in [0,1]}\subset Γ_m(B_0)$ denote a path from $B_0$ to $B_1$ in $Γ_m(B_0)$ depending continuously on $τ\in [0,1]$ with respect to the topology on $Γ_m(B_0)$. If $f \in L^{\infty}(\mathbb{R})$, then $$ \lim_{τ\to 0^+} \|ξ(\, \cdot \, ; B_τ, A_0) f - ξ(\, \cdot \, ; B_0, A_0) f\|_{L^1(\mathbb{R}; (|ν|^{m+1} + 1)^{-1}dν)} = 0. $$
We revisit the Callias index formula for Dirac-type operators $L$ in odd space dimension $n$, and prove that \begin{align} \text{ind} \, (L) =\bigg(\frac{i}{8π}\bigg)^{\frac{n-1}{2}}\frac{1}{2(\frac{n-1}{2})!} \lim_{Λ\to\infty}\frac{1}{Λ}\sum_{i_{1},\dots,i_{n} = 1}^n \varepsilon_{i_{1}\dots i_{n}} \int_{ΛS_{n-1}}\text{tr}_{\mathbb{C}^d}\, (U(x)(\partial_{i_{1}}U)(x)\dots (\partial_{i_{n-1}}U)(x)) x_{i_{n}}\, d^{n-1} σ(x), \, (*) \end{align} where $U(x) = \text{sgn} \,(Φ(x))$ and $L$ in $L^{2}(\mathbb{R}^{n})^{2^{\widehat n}d}$ is of the form \[ L= \mathcal{Q} + Φ, \] where \[ \mathcal{Q} = \bigg(\sum_{j=1}^{n}γ_{j,n}\partial_{j}\bigg) I_d, \] with $γ_{j,n}$ elements of the Euclidean Dirac algebra, and $n=2{\widehat n}$ or $n=2{\widehat n}+1$. Here $Φ$ is assumed to satisfy the following conditions: \begin{align} & Φ\in C_{b}^{2}\big(\mathbb{R}^{n};\mathbb{C}^{d\times d}\big), \quad d \in \mathbb{N}, \\ & Φ(x)=Φ(x)^{*}, \end{align} there exists $c>0$, $R\geq0$ such that \[ |Φ(x)|\geq c I_d, \quad x\in\mathbb{R}^{n}\backslash B(0,R), \] and there exists $\varepsilon> 1/2$ such that for all $α\in\mathbb{N}_{0}^{n}$, $|α|<3$, there is $κ>0$ such that \[ \|(\partial^αΦ)(x)\|\leq \begin{cases} κ(1+|x|)^{-1}, & |α|=1,\\ κ(1+ |x|)^{-1-\varepsilon}, & |α|=2, \end{cases}\quad x\in\mathbb{R}^{n}. \] These conditions on $Φ$ render $L$ a Fredholm operator, and appear to be the most general conditions known to date for which Callias' index formula has been derived. Generalizations of the index formula $(*)$ to certain classes of non-Fredholm operators $L$ invoking the (generalized) Witten index are also discussed.
We provide a construction of the two-component Camassa-Holm (CH-2) hierarchy employing a new zero-curvature formalism and identify and describe in detail the isospectral set associated to all real-valued, smooth, and bounded algebro-geometric solutions of the $n$th equation of the stationary CH-2 hierarchy as the real $n$-dimensional torus $\mathbb{T}^n$. We employ Dubrovin-type equations for auxiliary divisors and certain aspects of direct and inverse spectral theory for self-adjoint singular Hamiltonian systems. In particular, we employ Weyl-Titchmarsh theory for singular (canonical) Hamiltonian systems. While we focus primarily on the case of stationary algebro-geometric CH-2 solutions, we note that the time-dependent case subordinates to the stationary one with respect to isospectral torus questions.
Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [9]. Natural examples arise from $(1+1)$-dimensional differential operators using the model operator $D_A$ in $L^2(\mathbb{R}^2; dt dx)$ of the type $D_A = (d/dt) + A$, where $A = \int^{\oplus}_{\mathbb{R}} dt \, A(t)$, and the family of self-adjoint operators $A(t)$ in $L^2(\mathbb{R}; dx)$ is explicitly given by $A(t) = - i (d/dx) + θ(t) ϕ(\cdot)$, $t \in \mathbb{R}$. Here $ϕ: \mathbb{R} \to \mathbb{R}$ has to be integrable on $\mathbb{R}$ and $θ: \mathbb{R} \to \mathbb{R}$ tends to zero as $t \to - \infty$ and to $1$ as $t \to + \infty$. In particular, $A(t)$ has asymptotes in the norm resolvent sense $A_- = - i (d/dx)$, $A_+ = - i (d/dx) + ϕ(\cdot)$ as $t \to \mp \infty$. Since $D_A$ violates the relative trace class condition introduced in [9], we now employ a new approach based on an approximation technique. The approximants do fit the framework of [9] and lead to the following results: Introducing $H_1 = {D_A}^* D_A$, $H_2 = D_A {D_A}^*$, we recall that the resolvent regularized Witten index of $D_A$, denoted by $W_r(D_A)$, is defined by $$ W_r(D_A) = \lim_{λ\to 0} (- λ) {\rm tr}_{L^2(\mathbb{R}^2; dtdx)}((H_1 - λI)^{-1} - (H_2 - λI)^{-1}). $$ In the concrete example at hand, we prove $$ W_r(D_A) = ξ(0_+; H_2, H_1) = ξ(0; A_+, A_-) = 1/(2 π) \int_{\mathbb{R}} dx \, ϕ(x). $$ Here $ξ(\, \cdot \, ; S_2, S_1)$, denotes the spectral shift operator for the pair $(S_2,S_1)$, and we employ the normalization, $ξ(λ; H_2, H_1) = 0$, $λ< 0$.
Let $\{A(t)\}_{t \in \mathbb{R}}$ be a path of self-adjoint Fredholm operators in a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$ as $t \to \pm \infty$. Computing the index of the operator $D_A= (d/d t) + A$ acting in $L^2(\mathbb{R}; \mathcal{H})$, where $A = \int_{\mathbb{R}}^{\oplus} dt \, A(t)$, and its relation to spectral flow along this path, has a long history. While most of the latter focuses on the case where $A(t)$ all have purely discrete spectrum, we now particularly study situations permitting essential spectra. Introducing $H_1={D_A}^* D_A$ and $H_2=D_A {D_A}^*$, we consider spectral shift functions $ξ(\, \cdot \,; A_+, A_-)$ and $ξ(\, \cdot \, ; H_2, H_1)$ associated with the pairs $(A_+, A_-)$ and $(H_2,H_1)$. Assuming $A_+$ to be a relatively trace class perturbation of $A_-$ and $A_{\pm}$ to be Fredholm, the value $ξ(0; A_-, A_+)$ was shown in [14] to represent the spectral flow along the path $\{A(t)\}_{t\in \mathbb{R}}$ while that of $ξ(0_+; H_1,H_2)$ yields the Fredholm index of $D_A$. The fact, proved in [14], that these values of the two spectral functions are equal, resolves the index = spectral flow question in this case. When the path $\{A(t)\}_{t \in \mathbb{R}}$ consists of differential operators, the relatively trace class perturbation assumption is violated. The simplest assumption that applies (to differential operators in (1+1)-dimensions) is a relatively Hilbert-Schmidt perturbation. This is not just an incremental improvement. In fact, the approximation method we employ here to make this extension is of interest in any dimension. Moreover we consider $A_\pm$ which are not necessarily Fredholm and we establish that the relationships between the two spectral shift functions for the pairs $(A_+, A_-)$ and $(H_2,H_1)$ found in all of the previous papers [9], [14], and [22] can be proved in the non-Fredholm case.
Given a complex, separable Hilbert space $\mathcal{H}$, we consider self-adjoint $L^2$-realizations of differential expressions $τ= - (d^2/dx^2) I_{\mathcal{H}} + V(x)$, on half-lines and on the real line (assuming the limit-point property of $τ$ at $\pm \infty$). Here $V$ denotes a bounded operator-valued potential $V(\cdot) \in \mathcal{B}(\mathcal{H})$ such that $V(\cdot)$ is weakly measurable, the operator norm $\|V(\cdot)\|_{\mathcal{B}(\mathcal{H})}$ is locally integrable, and $V(\cdot) = V(\cdot)^*$ a.e. In a nutshell, a Donoghue-type $m$-function $M_{A,\mathcal{N}_i}^{Do}(\cdot)$ associated with self-adjoint extensions $A$ of a closed, symmetric operator $\dot A$ in $\mathcal{H}$ with deficiency spaces $\mathcal{N}_z = \ker \big({\dot A}^* - z I_{\mathcal{H}}\big)$ and corresponding orthogonal projections $P_{\mathcal{N}_z}$ onto $\mathcal{N}_z$ is given by $$ M_{A,\mathcal{N}_i}^{Do}(z) = zI_{\mathcal{N}_i} + (z^2+1) P_{\mathcal{N}_i} (A - z I_{\mathcal{H}})^{-1} P_{\mathcal{N}_i}\big\vert_{\mathcal{N}_i} \,, \quad {\rm Im}(z)\neq 0. $$ For half-line and full-line Schrödinger operators, the role of $\dot A$ is played by a suitably defined minimal Schrödinger operator which will be shown to be completely non-self-adjoint. The latter property is used to prove that the corresponding operator-valued measures in the Herglotz--Nevanlinna representations of the Donoghue-type $m$-functions corresponding to self-adjoint half-line and full-line Schrödinger operators encode the entire spectral information of the latter.
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).
We survey the notion of the spectral shift function of a pair of self-adjoint operators and recent progress on its connection with the Witten index. We also describe a proof of Krein's Trace Theorem that does not use complex analysis [53] and develop its extension to general $σ$-finite von Neumann algebras $\mathcal{M}$ of type II and unbounded perturbations from the predual of $\mathcal{M}$. We also discuss the connection between the theory of the spectral shift function and index theory for certain model operators. We start by introducing various definitions of the Witten index, (an extension of the notion of Fredholm index to non-Fredholm operators). Then we study the model operator $D_{A^{}} = (d/dt) + A$ in $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(A f)(t) = A(t) f(t)$ for a.e. $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$. The setup permits the operator family $A(t)$ on $\mathcal{H}$ to be an unbounded relatively trace class perturbation of the unbounded self-adjoint operator $A_-$, and no discrete spectrum assumptions are made on the asymptotes $A_{\pm}$. When $A_{\pm}$ are boundedly invertible, it is shown that $D_{A^{}}$ is Fredholm and its index can be computed as $ξ(0; A_+, A_-)$. When $0\inσ(A_+)$ (or $0\inσ(A_-)$), the operator $D_{A^{}}$ ceases to be Fredholm. However, if $0$ is a right and a left Lebesgue point of $ξ(\,\cdot\,\, ; A_+, A_-)$, the resolvent regularized Witten index $W_r(D_{A^{}})$ is given by $W_r(D_{A^{}}) = ξ(0_+; |D_{A^{*}}|^2, |D_{A^{}}|^2) = [ξ(0_+; A_+,A_-) + ξ(0_-; A_+, A_-)]/2$. We also study a special example, when the perturbation of the unbounded self-adjoint operator $A_-$ is not assumed to be relatively trace class.
The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schrödinger-type operators on a half-line from the underlying Weyl-Titchmarsh function.
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 $λ_0$ 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 $λ_0$ 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.
We compute the Fredholm index, ${\rm ind}(D_A)$, of the operator $D_A = (d/dt) + A$ on $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(A f)(t) = A(t) f(t)$ for a.e. $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$, via the spectral shift function $ξ(\, \cdot \,;A_+,A_-)$ associated with the pair $(A_+, A_-)$ of asymptotic operators $A_{\pm}=A(\pm\infty)$ on the separable complex Hilbert space $\mathcal{H}$ in the case when $A(t)$ is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator $A_-$. We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function $ξ(\, \cdot \,;A_+,A_-)$ for the pair $(A_+, A_-)$, and the corresponding spectral shift function $ξ(\, \cdot \,;H_2,H_1)$ for the pair of operators $(H_2,H_1)=(D_A {D_A}^*, {D_A}^* D_A)$ in this relative trace class context. This formula is then used to identify the Fredholm index of $D_A$ with $ξ(0;A_+,A_-)$. In addition, we prove that ${\rm ind}(D_A)$ coincides with the spectral flow SpFlow$(\{A(t)\}_{t=-\infty}^\infty)$ of the family $\{A(t)\}_{t\in\mathbb{R}}$ and also relate it to the (Fredholm) perturbation determinant for the pair $(A_+, A_-)$. We also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.
Building on work on Miura's transformation by Kappeler, Perry, Shubin, and Topalov, we develop a detailed spectral theoretic treatment of Schrödinger operators with matrix-valued potentials, with special emphasis on distributional potential coefficients. Our principal method relies on a supersymmetric (factorization) formalism underlying Miura's transformation, which intimately connects the triple of operators $(D, H_1, H_2)$ of the form [D= (0 & A^*, A & 0) \text{in} L^2(\mathbb{R})^{2m} \text{and} H_1 = A^* A, H_2 = A A^* \text{in} L^2(\mathbb{R})^m.] Here $A= I_m (d/dx) + ϕ$ in $L^2(\mathbb{R})^m$, with a matrix-valued coefficient $ϕ= ϕ^* \in L^1_{\text{loc}}(\mathbb{R})^{m \times m}$, $m \in \mathbb{N}$, thus explicitly permitting distributional potential coefficients $V_j$ in $H_j$, $j=1,2$, where [H_j = - I_m \frac{d^2}{dx^2} + V_j(x), \quad V_j(x) = ϕ(x)^2 + (-1)^{j} ϕ'(x), j=1,2.] Upon developing Weyl--Titchmarsh theory for these generalized Schrödinger operators $H_j$, with (possibly, distributional) matrix-valued potentials $V_j$, we provide some spectral theoretic applications, including a derivation of the corresponding spectral representations for $H_j$, $j=1,2$. Finally, we derive a local Borg--Marchenko uniqueness theorem for $H_j$, $j=1,2$, by employing the underlying supersymmetric structure and reducing it to the known local Borg--Marchenko uniqueness theorem for $D$.
In this paper we study the self-adjoint Krein-von Neumann realization $A_K$ of the perturbed Laplacian $-Δ+V$ in a bounded Lipschitz domain $Ω\subset\mathbb{R}^n$. We provide an explicit and self-contained description of the domain of $A_K$ in terms of Dirichlet and Neumann boundary traces, and we establish a Weyl asymptotic formula for the eigenvalues of $A_K$.
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 $L^2(Ω)$ on certain open sets $Ω\subseteq \mathbb{R}^n$, $n \in \mathbb{N}$, with Dirichlet, Neumann, and mixed boundary conditions on $\partial Ω$, under general hypotheses on the (typically, nonsmooth, unbounded) coefficients and on $\partialΩ$.