Self-adjoint Realizations Research Articles

On the Resolvent of H+A∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{*}$$\\end{document}+A

We present a much shorter and streamlined proof of an improved version of the results previously given in [A. Posilicano: On the Self-Adjointness of H+A∗+A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H+A^{*}+A$$\\end{document}. Math. Phys. Anal. Geom.23 (2020)] concerning the self-adjoint realizations of formal QFT-like Hamiltonians of the kind H+A∗+A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H+A^{*}+A$$\\end{document}, where H and A play the role of the free field Hamiltonian and of the annihilation operator respectively. We give explicit representations of the resolvent and of the self-adjointness domain; the consequent Kreĭn-type resolvent formula leads to a characterization of these self-adjoint realizations as limit (with respect to convergence in norm resolvent sense) of cutoff Hamiltonians of the kind H+An∗+An-En\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H+A^{*}_{n}+A_{n}-E_{n}$$\\end{document}, the bounded operator En\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{n}$$\\end{document} playing the role of a renormalizing counter term. These abstract results apply to various concrete models in Quantum Field Theory.

Non-local relativistic δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-shell interactions

In this paper, new self-adjoint realizations of the Dirac operator in dimension two and three are introduced. It is shown that they may be associated with the formal expression D0+|FδΣ⟩⟨GδΣ|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {D}_0+|F\\delta _\\Sigma \\rangle \\langle G\\delta _\\Sigma |$$\\end{document}, where D0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {D}_0$$\\end{document} is the free Dirac operator, F and G are matrix valued coefficients, and δΣ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta _\\Sigma $$\\end{document} stands for the single layer distribution supported on a hypersurface Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}, and that they can be understood as limits of the Dirac operators with scaled non-local potentials. Furthermore, their spectral properties are analysed.

Comparing the spectrum of Schrödinger operators on quantum graphs

We study Schrödinger operators on compact finite metric graphs subject to δ \delta -coupling and standard boundary conditions. We compare the n n -th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the mean value of deviations. By doing this, we generalize recent results from Rudnick et al. [Comm. Math. Phys. 388 (2021), pp. 1603–1635] obtained for domains in R 2 \mathbb {R}^2 to the setting of quantum graphs. This also leads to a generalization of related results previously and independently obtained by Sofer [Spectral curves of quantum graphs with δ s \delta _s type vertex conditions, arXiv:2212.09143, 2022] and Band et al. [Differences between Robin and Neumann eigenvalues on metric graphs, arXiv:2212.12531, 2022] for metric graphs. In addition, based on our main result, we introduce some surface measures for a (quantum) graph which might prove useful in the future.

A growth estimate for the monodromy matrix of a canonical system

We investigate the spectrum of 2-dimensional canonical systems in the limit circle case. It is discrete and, by the Krein–de Branges formula, cannot be more dense than the integers. But in many cases it will be more sparse. The spectrum of a particular selfadjoint realisation coincides with the zeroes of one entry of the monodromy matrix of the system. Classical function theory thus establishes an immediate connection between the growth of the monodromy matrix and the distribution of the spectrum. We prove a general and flexible upper estimate for the monodromy matrix, use it to prove a bound for the case of a continuous Hamiltonian, and construct examples which show that this bound is sharp. The first two results run along the lines of earlier work by R. Romanov, but significantly improve upon these results. This is seen even on the rough scale of exponential order.

On linear adiabatic perturbations of spherically symmetric gaseous stars governed by the Euler–Poisson equations

We analyze the linearized operator for nonradial oscillations of spherically symmetric self-gravitating gaseous stars in view of the functional analysis. The evolution of the star is supposed to be governed by the Euler–Poisson equations under the equation of state of the ideal gas, and the motion is supposed to be adiabatic. We consider the case of not necessarily isentropic, that is, not barotropic motions. Basic theory of self-adjoint realization of the linearized operator is established. Some problems in the investigation of the concrete properties of the spectrum of the linearized operator are proposed. The existence of eigenvalues which accumulate to zero is proved in a mathematically rigorous fashion. The absence of continuous spectra and the completeness of eigenfunctions for the operators reduced by spherical harmonics is discussed.

Quantum Graphs: Coulomb-Type Potentials and Exactly Solvable Models

We study the Schr\"{o}dinger operators on a non-compact star graph with the Coulomb-type potentials having singularities at the vertex. The convergence of regularized Hamiltonians $H_\varepsilon$ with cut-off Coulomb potentials coupled with $(\alpha \delta+\beta\delta')$-like ones is investigated.The 1D Coulomb potential and the $\delta'$-potential are very sensitive to their regularization method. The conditions of the norm resolvent convergence of $H_\varepsilon$ depending on the regularization are established. The limit Hamiltonians give the Schr\"{o}dinger operators with the Coulomb-type potentials a mathematically precise meaning, ensuring the correct choice of vertex conditions. We also describe all self-adjoint realizations of the formal Coulomb Hamiltonians on the star graph.

Inverse wave scattering in the time domain for point scatterers

Let Δα,Y be the bounded from above self-adjoint realization in L2(R3) of the Laplacian with n point scatterers placed at Y={y1,…,yn}⊂R3, the parameters (α1,…αn)≡α∈Rn being related to the scattering properties of the obstacles. Let ufϵα,Y and ufϵ∅ denote the solutions of the wave equations corresponding to Δα,Y and to the free Laplacian Δ respectively, with a source term given a pulse fϵ supported in ϵ-neighborhoods of the points in XN={x1,…,xN}, XN∩Y=∅. We show that, for any fixed λ>sup⁡σ(Δα,Y)≥0, there exists N∘≥1 such that the locations of the points in Y can be determined by the knowledge of a finite-dimensional scattering data operator FλN:RN→RN, N≥N∘. Such an operator is defined in terms of the limit as ϵ↘0 of the Laplace transform of ufϵα,Y(t,xk)−ufϵ∅(t,xk), k=1,…,N. We exploit the factorized form of the resolvent difference (−Δα,Y+λ)−1−(−Δ+λ)−1 and a variation on the finite-dimensional factorization in the MUSIC algorithm. Multiple scattering effects are not neglected; our model can be interpreted as the time-domain version of a frequency-domain scattering from an array of Foldy's point-like obstacles.

Holomorphic Family of Dirac–Coulomb Hamiltonians in Arbitrary Dimension

AbstractWe study massless one-dimensional Dirac–Coulomb Hamiltonians, that is, operators on the half-line of the form $$D_{\omega ,\lambda }:=\begin{bmatrix} -\frac{\lambda +\omega }{x} &{} - \partial _x \\ \partial _x &{} -\frac{\lambda -\omega }{x} \end{bmatrix}$$ D ω , λ : = - λ + ω x - ∂ x ∂ x - λ - ω x . We describe their closed realizations in the sense of the Hilbert space $$L^2(\mathbb {R}_+,\mathbb {C}^2)$$ L 2 ( R + , C 2 ) , allowing for complex values of the parameters $$\lambda ,\omega $$ λ , ω . In physical situations, $$\lambda $$ λ is proportional to the electric charge and $$\omega $$ ω is related to the angular momentum. We focus on realizations of $$D_{\omega ,\lambda }$$ D ω , λ homogeneous of degree $$-1$$ - 1 . They can be organized in a single holomorphic family of closed operators parametrized by a certain two-dimensional complex manifold. We describe the spectrum and the numerical range of these realizations. We give an explicit formula for the integral kernel of their resolvent in terms of Whittaker functions. We also describe their stationary scattering theory, providing formulas for a natural pair of diagonalizing operators and for the scattering operator. We describe the point spectrum of their nonhomogeneous realizations. It is well-known that $$D_{\omega ,\lambda }$$ D ω , λ arise after separation of variables of the Dirac–Coulomb operator in dimension 3. We give a simple argument why this is still true in any dimension. Furthermore, we explain the relationship of spherically symmetric Dirac operators with the Dirac operator on the sphere and its eigenproblem. Our work is mainly motivated by a large literature devoted to distinguished self-adjoint realizations of Dirac–Coulomb Hamiltonians. We show that these realizations arise naturally if the holomorphy is taken as the guiding principle. Furthermore, they are infrared attractive fixed points of the scaling action. Beside applications in relativistic quantum mechanics, Dirac–Coulomb Hamiltonians are argued to provide a natural setting for the study of Whittaker (or, equivalently, confluent hypergeometric) functions.

Hearing the shape of a quantum boundary condition

In this paper, we study the isospectrality problem for a free quantum particle confined in a ring with a junction, analyzing all the self-adjoint realizations of the corresponding Hamiltonian in terms of a boundary condition at the junction. In particular, by characterizing the energy spectrum in terms of a spectral function, we classify the self-adjoint realizations in two classes, identifying all the families of isospectral Hamiltonians. These two classes turn out to be discerned by the action of parity (i.e. space reflection), which plays a central role in our discussion.

Quantum geometric confinement and dynamical transmission in Grushin cylinder

We classify the self-adjoint realizations of the Laplace–Beltrami operator minimally defined on an infinite cylinder equipped with an incomplete Riemannian metric of Grushin type, in the class of metrics yielding an infinite deficiency index. Such realizations are naturally interpreted as Hamiltonians governing the geometric confinement of a Schrödinger quantum particle away from the singularity, or the dynamical transmission across the singularity. In particular, we characterize all physically meaningful extensions qualified by explicit local boundary conditions at the singularity. Within our general classification we retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting one.

The semi-classical limit with a delta-prime potential

We consider the quantum evolution [Formula: see text] of a Gaussian coherent state [Formula: see text] localized close to the classical state [Formula: see text], where [Formula: see text] denotes a self-adjoint realization of the formal Hamiltonian [Formula: see text], with [Formula: see text] the derivative of Dirac’s delta distribution at [Formula: see text] and [Formula: see text] a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (with respect to the [Formula: see text]-norm, uniformly for any [Formula: see text] away from the collision time) by [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] is a suitable self-adjoint extension of the restriction to [Formula: see text], [Formula: see text], of ([Formula: see text] times) the generator of the free classical dynamics. While the operator [Formula: see text] here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi and A. Posilicano, The semi-classical limit with a delta potential, Ann. Mat. Pura Appl. 200 (2021) 453–489], in the present case the approximation gives a smaller error: it is of order [Formula: see text], [Formula: see text], whereas it turns out to be of order [Formula: see text], [Formula: see text], for the delta potential. We also provide similar approximation results for both the wave and scattering operators.

On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction

We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schrödinger equation with a point interaction and a focusing power nonlinearity. The Schrödinger operator with a point interaction (−Δα)α∈R describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The operator −Δα always has a unique simple negative eigenvalue eα. We prove that if the frequency of the standing wave is close to −eα, it is stable. Moreover, if the frequency is sufficiently large, we have the stability in the L2-subcritical or critical case, while the instability in the L2-supercritical case.

All self-adjoint extensions of the magnetic Laplacian in nonsmooth domains and gauge transformations

We use boundary triples to find a parametrization of all self-adjoint extensions of the magnetic Schrodinger operator, in a quasi-convex domain~$\Omega$ with compact boundary, and magnetic potentials with components in $\textrm{W}^{1}_{\infty}(\overline{\Omega})$. This gives also a new characterization of all self-adjoint extensions of the Laplacian in nonregular domains. Then we discuss gauge transformations for such self-adjoint extensions and generalize a characterization of the gauge equivalence of the Dirichlet magnetic operator for the Dirichlet Laplacian; the relation to the Aharonov-Bohm effect, including irregular solenoids, is also discussed. In particular, in case of (bounded) quasi-convex domains it is shown that if some extension is unitarily equivalent (through the multiplication by a smooth unit function) to a realization with zero magnetic potential, then the same occurs for all self-adjoint realizations.

Second-Order Differential Operators in the Limit Circle Case

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

Self-adjointness of two-dimensional Dirac operators on corner domains

We investigate the self-adjointness of the two-dimensional Dirac operator D , with quantum - dot and Lorentz - scalar \delta - shell boundary conditions, on piecewise C^2 domains (with finitely many corners). For both models, we prove the existence of a unique self-adjoint realization whose domain is included in the Sobolev space H^{1/2} , the formal form domain of the free Dirac operator. The main part of our paper consists of a description of the domain of the adjoint operator D^* in terms of the domain of D and the set of harmonic functions that verify some mixed boundary conditions. Then, we give a detailed study of the problem on an infinite sector, where explicit computations can be made: we find the self-adjoint extensions for this case. The result is then translated to general domains by a coordinate transformation.

Magnetic perturbations of anyonic and Aharonov–Bohm Schrödinger operators

We study the Hamiltonian describing two anyons moving in a plane in the presence of an external magnetic field and identify a one-parameter family of self-adjoint realizations of the corresponding Schrödinger operator. We also discuss the associated model describing a quantum particle immersed in a magnetic field with a local Aharonov–Bohm singularity. For a special class of magnetic potentials, we provide a complete classification of all possible self-adjoint extensions.

Coupling constant dependence for the Schrödinger equation with an inverse-square potential

We consider the one-dimensional Schr\"odinger equation $-f''+q_\alpha f = Ef$ on the positive half-axis with the potential $q_\alpha(r)=(\alpha-1/4)r^{-2}$. It is known that the value $\alpha=0$ plays a special role in this problem: all self-adjoint realizations of the formal differential expression $-\partial^2_r + q_\alpha(r)$ for the Hamiltonian have infinitely many eigenvalues for $\alpha<0$ and at most one eigenvalue for $\alpha\geq 0$. We find a parametrization of self-adjoint boundary conditions and eigenfunction expansions that is analytic in $\alpha$ and, in particular, is not singular at $\alpha = 0$. Employing suitable singular Titchmarsh--Weyl $m$-functions, we explicitly find the spectral measures for all self-adjoint Hamiltonians and prove their smooth dependence on $\alpha$ and the boundary condition. Using the formulas for the spectral measures, we analyse in detail how the "phase transition" through the point $\alpha=0$ occurs for both the eigenvalues and the continuous spectrum of the Hamiltonians.

On properties of real selfadjoint operators

In spite of the important applications of real selfadjoint operators and monotone operators, very few papers have dealt in depth with the properties of such operators. In the present paper, we follow A. Rhodius to define the spectrum $$\sigma _{\mathbb {F}}(T)$$ and the numerical range $$W_{\mathbb {F}}(T)$$ of a selfadjoint operator T acting on a Hilbert space H over the real/complex field $$\mathbb {F}$$ , and study their topological and geometrical properties which are well known in the complex case $${\mathbb {F}}={\mathbb {C}}$$ . If $$\mathbb F={\mathbb {R}}$$ , the results are new; if $${\mathbb {F}}={\mathbb {C}}$$ , the results constitute an expository body of results containing simple and short proofs of the known facts. The results are then applied to real selfadjoint operators and then to complex normal operators to sharpen their Borel functional calculi with new and shorter proofs avoiding the classical sophisticated Gelfand–Naimark theorem or the Berberian’s amalgamation theory. For such a real selfadjoint or complex normal operator N, a normed functional algebra $$L^\infty _{\mathbb {F}}(N)$$ consisting of certain Borel functions defined on $$\sigma _{\mathbb {F}}(N)$$ is constructed which inherits the isometric properties of the continuous functional calculus $$f\mapsto f(N):C_{\mathbb {F}}(\sigma _{\mathbb {F}}(N))\rightarrow B(H)$$ .

Non-compact Quantum Graphs with Summable Matrix Potentials

Let $\mathcal{G}$ be a metric noncompact connected graph with finitely many edges. The main object of the paper is the Hamiltonian ${\bf H}_{\alpha}$ associated in $L^2(\mathcal{G};\mathbb{C}^m)$ with a matrix Sturm-Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian ${\bf H}_{\alpha}$ as well as any other self-adjoint realization of the Sturm-Liouville expression is empty. We also indicate conditions on the graph ensuring pure absolute continuity of the positive part of ${\bf H}_{\alpha}$. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of ${\bf H}_{\alpha}$ is obtained. Additionally, for a star graph $\mathcal{G}$ a formula is found for the scattering matrix of the pair $\{{\bf H}_{\alpha}, {\bf H}_D\}$, where ${\bf H}_D$ is the Dirichlet operator on $\mathcal{G}$.

On the Self-Adjointness of H+A\u2217+A

Let H:text {dom}(H)subseteq mathfrak {F}to mathfrak {F} be self-adjoint and let A:text {dom}(H)to mathfrak {F} (playing the role of the annihilation operator) be H-bounded. Assuming some additional hypotheses on A (so that the creation operator A∗ is a singular perturbation of H), by a twofold application of a resolvent Kreı̆n-type formula, we build self-adjoint realizations widehat H of the formal Hamiltonian H + A∗ + A with text {dom}(H)cap text {dom}(widehat H)={0}. We give an explicit characterization of text {dom}(widehat H) and provide a formula for the resolvent difference (-widehat H+z)^{-1}-(-H+z)^{-1}. Moreover, we consider the problem of the description of widehat H as a (norm resolvent) limit of sequences of the kind H+A^{*}_{n}+A_{n}+E_{n}, where the An’s are regularized operators approximating A and the En’s are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Kreı̆n’s resolvent formula and nonperturbative theory of renormalizable models in Quantum Field Theory; in particular, as an explicit example, we consider the Nelson model.