Self-adjoint Extensions Research Articles

Vacuum Polarization in the Point Impurity Background

The vacuum polarization effect of a massive scalar field φ(x) near the point δ-like source is considered. The corresponding interaction is introduced within a technique of self-adjoint extension of the Laplace operator. This method has been widely discussed within the framework of quantum mechanics. We propose to use it to investigate vacuum field effects. This approach allows computing the renormalized Hadamard function and the renormalized vacuum energy density hT00(x)iren for massive real scalar field with minimal curvature coupling. The dependence of the vacuum polarization effect upon the fields’s mass is analyzed.

Friedrichs and Kreĭn type extensions in terms of representing maps

A semibounded operator or relation S in a Hilbert space with lower bound γ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \\in {{\\mathbb {R}}}$$\\end{document} has a symmetric extension Sf=S+^({0}×mulS∗)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_\ extrm{f}=S \\, \\widehat{+} \\,(\\{0\\} \ imes \\mathrm{mul\\,}S^*)$$\\end{document}, the weak Friedrichs extension of S, and a selfadjoint extension SF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\ extrm{F}}$$\\end{document}, the Friedrichs extension of S, that satisfy S⊂Sf⊂SF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S \\subset S_{\ extrm{f}} \\subset S_\ extrm{F}$$\\end{document}. The Friedrichs extension SF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\ extrm{F}}$$\\end{document} has lower bound γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} and it is the largest semibounded selfadjoint extension of S. Likewise, for each c≤γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c \\le \\gamma $$\\end{document}, the relation S has a weak Kreĭn type extension Sk,c=S+^(ker(S∗-c)×{0})\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\ extrm{k},c}=S \\, \\widehat{+} \\,(\\mathrm{ker\\,}(S^*-c) \ imes \\{0\\})$$\\end{document} and Kreĭn type extension SK,c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\ extrm{K},c}$$\\end{document} of S, that satisfy S⊂Sk,c⊂SK,c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S \\subset S_{\ extrm{k},c} \\subset S_{\ extrm{K},c}$$\\end{document}. The Kreĭn type extension SK,c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\ extrm{K},c}$$\\end{document} has lower bound c and it is the smallest semibounded selfadjoint extension of S which is bounded below by c. In this paper these special extensions and, more generally, all extremal extensions of S are constructed via the semibounded sesquilinear form t(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathfrak {t}}}(S)$$\\end{document} that is associated with S; the representing map for the form t(S)-c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathfrak {t}}}(S)-c$$\\end{document} plays an essential role here.

The L2 Aeppli-Bott-Chern Hilbert complex

We analyse the L2 Hilbert complexes naturally associated to a non-compact complex manifold, namely the ones which originate from the Dolbeault and the Aeppli-Bott-Chern complexes. In particular we define the L2 Aeppli-Bott-Chern Hilbert complex and examine its main properties on general Hermitian manifolds, on complete Kähler manifolds and on Galois coverings of compact complex manifolds. The main results are achieved through the study of self-adjoint extensions of various differential operators whose kernels, on compact Hermitian manifolds, are isomorphic to either Aeppli or Bott-Chern cohomology.

Symmetric operator extensions of composites of higher order difference operators

In this paper we have considered two higher order difference operators generated by two higher order difference functions on the Hilbert space of square summable functions. By allowing the leading coefficients to be unbounded and the other coefficients as constant functions, we have shown that the composites of two symmetric difference operators are symmetric if the leading coefficients are scalar multiple of each other and the common divisor of their orders is 1. Using examples, we have shown that these conditions of symmetry cannot be weakened. Furthermore, We have shown that the deficiency indices of the composites is equal to the sum of the deficiency indices of the individual operators and that the spectra of the self-adjoint operator extensions is the whole of the real line.

Regularized quantum motion in a bounded set: Hilbertian aspects

It is known that the momentum operator canonically conjugated to the position operator for a particle moving in some bounded interval of the line (with Dirichlet boundary conditions) is not essentially self-adjoint: it has a continuous set of self-adjoint extensions. We prove that essential self-adjointness can be recovered by symmetrically weighting the momentum operator with a positive bounded function approximating the indicator function of the considered interval. This weighted momentum operator is consistently obtained from a similarly weighted classical momentum through the so-called Weyl-Heisenberg covariant integral quantization of functions or distributions.

Quantum approach to bound states in field theory

It is well known that (possibly nonunique) suitable field dynamics can be prescribed in spacetimes with timelike boundaries by means of appropriate boundary conditions. In [R. M. Wald, ], Wald derived a conserved energy functional for each prescribed dynamics. This conserved energy is related to the positive self-adjoint extensions of the spatial part A of the wave equation ∂2Φ/∂t2=−AΦ (A may not be, in principle, essentially self-adjoint). This is quite surprising since the canonical energy is not conserved in these cases. In this paper, we rederive this energy functional from an action principle (with appropriate boundary terms) following [A. A. Saharian, ] and consider field dynamics arising from nonpositive self-adjoint extensions of A. The spectrum of the resulting theory fails to be positive and unstable mode solutions for classical fields come to light. By studying fields in half-Minkowski spacetime, we illustrate that these unstable classical solutions come as a consequence of an inverted parabolic potential governing their dynamics. From the quantum mechanical point of view, this leads to an effective inverted harmonic oscillator at the boundary. We then explore these unstable modes behavior, as well as their instabilities, at the quantum level. Published by the American Physical Society 2024

Stability of Krein-von Neumann self-adjoint operator extension under unbounded perturbations

We have considered a fourth order difference operator defined on the Hilbert space of square summable sequences on N. We investigated the stability of existence of Krein-von Neumann self-adjoint extension of difference operators under bounded and unbounded coefficients. Using asymptotic summation based on discretised Levinson's theorem and appropriate smoothness and decay conditions, we have shown that unlike the case of deficiency indices and discrete spectrum, the existence of positive self-adjoint operator extensions is stable under unbounded perturbations. These results now exhaustively characterise the spectral and structural properties of difference operators with unbounded coefficients.

Harmonic oscillator on Weyl chambers

Let C+ denote a Weyl chamber distinguished in the framework of a finite reflection group W acting on Rd. We consider the harmonic oscillator −Δ+|x|2 as an operator on L2(C+) with appropriately chosen domains and construct families of corresponding self-adjoint extensions. These extensions are parametrized by the homomorphisms η∈Hom(W,Zˆ2) and are based on an η-symmetrization procedure of functions on the whole Rd. The analysis of the associated η-heat semigroups is included and positivity on C+ of the corresponding η-heat kernels is established.

Contact interactions, self-adjoint extensions, and low-energy scattering

Low-energy scattering is well described by the effective-range expansion. In quantum mechanics, a tower of contact interactions can generate terms in this expansion after renormalization. Scattering parameters are also encoded in the self-adjoint extension of the Hamiltonian. We briefly review this well-known result for two particles with s-wave interactions using impenetrable self-adjoint extensions, including the case of harmonically trapped two-particle states. By contrast, the one-dimensional scattering problem is surprisingly intricate. We show that the families of self-adjoint extensions correspond to a coupled system of symmetric and antisymmetric outgoing waves, which is diagonalized by an SU(2) transformation that accounts for mixing and a relative phase. This is corroborated by an effective theory computation that includes all four energy-independent contact interactions. The equivalence of various one-dimensional contact interactions is discussed and scrutinized from the perspective of renormalization. As an application, the spectrum of a general point interaction with a harmonic trap is solved in one dimension.

Boundary Value Problems for Dirac Operators on Graphs

We carry the index theory for manifolds with boundary of Bär and Ballmann over to first order differential operators on metric graphs. This approach results in a short proof for the index of such operators. Then the self-adjoint extensions and the spectrum of the Dirac operator on the complex line bundle are studied. We also introduce two types of boundary conditions for the Dirac operator, whose spectrum encodes information of the underlying topology of the graph.

Vacuum energy of scalar fields on spherical shells with general matching conditions

In this work we analyze the spectral zeta function for massless scalar fields propagating in a D-dimensional flat space under the influence of a shell potential. The static nature of the potential, and the spherical symmetry, allows us to focus on the spatial part of the field which satisfies a one-dimensional Schrodinger equation endowed with a point potential. The shell potential is defined in terms of the two-interval self-adjoint extensions of the one-dimensional Schrodinger equation that describes the radial part of the scalar field. After performing the necessary analytic continuation, we utilize the spectral zeta function of the system to compute the vacuum energy of the field.

On creating new essential spectrum by self-adjoint extension of gapped operators

On creating new essential spectrum by self-adjoint extension of gapped operators

ON DIFFERENT EXTENSIONS WITH EXIT FROM SPACE OF MINIMAL OPERATOR ASSOCIATED WITH SECOND ORDER OPERATOR DIFFERENTIAL EXPRESSION

Abstract. Our aim in that paper is defining domains of definitions of accumulative, dissipative and self-adjoint extensions of minimal operator associated with boundary value problems for operator Sturm-Liouville equation with boundary conditions, containing unbounded operator and spectral parameter λ.

Reduction of positive self-adjoint extensions

Reduction of positive self-adjoint extensions

Influence of Winkler - Steklov conditions on natural oscillations of elastic weighty body

We consider the spectral problem for the spacial system of equations of the elasticity theory. Small parts of the body surface are supplied with the Winkler - Steklov conditions, which model spring mount, while the remaining part of the boundary is traction-free. In several cases (the relative stiffness of springs and their positions are varied) we construct asymptotics for eigenfrequencies of the body and for corresponding eigenmodes. The limiting problems are ones for the body (spectral or stationary in some case) and problems of the elasticity theory for the half-spaces with the Winkler - Steklov conditions on flat sets (separated or joined into a single spectral theory in some cases). The discreteness of the spectrum of the problem in the half-space is ensured by a polynomial property of the system of equations of the elasticity theory. We study particular cases, formulate open questions and discuss patological situations, in which the spectrum loses usual properties. We construct asymptotic models of the problem, which provide two-terms asymptotics for the eigenpairs of the initial problem and which use the technique of self-adjoint extensions of differential operators or Hilbers spaces with separated asymptotics.

Convergence of operators with deficiency indices (k, k) and of their self-adjoint extensions

We consider an abstract sequence {An}n=1∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{A_n\\}_{n=1}^\\infty $$\\end{document} of closed symmetric operators on a separable Hilbert space H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}$$\\end{document}. It is assumed that all An\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_n$$\\end{document}’s have equal deficiency indices (k, k) and thus self-adjoint extensions {Bn}n=1∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{B_n\\}_{n=1}^\\infty $$\\end{document} exist and are parametrized by partial isometries {Un}n=1∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{U_n\\}_{n=1}^\\infty $$\\end{document} on H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}$$\\end{document} according to von Neumann’s extension theory. Under two different convergence assumptions on the An\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_n$$\\end{document}’s we give the precise connection between strong resolvent convergence of the Bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_n$$\\end{document}’s and strong convergence of the Un\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_n$$\\end{document}’s.

Vector valued de Branges spaces of entire functions based on pairs of Fredholm operator valued functions and functional model

In this paper, we have considered vector valued reproducing kernel Hilbert spaces (RKHS) H of entire functions associated with operator valued kernel functions. de Branges operators E=(E−,E+) analogous to de Branges matrices have been constructed with the help of pairs of Fredholm operator valued entire functions on X, where X is a complex separable Hilbert space. A few explicit examples of these de Branges operators are also discussed. The newly defined RKHS B(E) based on the de Branges operator E=(E−,E+) has been characterized under some special restrictions. The complete parametrizations and canonical descriptions of all selfadjoint extensions of the closed, symmetric multiplication operator by the independent variable have been given in terms of unitary operators between ranges of reproducing kernels. A sampling formula for the de Branges spaces B(E) has been discussed. A particular class of entire operators with infinite deficiency indices has been dealt with and shown that they can be considered as the multiplication operator for a specific class of these de Branges spaces. Finally, a brief discussion on the connection between the characteristic function of a completely nonunitary contraction operator and the de Branges spaces B(E) has been given.

Oversampling on a class of symmetric regular de Branges spaces

A de Branges space is regular if the constants belong to its space of associated functions and is symmetric if it is isometrically invariant under the map . Let be the reproducing kernel in and be the operator of multiplication by the independent variable with maximal domain in . Loosely speaking, we say that has the -oversampling property relative to a proper subspace of it, with , if there exists such that for all , for all and almost every self-adjoint extension of . This definition is motivated by the well-known oversampling property of Paley-Wiener spaces. In this paper, we provide sufficient conditions for a symmetric, regular de Branges space to have the -oversampling property relative to a chain of de Branges subspaces of it.

The spectrum of self-adjoint extensions associated with exceptional Laguerre differential expressions

Exceptional Laguerre-type differential expressions form an infinite class of Schrödinger operators having rational potentials and one limit-circle endpoint. In this manuscript, the spectrum of all self-adjoint extensions for a general exceptional Laguerre-type differential expression is given in terms of the Darboux transformations which relate the expression to the classical Laguerre differential expression. The spectrum is extracted from an explicit Weyl m -function, up to a sign. The construction relies primarily on two tools: boundary triples, which parameterize the self-adjoint extensions and produce the Weyl m -functions, and manipulations of Maya diagrams and partitions, which classify the seed functions defining the relevant Darboux transforms. Several examples are presented.

Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac–Coulomb operators

We consider a linear symmetric operator in a Hilbert space that is neither bounded from above nor from below, admits a block decomposition corresponding to an orthogonal splitting of the Hilbert space and has a variational gap property associated with the block decomposition. A typical example is the Dirac–Coulomb operator defined on C^\infty_c(\mathbb{R}^3\setminus\{0\}, \mathbb{C}^4) . In this paper we define a distinguished self-adjoint extension with a spectral gap and characterize its eigenvalues in that gap by a min-max principle. This has been done in the past under technical conditions. Here we use a different, geometric strategy, to achieve that goal by making only minimal assumptions. Our result applied to the Dirac–Coulomb-like Hamitonians covers sign-changing potentials as well as molecules with an arbitrary number of nuclei having atomic numbers less than or equal to 137