Self-adjoint Extensions Research Articles

Geometry of quantum dynamics in infinite-dimensional Hilbert space

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional, i.e. we do not restrict considerations to finite-dimensional Hilbert spaces, contrary to many other works on the geometry of quantum mechanics, and include a Lagrangian formalism in which self-adjoint (Schrödinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we also obtain results concerning coadjoint orbits of the unitary group in infinite dimensions, embedding of pure states in the unitary group, and self-adjoint extensions of symmetric relations.

Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors

This paper deals with the study of the two-dimensional Dirac operatorwith infinite mass boundary condition in a sector. We investigate the question ofself-adjointness depending on the aperture of the sector: when the sector is convexit is self-adjoint on a usual Sobolev space whereas when the sector is non-convexit has a family of self-adjoint extensions parametrized by a complex number of theunit circle. As a byproduct of this analysis we are able to give self-adjointnessresults on polygones. We also discuss the question of distinguished self-adjointextensions and study basic spectral properties of the operator in the sector.

Scalar problems in junctions of rods and a plate

In this work we deal with a scalar spectral mixed boundary value problem in a spacial junction of thin rods and a plate. Constructing asymptotics of the eigenvalues, we employ two equipollent asymptotic models posed on the skeleton of the junction, that is, a hybrid domain. We, first, use the technique of self-adjoint extensions and, second, we impose algebraic conditions at the junction points in order to compile a problem in a function space with detached asymptotics. The latter problem is involved into a symmetric generalized Green formula and, therefore, admits the variational formulation. In comparison with a primordial asymptotic procedure, these two models provide much better proximity of the spectra of the problems in the spacial junction and in its skeleton. However, they exhibit the negative spectrum of finite multiplicity and for these “parasitic” eigenvalues we derive asymptotic formulas to demonstrate that they do not belong to the service area of the developed asymptotic models.

The Maslov index and the spectra of second order elliptic operators

We consider second order elliptic differential operators on a bounded Lipschitz domain Ω. Firstly, we establish a natural one-to-one correspondence between their self-adjoint extensions, with domains of definition containing in H1(Ω), and Lagrangian planes in H1/2(∂Ω)×H−1/2(∂Ω). Secondly, we derive a formula relating the spectral flow of the one-parameter families of such operators to the Maslov index, the topological invariant counting the signed number of conjugate points of paths of Lagrangian planes in H1/2(∂Ω)×H−1/2(∂Ω). Furthermore, we compute the Morse index, the number of negative eigenvalues, in terms of the Maslov index for several classes of the second order operators: the θ→-periodic Schrödinger operators on a period cell Q⊂Rn, the elliptic operators with Robin-type boundary conditions, and the abstract self-adjoint extensions of the Schrödinger operators on star-shaped domains. Our work is built on the techniques recently developed by B. Booß-Bavnbek, K. Furutani, and C. Zhu, and extends the scope of validity of their spectral flow formula by incorporating the self-adjoint extensions of the second order operators with domains in the first order Sobolev space H1(Ω). In addition, we generalize the results concerning relations between the Maslov and Morse indices quite recently obtained by G. Cox, J. Deng, C. Jones, J. Marzuola, A. Sukhtayev and the authors. Finally, we describe and study a link between the theory of abstract boundary triples and the Lagrangian description of self-adjoint extensions of abstract symmetric operators.

Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei

We derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the Kreĭn-Višik-Birman extension scheme, or also on Grubb’s universal classification theory, as opposite to previous works within the standard von Neumann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the extension, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension.

Surface spectra of Weyl semimetals through self-adjoint extensions

We apply the method of self-adjoint extensions of Hermitian operators to the low-energy, continuum Hamiltonians of Weyl semimetals in bounded geometries and derive the spectrum of the surface states on the boundary. This allows for the full characterization of boundary conditions and the surface spectra on surfaces both normal to the Weyl node separation as well as parallel to it. We show that the boundary conditions for quadratic bulk dispersions are, in general, specified by a $\mathbb{U}(2)$ matrix relating the wavefunction and its derivatives normal to the surface. We give a general procedure to obtain the surface spectra from these boundary conditions and derive them in specific cases of bulk dispersion. We consider the role of global symmetries in the boundary conditions and their effect on the surface spectrum. We point out several interesting features of the surface spectra for different choices of boundary conditions, such as a Mexican-hat shaped dispersion on the surface normal to Weyl node separation. We find that the existence of bound states, Fermi arcs, and the shape of their dispersion, depend on the choice of boundary conditions. This illustrates the importance of the physics at and near the boundaries in the general statement of bulk-boundary correspondence.

Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan—Skornyakov type

We reconstruct the whole family of self-adjoint Hamiltonians of Ter-Martirosyan—Skornyakov type for a system of two identical fermions coupled with a third particle of different nature through an interaction of zero range. We proceed through an operator-theoretic approach based on the self-adjoint extension theory of Krein, Visik, and Birman. We identify the explicit ‘Krein—Visik-Birman extension parameter’ as an operator on the ‘space of charges’ for this model (the ‘Krein space’) and we come to formulate a sharp conjecture on the dimensionality of its kernel. Based on our conjecture, for which we also discuss an amount of evidence, we explain the emergence of a multiplicity of extensions in a suitable regime of masses and we reproduce for the first time the previous partial constructions obtained by means of an alternative quadratic form approach.

On the adjoint of Hilbert space operators

In general, it is a non-trivial task to determine the adjoint of an unbounded operator S acting between two Hilbert spaces. We provide necessary and sufficient conditions for a given operator T to be identical with . In our considerations, a central role is played by the operator matrix . Our approach has several consequences such as characterizations of closed, normal, skew- and selfadjoint, unitary and orthogonal projection operators in real or complex Hilbert spaces. We also give a self-contained proof of the fact that always has a positive selfadjoint extension.

On extensions of singular fourth order dynamic operators on time scales

In this work, we consider a singular fourth order dynamic operator on time scales. We construct a space of boundary values. Later, we give a description of all maximal dissipative, self-adjoint and other extensions of singular fourth order differential operators on unbounded time scales.

SELFADJOINT SINGULAR DIFFERENTIAL OPERATORS FOR FIRST ORDER

The parametrization of all selfadjoint extensions of the minimal operator generated by frst order linear symmetric singular differential-operator expression in the Hilbert space of vector-functions defined at the right semi-axis has been given. To this end we use the Calkin-Gorbachuk method. Finally, the structure of spectrum set of such extensions is researched

Quasi-stationary states and fermion pair creation from a vacuum in supercritical Coulomb field

Creation of charged fermion pair from a vacuum in so-called supercritical Coulomb potential is examined for the case when fermions can move only in the same (one) plane. In which case, quantum dynamics of charged massive or massless fermions can be described by the two-dimensional Dirac Hamiltonians with an usual [Formula: see text] Coulomb potential. These Hamiltonians are singular and require the additional definition in order for them to be treated as self-adjoint quantum-mechanical operators. We construct the self-adjoint two-dimensional Dirac Hamiltonians with a Coulomb potential and determine the quantum-mechanical states for such Hamiltonians in the corresponding Hilbert spaces of square-integrable functions. We determine the scattering amplitude in which the self-adjoint extension parameter is incorporated and then obtain equations implicitly defining possible discrete energy spectra of the self-adjoint Dirac Hamiltonians with a Coulomb potential. It is shown that this quantum system becomes unstable in the presence of a supercritical Coulomb potential which manifests in the appearance of quasi-stationary states in the lower (negative) energy continuum. The energy spectrum of those states is quasi-discrete, consists of broadened levels with widths related to the inverse lifetimes of the quasi-stationary states as well as the probability of creation of charged fermion pair by a supercritical Coulomb field. Explicit analytical expressions for the creation probabilities of charged (massive or massless) fermion pair are obtained in a supercritical Coulomb field.

Relativistic quantum dynamics of vector bosons in an Aharonov–Bohm potential

The Aharonov–Bohm (AB) problem for vector bosons by the Duffin–Kemmer–Petiau (DKP) formalism is analyzed. Depending on the values of the spin projection, the relevant eigenvalue equation coming from the DKP formalism reveals an equivalence to the spin- AB problem. Using a boundary condition at the origin based on the self-adjoint extension method, we study the scattering and bound states problems. Expressions for matrix, phase shift and bound states are obtained.

Regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint

This paper is concerned with regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint. For any given self-adjoint subspace extension (SSE) of the corresponding minimal subspace, its spectrum can be approximated by eigenvalues of a sequence of induced regular SSEs, generated by the same difference expression on smaller finite intervals. It is shown that every SSE of the minimal subspace has a pure discrete spectrum, and the k-th eigenvalue of any given SSE is exactly the limit of the k-th eigenvalues of the induced regular SSEs; that is, spectral exactness holds, in the limit circle case. Furthermore, error estimates for the approximations of eigenvalues are given in this case. In addition, in the limit point and intermediate cases, spectral inclusive holds.

Trotter’s limit formula for the Schrödinger equation with singular potential

We discuss the Schrödinger equation with singular potentials. Our focus is non-relativistic Schrödinger operators H with scalar potentials V defined on Rd, hence covering such quantum systems as atoms, molecules, and subatomic particles whether free, bound, or localized. By a “singular potential” V, we refer to the case when the corresponding Schrödinger operators H, with their natural minimal domain in L2(Rd), are not essentially self-adjoint. Since V is assumed real valued, the corresponding Hermitian symmetric operator H commutes with the conjugation in L2(Rd), and so (by von Neumann’s theorem), H has deficiency indices (n, n). The case of singular potentials V refers to when n > 0. Hence, by von Neumann’s theory, we know the full variety of all the self-adjoint extensions. Since the Trotter formula is restricted to the case when n = 0, and here n > 0, two questions arise: (i) existence of the Trotter limit and (ii) the nature of this limit. We answer (i) affirmatively. Our answer to (ii) is that when n > 0, the Trotter limit is a strongly continuous contraction semigroup; so it is not time-reversible.

Scalar field dynamics in a BTZ background with generic boundary conditions

We revisit the dynamics of a massive scalar field in a Banados, Teitelboim, and Zanelli background taking into account the lack of global hyperbolicity of the spacetime. We approach this issue using the strategy of Ishibashi and Wald which finds a unique smooth solution as the causal evolution of initial data, each possible evolution corresponding to a positive self-adjoint extension of certain operator in a Hilbert space on the initial surface. Moreover, solutions obtained this way are the most general ones satisfying a few physically sensible requirements. This procedure is intimately related to the choice of boundary conditions and the existence of bound states. We find that the scalar field dynamics in the (effective) mass window -3/4le m_e^2ell ^2<0 can be well defined within a one-parametric family of distinct boundary conditions (-3/4 being the conformally coupled case), while for m_e^2ell ^2ge 0 the boundary condition is unique (only one self-adjoint extension is possible). It is argued that there is no sensible evolution possible for m_e^2ell ^2<-1, and also it is shown that in the range m_e^2ell ^2 in [-1,-3/4) there is a U(1) family of allowed boundary conditions, however, the positivity of the self-adjoint extensions is only motivated but not proven. We focus mainly on describing the dynamics of such evolutions given the initial data and all possible boundary conditions, and in particular we show the energy is always positive and conserved.

Gravitational waves propagation in nondynamical Chern–Simons gravity

We investigate the propagation of gravitational waves in linearized Chern–Simons (CS) modified gravity by considering two nondynamical models for the coupling field [Formula: see text]: (i) a domain wall and (ii) a surface layer of [Formula: see text], motivated by their relevance in condensed matter physics. We demonstrate that the metric and its first derivative become discontinuous for a domain wall of [Formula: see text], and we determine the boundary conditions by realizing that the additional contribution to the wave equation corresponds to one of the self-adjoint extensions of the D'Alembert operator. Nevertheless, such discontinuous metric satisfies the area matching conditions introduced by Barrett. On the other hand, the propagation through a surface layer of [Formula: see text] behaves similarly to the propagation of electromagnetic waves in CS extended electrodynamics. In both cases, we calculate the corresponding reflection and transmission amplitudes. As a consequence of the distributional character of the additional terms in the equations that describe wave propagation, the results obtained for the domain wall are not reproduced when the thickness of the surface layer goes to zero, as one could naively expect.

Self-adjoint extensions and unitary operators on the boundary

We establish a bijection between the self-adjoint extensions of the Laplace operator on a bounded regular domain and the unitary operators on the boundary. Each unitary encodes a specific relation between the boundary value of the function and its normal derivative. This bijection sets up a characterization of all physically admissible dynamics of a nonrelativistic quantum particle confined in a cavity. More- over, this correspondence is discussed also at the level of quadratic forms. Finally, the connection between this parametrization of the extensions and the classical one, in terms of boundary self-adjoint operators on closed subspaces, is shown.

Spaces of Smooth and Generalized Vectors of the Generator of an Analytic Semigroup and Their Applications

For a strongly continuous analytic semigroup {e tA }t ≥ 0 of linear operators in a Banach space B, we study some locally convex spaces of smooth and generalized vectors of its generator A, as well as the extensions and restrictions of this semigroup to these spaces. We extend the Lagrange result on the representation of a translation group in the form of exponential series to the case of these semigroups and solve the Hille problem on the description of the set of all vectors x∈𝔅 for which the limit $$ {\lim}_{n\to \infty }{\left(I+\frac{tA}{n}\right)}^nx $$ exists and coincides with e tA x. Moreover, we present a brief survey of particular problems whose solutions are required for the introduction of the above-mentioned spaces, namely, the description of all maximal dissipative (self-adjoint) extensions of a dissipative (symmetric) operator, the representation of solutions of operator-differential equations on an open interval and the investigation of their boundary values, and the existence of solutions of an abstract Cauchy problem in various classes of analytic vector-valued functions.

On the almost everywhere convergence of the eigenfunction expansions from Liouville classes

The relevance of waves in quantum mechanics naturally implies that the decomposition of arbitrary wave packets in terms of monochromatic waves plays an important role in applications of the theory. When eigenfunction expansions does not converge, then the expansions of the functions with certain smoothness should be considered. Such functions gained prominence primarily through their application in quantum mechanics. In this work we study the almost everywhere convergence of the eigenfunction expansions from Liouville classes , related to the self-adjoint extension of the Laplace operator in torus TN. The sufficient conditions for summability is obtained using the modified Poisson formula. Isomorphism properties of the elliptic differential operators is applied in order to obtain estimation for the Fourier series of the functions from the classes of Liouville .

Finite element method to solve the spectral problem for arbitrary self-adjoint extensions of the Laplace–Beltrami operator on manifolds with a boundary

A numerical scheme to compute the spectrum of a large class of self-adjoint extensions of the Laplace–Beltrami operator on manifolds with boundary in any dimension is presented. The algorithm is based on the characterisation of a large class of self-adjoint extensions of Laplace–Beltrami operators in terms of their associated quadratic forms. The convergence of the scheme is proved. A two-dimensional version of the algorithm is implemented effectively and several numerical examples are computed showing that the algorithm treats in a unified way a wide variety of boundary conditions.