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Journal articles on the topic 'Essentially self-adjoint'

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Published: 10 December 2022

Last updated: 29 January 2023

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1

Sebestyén, Zoltán, and Zsigmond Tarcsay. "Characterizations of essentially self-adjoint and skew-adjoint operators." Studia Scientiarum Mathematicarum Hungarica 52, no. 3 (September 2015): 371–85. http://dx.doi.org/10.1556/012.2015.52.3.1300.

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An extension of von Neumann’s characterization of essentially selfadjoint operators is given among not necessarily densely defined symmetric operators which are assumed to be closable. Our arguments are of algebraic nature and follow the idea of [1].

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2

Khrushchev, S. V. "Uniqueness theorems and essentially self-adjoint operators." Journal of Soviet Mathematics 36, no. 3 (February 1987): 403–8. http://dx.doi.org/10.1007/bf01839612.

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3

Kalf, H., and F. S. Rofe-Beketov. "On the essential self-adjointness of Schrödinger operators with locally integrable potentials." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 1 (1998): 95–106. http://dx.doi.org/10.1017/s0308210500027177.

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Results by Simader, Brézis and Cycon of the genre ‘locally essentially self-adjoint implies globally essentially self-adjoint’ are generalised to Schrödinger operators that are not necessarily bounded from below.

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4

Falomir, H. A., and P. A. G. Pisani. "Spectral functions of non-essentially self-adjoint operators." Journal of Physics A: Mathematical and Theoretical 45, no. 37 (September 4, 2012): 374017. http://dx.doi.org/10.1088/1751-8113/45/37/374017.

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5

Kappeler, Th. "Positive perturbations of self-adjoint Schrödinger operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 99, no. 3-4 (1985): 241–48. http://dx.doi.org/10.1017/s0308210500014268.

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SynopsisIn this paper, we prove that a positive perturbation T = T0 + q (q ≧ 0 and in ) of an essentially self-adjoint Schrödinger operator T0 = −Δ + q0 on is again essentially self-adjoint if T is relatively bounded with respect to T0. An application of the method of the proof to positive approximations of elements u ≧ 0 in D(T) by a positive sequence in is given.

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6

Fatehi, Mahsa, and Mahmood Haji Shaabani. "Certain nontrivially essentially self-adjoint weighted composition operators onH2and." Complex Variables and Elliptic Equations 59, no. 12 (January 28, 2014): 1626–35. http://dx.doi.org/10.1080/17476933.2013.870560.

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7

Neidhardt, Hagen, and Valentin Zagrebnov. "On the Right Hamiltonian for Singular Perturbations: General Theory." Reviews in Mathematical Physics 09, no. 05 (July 1997): 609–33. http://dx.doi.org/10.1142/s0129055x97000221.

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Let the pair of self-adjoint operators {A≥0,W≤0} be such that: (a) there is a dense domain [Formula: see text] such that [Formula: see text] is semibounded from below (stability domain), (b) the symmetric operator [Formula: see text] is not essentially self-adjoint (singularity of the perturbation), (c) the Friedrichs extension [Formula: see text] of [Formula: see text] is maximal with respect to W, i.e., [Formula: see text]. [Formula: see text]. Let [Formula: see text] be a regularizing sequence of bounded operators which tends in the strong resolvent sense to W. The abstract problem of the right Hamiltonian is: (i) to give conditions such that the limit H of self-adjoint regularized Hamiltonians [Formula: see text] exists and is unique for any self-adjoint extension [Formula: see text] of [Formula: see text], (ii) to describe the limit H. We show that under the conditions (a)–(c) there is a regularizing sequence [Formula: see text] such that [Formula: see text] tends in the strong resolvent sense to unique (right Hamiltonian) [Formula: see text], otherwise the limit is not unique.

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8

ICHINOSE, TAKASHI, and WATARU ICHINOSE. "ON THE ESSENTIAL SELF-ADJOINTNESS OF THE RELATIVISTIC HAMILTONIAN WITH A NEGATIVE SCALAR POTENTIAL." Reviews in Mathematical Physics 07, no. 05 (July 1995): 709–21. http://dx.doi.org/10.1142/s0129055x95000281.

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The relativistic quantum Hamiltonian H describing a spinless particle in an electromagnetic field is considered. H is associated with the classical Hamiltonian [Formula: see text] via Weyl’s correspondence. In the previous papers the second author has proved that H is essentially self-adjoint on [Formula: see text] if the scalar potential V(x) is a function bounded from below by a polynomial in x. In the present paper this result will be extended to show that H is essentially self-adjoint there if V(x) is bounded from below by -C exp a|x| for some positive constants C and a. Ameliorated is also the condition on the vector potential A(x). The result of this kind is quite different from that on the non-relativistic operator, i.e. the Schrödinger operator, but much closer to that on the Dirac operator.

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9

ALBEVERIO, SERGIO, and VOLODYMYR KOSHMANENKO. "ON THE PROBLEM OF THE RIGHT HAMILTONIAN UNDER SINGULAR FORM-SUM PERTURBATIONS." Reviews in Mathematical Physics 12, no. 01 (January 2000): 1–24. http://dx.doi.org/10.1142/s0129055x00000022.

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Let a perturbation of the self-adjoint operator H0>0 in the Hilbert space ℋ be given by an operator V (or by a quadratic form ν) which is possibly singular and in general nonpositive, so H0+V on [Formula: see text] is only a symmetric operator with nontrivial deficiency indices. The definition of the sum [Formula: see text] in the sense of quadratic forms is extended to cases which are not covered by the well-known KLMN-theorem and conditions are found which ensure the unique self-adjoint realization of H in ℋ. It is also shown that ℋ coincides with the strong resolvent limit of the approximating sequence Hn = H0+Vn, where Vn are bounded self-adjoint operators such that Vn → V in a suitable sense. Essentially that operator V might be strongly singular and acts in the H0-scale of spaces, V:ℋ+→ℋ-.

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10

MILATOVIC, OGNJEN. "POSITIVE PERTURBATIONS OF SELF-ADJOINT SCHRÖDINGER OPERATORS ON RIEMANNIAN MANIFOLDS." International Journal of Geometric Methods in Modern Physics 02, no. 04 (August 2005): 543–52. http://dx.doi.org/10.1142/s0219887805000715.

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We consider a Schrödinger differential expression L0 = ΔM + V0 on a Riemannian manifold (M,g) with metric g, where ΔM is the scalar Laplacian on M and V0 is a real-valued locally square integrable function on M. We consider a perturbation L0 + V, where V is a non-negative locally square-integrable function on M, and give sufficient conditions for L0 + V to be essentially self-adjoint on [Formula: see text]. This is an extension of a result of T. Kappeler.

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11

Voronin, A. V. "Discrete vacuum superselection rule in Wightman theory with essentially self-adjoint field operators." Theoretical and Mathematical Physics 66, no. 1 (January 1986): 8–19. http://dx.doi.org/10.1007/bf01028934.

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12

Gadella, Manuel, José Hernández-Muñoz, Luis Miguel Nieto, and Carlos San Millán. "Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian." Symmetry 13, no. 2 (February 21, 2021): 350. http://dx.doi.org/10.3390/sym13020350.

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We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator −d2/dx2 on L2[−a,a], a>0, that is, the one dimensional infinite square well. First of all, we classify these self-adjoint extensions in terms of several choices of the parameters determining each of the extensions. There are essentially two big groups of extensions. In one, the ground state has strictly positive energy. On the other, either the ground state has zero or negative energy. In the present paper, we show that each of the extensions belonging to the first group (energy of ground state strictly positive) has an infinite sequence of supersymmetric partners, such that the ℓ-th order partner differs in one energy level from both the (ℓ−1)-th and the (ℓ+1)-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of −d2/dx2 come from a transcendental equation and are all infinite. For the case under our study, we determine the eigenvalues, which are also infinite, all the extensions have a purely discrete spectrum, and their respective eigenfunctions for all of its ℓ-th supersymmetric partners of each extension.

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13

TAMURA, HIDEO. "NORM RESOLVENT CONVERGENCE TO MAGNETIC SCHRÖDINGER OPERATORS WITH POINT INTERACTIONS." Reviews in Mathematical Physics 13, no. 04 (April 2001): 465–511. http://dx.doi.org/10.1142/s0129055x01000697.

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The Schrödinger operator with δ-like magnetic field at the origin in two dimensions is not essentially self-adjoint. It has the deficiency indices (2, 2) and each self-adjoint extension is realized as a differential operator with some boundary conditions at the origin. We here consider Schrödinger operators with magnetic fields of small support and study the norm resolvent convergence to Schrödinger operator with δ-like magnetic field. We are concerned with the boundary conditions realized in the limit when the support shrinks. The results obtained heavily depend on the total flux of magnetic field and on the resonance space at zero energy, and the proof is based on the analysis at low energy for resolvents of Schrödinger operators with magnetic potentials slowly falling off at infinity.

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14

Faierman, M. "An elliptic boundary problem involving an indefinite weight." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 2 (April 2000): 287–305. http://dx.doi.org/10.1017/s0308210500000160.

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The spectral theory for non-self-adjoint elliptic boundary problems involving an indefinite weight function has only been established for the case of higher-order operators under the assumption that the reciprocal of the weight function is essentially bounded. In this paper we are concerned with the spectral theory for a case where the weight function vanishes on a set of positive measure.

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15

Khalilov, V. R. "Quantum states of a neutral massive fermion with an anomalous magnetic moment in an Aharonov–Casher field." International Journal of Modern Physics A 32, no. 18 (June 28, 2017): 1750111. http://dx.doi.org/10.1142/s0217751x17501111.

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The planar nonrelativistic quantum dynamics of a neutral massive fermion with an anomalous magnetic moment (AMM) in the electric field of infinitely long and thin thread with a charge density distributed uniformly along it (an Aharonov–Casher field) is examined. The relevant Hamiltonian is singular and requires additional specification of a one-parameter self-adjoint extension, which can be given in terms of physically acceptable boundary conditions. We find all possible self-adjoint Hamiltonians with an Aharonov–Casher field (ACF) by constructing the corresponding Hilbert space of square-integrable functions, including the [Formula: see text] region, for all their Hamiltonians. We determine the most relevant physical quantities, such as energy spectrum and wave functions and discuss their correspondence with those obtained by the physical regularization procedure. We show that energy levels of bound states are simple poles of the scattering amplitude. It is shown that the scattering amplitudes and cross-sections depend essentially on the initial-state spin of fermions.

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16

Goodman, F. M., P. E. T. Jorgensen, and C. Peligrad. "Smooth derivations commuting with Lie group actions." Mathematical Proceedings of the Cambridge Philosophical Society 99, no. 2 (March 1986): 307–14. http://dx.doi.org/10.1017/s0305004100064227.

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N. S. Poulsen, motivated in part by questions from relativistic quantum scattering theory, studied symmetric operators S in Hilbert space commuting with a unitary representation U of a Lie group G. (The group of interest in the physical setting is the Poincaré group.) He proved ([17], corollary 2·2) that if S is defined on the space of C∞-vectors for U (i.e. D(S) ⊇ ℋ∞(U)), then S is essentially self-adjoint.

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17

MUHLY, PAUL S., and BARUCH SOLEL. "DILATIONS AND COMMUTANT LIFTING FOR SUBALGEBRAS OF GROUPOID C*-ALGEBRAS." International Journal of Mathematics 05, no. 01 (February 1994): 87–123. http://dx.doi.org/10.1142/s0129167x9400005x.

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Let B be a nuclear C*-algebra that has a diagonal subalgebra D in the sense of Kumjian and let A be a closed, not necessarily self-adjoint subalgebra of B that contains D such that A + A* is dense in B. We show that every contractive representation of A has an essentially unique minimal dilation to a C*-representation of B and that the commutant of the representation of A can be lifted to the commutant of the dilation without increasing norms.

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18

GREGORATTI, M. "ON THE HAMILTONIAN OPERATOR ASSOCIATED TO SOME QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 03, no. 04 (December 2000): 483–503. http://dx.doi.org/10.1142/s0219025700000327.

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We consider the quantum stochastic differential equation introduced by Hudson and Parthasarathy to describe the stochastic evolution of an open quantum system together with its environment. We study the (unbounded) Hamiltonian operator generating the unitary group connected, as shown by Frigerio and Maassen, to the solution of the equation. We find a densely defined restriction of the Hamiltonian operator; in some special cases we prove that this restriction is essentially self-adjoint and in one particular case we get the whole Hamiltonian with its full domain.

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19

Wong, M. W. "Minimal and Maximal Operator Theory With Applications." Canadian Journal of Mathematics 43, no. 3 (June 1, 1991): 617–27. http://dx.doi.org/10.4153/cjm-1991-036-7.

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AbstractLetXbe a complex Banach space andAa linear operator fromXintoXwith dense domain. We construct the minimal and maximal operators of the operatorAand prove that they are equal under reasonable hypotheses on the spaceXand operatorA. As an application, we obtain the existence and regularity of weak solutions of linear equations on the spaceX. As another application we obtain a criterion for a symmetric operator on a complex Hilbert space to be essentially self-adjoint. An application to pseudo-differential operators of the Weyl type is given.

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20

Dereziński, Jan, and Daniel Siemssen. "Feynman propagators on static spacetimes." Reviews in Mathematical Physics 30, no. 03 (March 6, 2018): 1850006. http://dx.doi.org/10.1142/s0129055x1850006x.

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We consider the Klein–Gordon equation on a static spacetime and minimally coupled to a static electromagnetic potential. We show that it is essentially self-adjoint on [Formula: see text]. We discuss various distinguished inverses and bisolutions of the Klein–Gordon operator, focusing on the so-called Feynman propagator. We show that the Feynman propagator can be considered the boundary value of the resolvent of the Klein–Gordon operator, in the spirit of the limiting absorption principle known from the theory of Schrödinger operators. We also show that the Feynman propagator is the limit of the inverse of the Wick rotated Klein–Gordon operator.

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21

WARD, A. D. "ON THE VARIATIONAL CONSTANT ASSOCIATED TO THE -HARDY INEQUALITY." Journal of the Australian Mathematical Society 102, no. 3 (September 23, 2016): 405–19. http://dx.doi.org/10.1017/s1446788716000276.

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Let$\unicode[STIX]{x1D6FA}$be a domain in$\mathbb{R}^{m}$with nonempty boundary. In Ward [‘On essential self-adjointness, confining potentials and the$L_{p}$-Hardy inequality’, PhD Thesis, NZIAS Massey University, New Zealand, 2014] and [‘The essential self-adjointness of Schrödinger operators on domains with non-empty boundary’,Manuscripta Math.150(3) (2016), 357–370] it was shown that the Schrödinger operator$H=-\unicode[STIX]{x1D6E5}+V$, with domain of definition$D(H)=C_{0}^{\infty }(\unicode[STIX]{x1D6FA})$and$V\in L_{\infty }^{\text{loc}}(\unicode[STIX]{x1D6FA})$, is essentially self-adjoint provided that$V(x)\geq (1-\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA}))/d(x)^{2}$. Here$d(x)$is the Euclidean distance to the boundary and$\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$is the nonnegative constant associated to the$L_{2}$-Hardy inequality. The conditions required for a domain to admit an$L_{2}$-Hardy inequality are well known and depend intimately on the Hausdorff or Aikawa/Assouad dimension of the boundary. However, there are only a handful of domains where the value of$\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$is known explicitly. By obtaining upper and lower bounds on the number of cubes appearing in the$k\text{th}$generation of the Whitney decomposition of$\unicode[STIX]{x1D6FA}$, we derive an upper bound on$\unicode[STIX]{x1D707}_{p}(\unicode[STIX]{x1D6FA})$, for$p>1$, in terms of the inner Minkowski dimension of the boundary.

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22

Turov, M. M., V. E. Fedorov, and B. T. Kien. "Linear Inverse Problems for Multi-term Equations with Riemann — Liouville Derivatives." Bulletin of Irkutsk State University. Series Mathematics 38 (2021): 36–53. http://dx.doi.org/10.26516/1997-7670.2021.38.36.

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The issues of well-posedness of linear inverse coefficient problems for multi-term equations in Banach spaces with fractional Riemann – Liouville derivatives and with bounded operators at them are considered. Well-posedness criteria are obtained both for the equation resolved with respect to the highest fractional derivative, and in the case of a degenerate operator at the highest derivative in the equation. Two essentially different cases are investigated in the degenerate problem: when the fractional part of the order of the second-oldest derivative is equal to or different from the fractional part of the order of the highest fractional derivative. Abstract results are applied in the study of inverse problems for partial differential equations with polynomials from a self-adjoint elliptic differential operator with respect to spatial variables and with Riemann – Liouville derivatives in time.

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23

Mangut, M., and O. Gurtug. "Quantum probe of time-like naked singularities for electrically and magnetically charged black holes in a model of nonlinear electrodynamics." Modern Physics Letters A 35, no. 29 (July 29, 2020): 2050242. http://dx.doi.org/10.1142/s0217732320502429.

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The time-like naked singularities of the electrically and magnetically charged black hole solutions obtained in a model of nonlinear electrodynamics proposed by Kruglov is investigated within the framework of quantum mechanics. In view of quantum mechanics, the spacetime is quantum regular provided that the time evolution of the test quantum wave packet uniquely propagates on an underlying background. Rigorous calculations have shown that when the singularity is probed with specific quantum wave/particle modes, the quantum wave operator turns out to be essentially self-adjoint. Thus, the time evolution of the quantum wave/particle is determined uniquely. In the case of electrically charged black hole background, the unique evolution is restricted to [Formula: see text]-wave only. For the two different magnetically charged black hole backgrounds, the time evolution is restricted to different modes for each case.

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24

Bessa, Gregório P., Luquésio P. Jorge, Barnabé P. Lima, and José F. Montenegro. "Fundamental tone estimates for elliptic operators in divergence form and geometric applications." Anais da Academia Brasileira de Ciências 78, no. 3 (September 2006): 391–404. http://dx.doi.org/10.1590/s0001-37652006000300001.

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We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 1 > 0 in terms of the r-th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).

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25

Bendikov, Alexander, and Wojciech Cygan. "Poisson approximation related to spectra of hierarchical Laplacians." Stochastics and Dynamics 20, no. 05 (December 30, 2019): 2050035. http://dx.doi.org/10.1142/s0219493720500355.

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Let [Formula: see text] be a locally compact separable ultrametric space. Given a measure [Formula: see text] on [Formula: see text] and a function [Formula: see text] defined on the set of all non-singleton balls [Formula: see text] of [Formula: see text], we consider the hierarchical Laplacian [Formula: see text]. The operator [Formula: see text] acts in [Formula: see text] is essentially self-adjoint and has a purely point spectrum. Choosing a sequence [Formula: see text] of i.i.d. random variables, we consider the perturbed function [Formula: see text] and the perturbed hierarchical Laplacian [Formula: see text] Under certain conditions, the density of states [Formula: see text] exists and it is a continuous function. We choose a point [Formula: see text] such that [Formula: see text] and build a sequence of point processes defined by the eigenvalues of [Formula: see text] located in the vicinity of [Formula: see text]. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity [Formula: see text].

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26

Shi, Feng, Guoping Liang, Yubo Zhao, and Jun Zou. "New Splitting Methods for Convection-Dominated Diffusion Problems and Navier-Stokes Equations." Communications in Computational Physics 16, no. 5 (November 2014): 1239–62. http://dx.doi.org/10.4208/cicp.031013.030614a.

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AbstractWe present a new splitting method for time-dependent convention-dominated diffusion problems. The original convention diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a convection problem and a diffusion problem are solved successively. A few important features of the scheme lie in the facts that the convection subproblem is solved explicitly and multistep techniques can be used to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; while the diffusion subproblem is always self-adjoint and coercive so that they can be solved efficiently using many existing optimal preconditioned iterative solvers. The scheme can be extended for solving the Navier-Stokes equations, where the nonlinearity is resolved by a linear explicit multistep scheme at the convection step, while only a generalized Stokes problem is needed to solve at the diffusion step and the major stiffness matrix stays invariant in the time marching process. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step and multistep variants of the new scheme.

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27

Bellino, Vito Flavio, and Giampiero Esposito. "Fractional linear maps in general relativity and quantum mechanics." International Journal of Geometric Methods in Modern Physics 18, no. 10 (June 24, 2021): 2150157. http://dx.doi.org/10.1142/s0219887821501577.

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This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi–Metzner–Sachs transformations in general relativity. The analogy therefore arising suggests that further investigations might be performed for a theory in which the role of fractional linear maps is viewed as a bridge between the quantum theory and general relativity. The second aspect to point out is the possibility of interpreting the limit-point condition at both ends of the positive real line, for a second-order singular differential operator, which occurs frequently in applied quantum mechanics, as the limiting procedure arising from a very particular Kleinian group which is the hyperbolic cyclic group. In this framework, this work finds that a consistent system of equations can be derived and studied. Hence, one is led to consider the entire transcendental functions, from which it is possible to construct a fundamental system of solutions of a second-order differential equation with singular behavior at both ends of the positive real line, which in turn satisfy the limit-point conditions. Further developments in this direction might also be obtained by constructing a fundamental system of solutions and then deriving the differential equation whose solutions are the independent system first obtained. This guarantees two important properties at the same time: the essential self-adjointness of a second-order differential operator and the existence of a conserved quantity which is an automorphic function for the cyclic group chosen.

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Rottensteiner, David, and Michael Ruzhansky. "Harmonic and anharmonic oscillators on the Heisenberg group." Journal of Mathematical Physics 63, no. 11 (November 1, 2022): 111509. http://dx.doi.org/10.1063/5.0106068.

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In this article, we present a notion of the harmonic oscillator on the Heisenberg group H n, which, under a few reasonable assumptions, forms the natural analog of a harmonic oscillator on [Formula: see text]: a negative sum of squares of operators on H n, which is essentially self-adjoint on L2(H n) with purely discrete spectrum and whose eigenvectors are Schwartz functions forming an orthonormal basis of L2(H n). The differential operator in question is determined by the Dynin–Folland group—a stratified nilpotent Lie group—and its generic unitary irreducible representations, which naturally act on L2(H n). As in the Euclidean case, our notion of harmonic oscillator on H n extends to a whole class of so-called anharmonic oscillators, which involve left-invariant derivatives and polynomial potentials of order greater or equal 2. These operators, which enjoy similar properties as the harmonic oscillator, are in one-to-one correspondence with positive Rockland operators on the Dynin–Folland group. The latter part of this article is concerned with spectral multipliers. We obtain useful L p- L q-estimates for a large class of spectral multipliers of the sub-Laplacian [Formula: see text] and, in fact, of generic Rockland operators on graded groups. As a by-product, we obtain explicit hypoelliptic heat semigroup estimates and recover the continuous Sobolev embeddings on graded groups, provided 1 < p ≤ 2 ≤ q < ∞.

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Herrmann, Lukas, Kristin Kirchner, and Christoph Schwab. "Multilevel approximation of Gaussian random fields: Fast simulation." Mathematical Models and Methods in Applied Sciences 30, no. 01 (December 30, 2019): 181–223. http://dx.doi.org/10.1142/s0218202520500050.

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We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain [Formula: see text] or, more generally, by a compact metric space [Formula: see text] such as a compact [Formula: see text]-manifold [Formula: see text]. A colored GRF [Formula: see text], admissible for our algorithms, solves the stochastic fractional-order equation [Formula: see text] for some [Formula: see text], where [Formula: see text] is a linear, local, second-order elliptic self-adjoint differential operator in divergence form and [Formula: see text] is white noise on [Formula: see text]. We thus consider GRFs on [Formula: see text] with covariance operators of the form [Formula: see text]. The proposed algorithms numerically approximate samples of [Formula: see text] on nested sequences [Formula: see text] of regular, simplicial partitions [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. Work and memory to compute one approximate realization of the GRF [Formula: see text] on the triangulation [Formula: see text] of [Formula: see text] with consistency [Formula: see text], for some consistency order [Formula: see text], scale essentially linearly in [Formula: see text], independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic “coloring” operator [Formula: see text] to white noise [Formula: see text]. For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms.

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30

Takaesu, Toshimitsu. "Essential Self-Adjointness of Anticommutative Operators." Journal of Mathematics 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/265349.

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The self-adjoint extensions of symmetric operators satisfying anticommutation relations are considered. It is proven that an anticommutative type of the Glimm-Jaffe-Nelson commutator theorem follows. Its application to an abstract Dirac operator is also considered.

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31

Klein, Markus, and Elke Rosenberger. "The tunneling effect for a class of difference operators." Reviews in Mathematical Physics 30, no. 04 (April 19, 2018): 1830002. http://dx.doi.org/10.1142/s0129055x18300029.

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We analyze a general class of self-adjoint difference operators [Formula: see text] on [Formula: see text], where [Formula: see text] is a multi-well potential and [Formula: see text] is a small parameter. We give a coherent review of our results on tunneling up to new sharp results on the level of complete asymptotic expansions (see [30–35]).Our emphasis is on general ideas and strategy, possibly of interest for a broader range of readers, and less on detailed mathematical proofs.The wells are decoupled by introducing certain Dirichlet operators on regions containing only one potential well. Then the eigenvalue problem for the Hamiltonian [Formula: see text] is treated as a small perturbation of these comparison problems. After constructing a Finslerian distance [Formula: see text] induced by [Formula: see text], we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by this distance to the well. It follows with microlocal techniques that the first [Formula: see text] eigenvalues of [Formula: see text] converge to the first [Formula: see text] eigenvalues of the direct sum of harmonic oscillators on [Formula: see text] located at several wells. In a neighborhood of one well, we construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low-lying eigenvalues of [Formula: see text]. These are obtained from eigenfunctions or quasimodes for the operator [Formula: see text], acting on [Formula: see text], via restriction to the lattice [Formula: see text].Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrödinger operator (see [22]), the remainder is exponentially small and roughly quadratic compared with the interaction matrix. We give weighted [Formula: see text]-estimates for the difference of eigenfunctions of Dirichlet-operators in neighborhoods of the different wells and the associated WKB-expansions at the wells. In the last step, we derive full asymptotic expansions for interactions between two “wells” (minima) of the potential energy, in particular for the discrete tunneling effect. Here we essentially use analysis on phase space, complexified in the momentum variable. These results are as sharp as the classical results for the Schrödinger operator in [22].

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32

Xu, Guixin, and Yuming Shi. "Essential spectra of self-adjoint relations under relatively compact perturbations." Linear and Multilinear Algebra 66, no. 12 (November 14, 2017): 2438–67. http://dx.doi.org/10.1080/03081087.2017.1399979.

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33

Ibrogimov, Orif O. "Essential spectrum of non-self-adjoint singular matrix differential operators." Journal of Mathematical Analysis and Applications 451, no. 1 (July 2017): 473–96. http://dx.doi.org/10.1016/j.jmaa.2017.02.017.

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34

Ando, Hiroshi, and Yasumichi Matsuzawa. "The Weyl–von Neumann theorem and Borel complexity of unitary equivalence modulo compacts of self-adjoint operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 6 (October 29, 2015): 1115–44. http://dx.doi.org/10.1017/s0308210515000293.

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The Weyl–von Neumann theorem asserts that two bounded self-adjoint operators A, B on a Hilbert space H are unitarily equivalent modulo compacts, i.e.uAu* + K = B for some unitary u 𝜖 u(H) and compact self-adjoint operator K, if and only if A and B have the same essential spectrum: σess (A) = σess (B). We study, using methods from descriptive set theory, the problem of whether the above Weyl–von Neumann result can be extended to unbounded operators. We show that if H is separable infinite dimensional, the relation of unitary equivalence modulo compacts for bounded self-adjoint operators is smooth, while the same equivalence relation for general self-adjoint operators contains a dense Gδ-orbit but does not admit classification by countable structures. On the other hand, the apparently related equivalence relation A ~ B ⇔ ∃u 𝜖 U(H) [u(A-i)–1u* - (B-i)–1 is compact] is shown to be smooth.

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35

Kaiqi, Yu. "Schrödinger operators with magnetic and electric potentials." Bulletin of the Australian Mathematical Society 50, no. 2 (October 1994): 299–312. http://dx.doi.org/10.1017/s0004972700013757.

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In the present paper, we consider Schrödinger operators which are formally given by . In Section 2 and 3 we prove that P has a regularly accretive extension which is a self-adjoint extension of P and it is the only self-adjoint realisation of P in L2 (RN) when satisfies = (a1, a2, …, aN) ∈ , aj, real-valued, , real-valued and the negative part V-:= max(0, -V) satisfys , with constants 0 ≤ C1 < 1, C2 ≥ 0 independent of V. In Section 4, we prove that P is essential self-adjoint on when , V sat0isfy ; V = V1 + V2, V real-valued, , i = 1, 2, V1(x) ≥ –C |x|2, for x ∈ RN with C ≥ 0 and 0 ≥ V2 ∈ KN.

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36

Balti, Marwa. "Non self-adjoint Laplacians on a directed graph." Filomat 31, no. 18 (2017): 5671–83. http://dx.doi.org/10.2298/fil1718671b.

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We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We analyse spectral properties of this Laplacian under a Kirchhoff assumption. Moreover we establish isoperimetric inequalities in terms of the numerical range to show the absence of the essential spectrum of the Laplacian on heavy end directed graphs.

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37

Shi, Yuming. "Stability of essential spectra of self-adjoint subspaces under compact perturbations." Journal of Mathematical Analysis and Applications 433, no. 2 (January 2016): 832–51. http://dx.doi.org/10.1016/j.jmaa.2015.08.017.

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38

Pushnitski, Alexander. "Spectral Theory of Discontinuous Functions of Self-Adjoint Operators: Essential Spectrum." Integral Equations and Operator Theory 68, no. 1 (March 24, 2010): 75–99. http://dx.doi.org/10.1007/s00020-010-1789-4.

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39

GEORGESCU, VLADIMIR, and ANDREI IFTIMOVICI. "LOCALIZATIONS AT INFINITY AND ESSENTIAL SPECTRUM OF QUANTUM HAMILTONIANS I: GENERAL THEORY." Reviews in Mathematical Physics 18, no. 04 (May 2006): 417–83. http://dx.doi.org/10.1142/s0129055x06002693.

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We isolate a large class of self-adjoint operators H whose essential spectrum is determined by their behavior at x ~ ∞ and we give a canonical representation of σ ess (H) in terms of spectra of limits at infinity of translations of H.

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40

ERCOLESSI, E., P. TEOTONIO-SOBRINHO, and G. BIMONTE. "DISCRETIZED LAPLACIANS ON AN INTERVAL AND THEIR RENORMALIZATION GROUP." International Journal of Modern Physics A 09, no. 25 (October 10, 1994): 4485–509. http://dx.doi.org/10.1142/s0217751x94001783.

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The Laplace operator admits infinite self-adjoint extensions when considered on a segment of the real line. They have different domains of essential self-adjointness characterized by a suitable set of boundary conditions on the wave functions. In this paper we show how these extensions can be recovered by studying the continuum limit of certain discretized versions of the Laplace operator on a lattice. Associated to this limiting procedure, there is a renormalization flow in the finite-dimensional parameter space describing the discretized operators. This flow is shown to have infinite fixed points, corresponding to the self-adjoint extensions characterized by scale-invariant boundary conditions. The other extensions are recovered by looking at the other trajectories of the flow.

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41

Wilson, Robert Howard. "Non-self-adjoint difference operators and their spectrum." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2057 (April 27, 2005): 1505–31. http://dx.doi.org/10.1098/rspa.2004.1416.

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Initially, this paper is a discrete analogue of the work of Brown et al. (1999 Proc. R. Soc. A 455 , 1235–1257) on second-order differential equations with complex coefficients. That is, we investigate the general non-self-adjoint second-order difference expression where the coefficients p n and q n are complex and Δ is the forward difference operator, i.e. Δ x n = x n +1 − x n . Properties of the so-called Hellinger–Nevanlinna m -function for the recurrence relation Mx n = λ w n x n , where the w n are real and positive, are examined, and relationships between the properties of the m -function and the spectrum of the associated operator are explored. However, an essential difference between the continuous and the discrete case arises in the way in which we define the operator natural to the problem. Nevertheless, analogous results regarding the spectrum of this operator are obtained.

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42

Takaesu, Toshimitsu. "Essential spectrum of a fermionic quantum field model." Infinite Dimensional Analysis, Quantum Probability and Related Topics 17, no. 04 (November 25, 2014): 1450024. http://dx.doi.org/10.1142/s0219025714500246.

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An interaction system of a fermionic quantum field is considered. The state space is defined by a tensor product space of a fermion Fock space and a Hilbert space. It is assumed that the total Hamiltonian is a self-adjoint operator on the state space and bounded from below. Then it is proven that a subset of real numbers is the essential spectrum of the total Hamiltonian. It is applied to a Yukawa interaction system, which is a system of a Dirac field coupled to a Klein–Gordon, and the HVZ theorem is obtained.

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43

Wong, M. W. "The spectrum of a one-dimensional pseudo-differential operator." Mathematical Proceedings of the Cambridge Philosophical Society 104, no. 3 (November 1988): 575–80. http://dx.doi.org/10.1017/s0305004100065762.

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AbstractWe describe the spectrum of a self-adjoint pseudo-differential operator on L2 (– ∞, ∞). We show that the essential spectrum coincides with the interval ([1, ∞) and give a lower bound for the lowest eigenvalue in (– ∞, 1). A sufficient condition for the existence of an eigenvalue in (– ∞, 1) is also given.

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44

Rasulov, Tulkin Husenovich. "THRESHOLD EIGENVALUES AND RESONANCES OF A FRIEDRICHS MODEL WITH RANK TWO PERTURBATION." Scientific Reports of Bukhara State University 3, no. 3 (March 30, 2019): 31–38. http://dx.doi.org/10.52297/2181-1466/2019/3/3/3.

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In this paper in the Hilbert space a bounded self-adjoint Friedrichs model with rank two perturbation is considered. Number and location of the eigenvalues of are studied. An existence conditions of these eigenvalues are found. Under some conditions we prove that the lower (upper) bound of the essential spectrum of is either threshold eigenvalue or virtual level of .

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45

Tulkin, Tulkin, and Shokhida Nematova. "INVESTIGATION OF THE SPECTRUM OF A GENERALIZED FRIEDRICHS MODEL: NON-INTEGRAL LATTICE CASE." Scientific Reports of Bukhara State University 3, no. 1 (January 30, 2019): 5–11. http://dx.doi.org/10.52297/2181-1466/2019/3/1/1.

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The article investigates the essential and discrete spectrum of the self-adjoint generalized Friedrichs model. This model corresponds to a system consisting of no more than two particles on a non-integral lattice, and operates in a truncated subspace of Fock space. The number and location of eigenvalues is determined according to the "interaction parameter". Anobvious form of the eigenvectors is found

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46

MĂNTOIU, MARIUS. "On the essential spectrum of phase-space anisotropic pseudodifferential operators." Mathematical Proceedings of the Cambridge Philosophical Society 154, no. 1 (July 2, 2012): 29–39. http://dx.doi.org/10.1017/s0305004112000321.

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AbstractA phase-space anisotropic operator in=L2(ℝn) is a self-adjoint operator whose resolvent family belongs to a naturalC*-completion of the space of Hörmander symbols of order zero. Equivalently, each member of the resolvent family is norm-continuous under conjugation with the Schrödinger unitary representation of the Heisenberg group. The essential spectrum of such a phase-space anisotropic operator is the closure of the union of usual spectra of all its “phase-space asymptotic localizations”, obtained as limits over diverging ultrafilters of ℝn×ℝn-translations of the operator. The result extends previous analysis of the purely configurational anisotropic operators, for which only the behavior at infinity in ℝnwas allowed to be non-trivial.

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47

Nazarov, S. A. "Gap in the essential spectrum of an elliptic formally self-adjoint system of differential equations." Differential Equations 46, no. 5 (May 2010): 730–41. http://dx.doi.org/10.1134/s0012266110050125.

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48

Smith, Dale T. "On the spectral analysis of self adjoint operators generated by second order difference equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 118, no. 1-2 (1991): 139–51. http://dx.doi.org/10.1017/s0308210500028973.

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SynopsisIn this paper, I shall consider operators generated by difference equations of the formwhere Δ is the forward difference operator, and a, p, and r are sequences of real numbers. The connection between the oscillation constant of this equation and the bottom of the essential spectrum of self-adjoint extensions of the operator generated by the equation is given, as well as various other information about the spectrum of such extensions. In particular, I derive conditions for the spectrum to have only countably many eigenvalues below zero, and a simple criterion for the invariance of the essential spectrum.

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49

KACHMAR, AYMAN. "WEYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS AND DE GENNES' BOUNDARY CONDITION." Reviews in Mathematical Physics 20, no. 08 (September 2008): 901–32. http://dx.doi.org/10.1142/s0129055x08003468.

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This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schrödinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an asymptotic expansion of the number of eigenvalues below the essential spectrum (Weyl-type asymptotics). The methods of proof rely on results concerning the asymptotic behavior of the first eigenvalue obtained in a previous work [10].

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50

N. Kuljanov, Utkir. "ON THE SPECTRUM OF THE TWO-PARTICLE SCHRÖDINGER OPERATOR WITH POINT POTENTIAL: ONE DIMENTIONAL CASE." Advances in Mathematics: Scientific Journal 10, no. 12 (December 3, 2021): 3569–78. http://dx.doi.org/10.37418/amsj.10.12.4.

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In the paper a one-dimensional two-particle quantum system interacted by two identical point interactions is considered. The corresponding Schr\"{o}\-dinger operator (energy operator) $h_\varepsilon$ depending on $\varepsilon,$ is constructed as a self-adjoint extension of the symmetric Laplace operator. The main results of the work are based to the study of the operator $h_\varepsilon.$ First the essential spectrum is described. The existence of unique negative eigenvalue of the Schr\"{o}dinger operator is proved. Further, this eigenvalue and corresponding eigenfunction are found.

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