We notice that, when a quantum system involves exceptional points, i.e. the special values of parameters where the Hamiltonian loses its self-adjointness and acquires the Jordan block structure, the corresponding classical system also exhibits a singular behaviour associated with restructuring of classical trajectories. The system with the crypto-Hermitian Hamiltonian H = (p + z)/2 − igz and hy...

In this paper we discussed the self-adjointness of the Maxwell’s equations with variable coefficients ε and μ. Three different Lagrangian are attained. By the Legendre transformation, a multisymplectic Bridge’s (Hamilton) form is obtained. Based on the multisymplectic structure, the multisymplectic conservation law of the system is derived and a nine-point Preissman multisymplectic scheme which...

Strong and Markov uniqueness problems in L for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on a–priori estimates is used. The extension of the problem to the L-setting is discussed. As a direct application essential self–adjointness and strong uniqueness in L is proved for the generator (with initial domain the bounded smooth cylinder functions) of the s...

The present paper introduces a mixed stress-displacement method for linear elasticity, as an extension of the work performed by Christiansen [15] for computational relativity. Moreover, we establish self-adjointness in a special sense and perform numerical simulations to assess the relevance of the method, including comparison with the recent TDNNS [30, 34] mixed formulation. Extensions to high...

We prove essential self-adjointness for semi-bounded below magnetic Schrödinger operators on complete Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. This is an extension of the Povzner–Wienholtz theorem. The proof uses the scheme of Wienholtz but requires a refined invariant integration by parts technique, as well as a use of a family of cu...

We fix a logical connection (Stone Pred : Set op −→ BA given by 2 as a schizophrenic object) and study coalgebraic modal logic that is induced by a functor T : Set −→ Set which is finitary, standard, preserves weak pullbacks and finite sets. We prove that for any such T , the cover modality nabla is a left (and its dual delta is a right) adjoint relative to Pω. We then consider monotone unary m...

A scale of Sobolev spaces is introduced to measure the global hypoellipticity of the twisted Laplacian. The essential self-adjointness of the twisted Laplacian is established and the domain of the unique self-adjoint extension is determined in terms of the Sobolev spaces. The global hypoellipticity of the twisted Laplacian in the Gelfand–Shilov spaces is also proved. 1. The twisted Laplacian Le...

We provide a systematic study on the possibility of supersymmetry (SUSY) for one dimensional quantum mechanical systems consisting of a pair of lines R or intervals [−l, l] each having a point singularity. We consider the most general singularities and walls (boundaries) at x = ±l admitted quantum mechanically, using a U(2) family of parameters to specify one singularity and similarly a U(1) fa...

In this note we prove that the maximally defined operator associated with the Dirac-type differential expression M(Q) = i ( d dx Im −Q −Q − d dx Im ) , where Q represents a symmetric m × m matrix (i.e., Q(x) = Q(x) a.e.) with entries in L loc (R), is J -self-adjoint, where J is the antilinear conjugation defined by J = σ1C, σ1 = ( 0 Im Im 0 ) and C(a1, . . . , am, b1, . . . , bm) = (a1, . . . ,...

1. Topological Vector Spaces 1 2. Banach Algebras 10 2.1. ∗–Algebras (over complexes) 15 2.2. Exercises 18 3. The Spectral Theorem 19 3.1. Problems on the Spectral Theorem (Multiplication Operator Form) 24 3.2. Integration with respect to a Projection Valued Measure 25 3.3. The Functional Calculus 32 4. Unbounded Operators 35 4.1. Closed, symmetric and self-adjoint operators 35 4.2. Differentia...