Letters in Mathematical Physics最新文献

We consider the 3D Mikhalev system,

which has first appeared in the context of KdV-type hierarchies. Under the reduction (w=f(u)), one obtains a pair of commuting first-order equations,

which govern simple wave solutions of the Mikhalev system. In this paper we study higher-order reductions of the form

which turn the Mikhalev system into a pair of commuting higher-order equations. Here the terms at (epsilon ^n) are assumed to be differential polynomials of degree n in the x-derivatives of u. We will view w as an (infinite) formal series in the deformation parameter (epsilon ). It turns out that for such a reduction to be non-trivial, the function f(u) must be quadratic, (f(u)=lambda u^2), furthermore, the value of the parameter (lambda ) (which has a natural interpretation as an eigenvalue of a certain second-order operator acting on an infinite jet space), is quantised. There are only two positive allowed eigenvalues, (lambda =1) and (lambda =3/2), as well as infinitely many negative rational eigenvalues. Two-component reductions of the Mikhalev system are also discussed. We emphasise that the existence of higher-order reductions of this kind is a reflection of linear degeneracy of the Mikhalev system, in particular, such reductions do not exist for most of the known 3D dispersionless integrable systems such as the dispersionless KP and Toda equations.

This short letter considers the case of acoustic scattering by several obstacles in (mathbb {R}^{d+r}) for (r,d ge 1) of the form (Omega times mathbb {R}^r), where (Omega ) is a smooth bounded domain in (mathbb {R}^d). As a main result, a von Neumann trace formula for the relative trace is obtained in this setting. As a special case, we obtain a dimensional reduction formula for the Casimir energy for the massive and massless scalar fields in this configuration (Omega times mathbb {R}^r) per unit volume in (mathbb {R}^r).

This paper continues with Part I. We define the module for a (frac{{mathbb Z}}{2})-graded meromorphic open-string vertex algebra that is twisted by an involution and show that the axioms are sufficient to guarantee the convergence of products and iterates of any number of vertex operators. A module twisted by the parity involution is called a canonically ({mathbb Z}_2)-twisted module. As an example, we give a fermionic construction of the canonically ({mathbb Z}_2)-twisted module for the (frac{{mathbb Z}}{2})-graded meromorphic open-string vertex algebra constructed in Part I. Similar to the situation in Part I, the example is also built on a universal ({mathbb Z})-graded non-anti-commutative Fock space where a creation operator and an annihilation operator satisfy the fermionic anti-commutativity relation, while no relations exist among the creation operators or among the zero modes. The Wick’s theorem still holds, though the actual vertex operator needs to be corrected from the naïve definition by normal ordering using the (exp (Delta (x)))-operator in Part I.

We derive an expression for the spectral determinant of a second-order elliptic differential operator ( mathcal {T} ) defined on the whole real line, in terms of the Wronskians of two particular solutions of the equation ( mathcal {T} u=0). Examples of application of the resulting formula include the explicit calculation of the determinant of harmonic and anharmonic oscillators with an added bounded potential with compact support.

In this paper, we consider the Bloch eigenvalues, band functions and bands of the self-adjoint differential operator L generated by the differential expression of odd order n with the (mtimes m) periodic matrix coefficients, where (n>1.) We study the localizations of the Bloch eigenvalues and continuity of the band functions and prove that each point of the set (left[ (2pi N)^{n},infty right) cup (-infty ,(-2pi N)^{n}]) belongs to at least m bands, where N is the smallest integer satisfying (Nge pi ^{-2}M+1) and M is the sum of the norms of the coefficients. Moreover, we prove that if (Mle pi ^{2}2^{-n+1/2}), then each point of the real line belong to at least m bands.

We investigate the capability of symplectic quandles to detect causality for (2+1)-dimensional globally hyperbolic spacetimes (X). Allen and Swenberg showed that the Alexander–Conway polynomial is insufficient to distinguish connected sum of two Hopf links from the links in the family of Allen–Swenberg 2-sky like links, suggesting that it cannot always detect causality in X. We find that symplectic quandles, combined with Alexander–Conway polynomial, can distinguish these two types of links, thereby suggesting their ability to detect causality in X. The fact that symplectic quandles can capture causality in the Allen–Swenberg example is intriguing since the theorem of Chernov and Nemirovski, which states that Legendrian linking equals causality, is proved using Contact Geometry methods.

We consider topological twists of four-dimensional (mathcal {N}=2) supersymmetric QCD with gauge group SU(2) and (N_fle 3) fundamental hypermultiplets. The twists are labelled by a choice of background fluxes for the flavour group, which provides an infinite family of topological partition functions. In this Part I, we demonstrate that in the presence of such fluxes the theories can be formulated for arbitrary gauge bundles on a compact four-manifold. Moreover, we consider arbitrary masses for the hypermultiplets, which introduce new intricacies for the evaluation of the low-energy path integral on the Coulomb branch. We develop techniques for the evaluation of these path integrals. In the forthcoming Part II, we will deal with the explicit evaluation.

A Landau–Zener-type formula for a degenerate avoided-crossing is studied in the non-coupled regime. More precisely, a (2times 2) system of first-order h-differential operator with (mathcal {O}(varepsilon )) off-diagonal part is considered in 1D. Asymptotic behavior as (varepsilon h^{m/(m+1)}rightarrow 0^+) of the local scattering matrix near an avoided-crossing is given, where m stands for the contact order of two curves of the characteristic set. A generalization including the cases with vanishing off-diagonals and non-Hermitian symbols is also given.

We define the (frac{{{mathbb {Z}}}}{2})-graded meromorphic open-string vertex algebra that is an appropriate noncommutative generalization of the vertex operator superalgebra. We also illustrate an example that can be viewed as a noncommutative generalization of the free fermion vertex operator superalgebra. The example is built upon a universal half-integer-graded non-anti-commutative Fock space where a creation operator and an annihilation operator satisfy the fermionic anti-commutativity relation, while no relations exist among the creation operators. The former feature allows us to define the normal ordering, while the latter feature allows us to describe interactions among the fermions. With respect to the normal ordering, Wick’s theorem holds and leads to a proof of weak associativity and a closed formula of correlation functions.