7,159 research outputs found
A Liouville theorem for some Bessel generalized operators
In this paper we establish a Liouville theorem in for a
wider class of operators in that generalizes the
-dimensional Bessel operator. We will present two different proofs, based in
two representation theorems for certain distributions "supported in zero"
Integral representations and Liouville theorems for solutions of periodic elliptic equations
The paper contains integral representations for certain classes of
exponentially growing solutions of second order periodic elliptic equations.
These representations are the analogs of those previously obtained by S. Agmon,
S. Helgason, and other authors for solutions of the Helmholtz equation. When
one restricts the class of solutions further, requiring their growth to be
polynomial, one arrives to Liouville type theorems, which describe the
structure and dimension of the spaces of such solutions. The Liouville type
theorems previously proved by M. Avellaneda and F.-H. Lin, and J. Moser and M.
Struwe for periodic second order elliptic equations in divergence form are
significantly extended. Relations of these theorems with the analytic structure
of the Fermi and Bloch surfaces are explained.Comment: 48 page
Super-Liouville - Double Liouville correspondence
The AGT motivated relation between the tensor product of the N = 1
super-Liouville field theory with the imaginary free fermion (SL) and a certain
projected tensor product of the real and the imaginary Liouville field theories
(LL) is analyzed. Using conformal field theory techniques we give a complete
proof of the equivalence in the NS sector. It is shown that the SL-LL
correspondence is based on the equivalence of chiral objects including suitably
chosen chiral structure constants of all the three Liouville theories involved.Comment: The Introduction expanded, main points of the paper clarified.
Misprints corrected and references added. Published in JHE
Direct and inverse spectral theorems for a class of canonical systems with two singular endpoints
Part I of this paper deals with two-dimensional canonical systems
, , whose Hamiltonian is non-negative and
locally integrable, and where Weyl's limit point case takes place at both
endpoints and . We investigate a class of such systems defined by growth
restrictions on H towards a. For example, Hamiltonians on of the
form where
are included in this class. We develop a direct and inverse spectral theory
parallel to the theory of Weyl and de Branges for systems in the limit circle
case at . Our approach proceeds via - and is bound to - Pontryagin space
theory. It relies on spectral theory and operator models in such spaces, and on
the theory of de Branges Pontryagin spaces.
The main results concerning the direct problem are: (1) showing existence of
regularized boundary values at ; (2) construction of a singular Weyl
coefficient and a scalar spectral measure; (3) construction of a Fourier
transform and computation of its action and the action of its inverse as
integral transforms. The main results for the inverse problem are: (4)
characterization of the class of measures occurring above (positive Borel
measures with power growth at ); (5) a global uniqueness theorem (if
Weyl functions or spectral measures coincide, Hamiltonians essentially
coincide); (6) a local uniqueness theorem.
In Part II of the paper the results of Part I are applied to Sturm--Liouville
equations with singular coefficients. We investigate classes of equations
without potential (in particular, equations in impedance form) and
Schr\"odinger equations, where coefficients are assumed to be singular but
subject to growth restrictions. We obtain corresponding direct and inverse
spectral theorems
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