We give a complete classification and present new exotic phenomena of the meromorphic structure of ζ -functions associated to general self-adjoint extensions of Laplace-type operators over conic manifolds. We show that the meromorphic extensions of these ζ -functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. The corresponding heat kernel and resolvent trace expansions also exhibit exotic behaviors with logarithmic terms of arbitrary positive and negative multiplicity. We also give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.
The quadratically damped oscillator: A case study of a non-linear equation of motion Am. J. Phys. 80, 816 (2012) Pointy ice-drops: How water freezes into a singular shape Am. J. Phys. 80, 764 (2012) Why no shear in "Div, grad, curl, and all that"? Am. J. Phys. 80, 519 (2012) Jacobi elliptic functions and the complete solution to the bead on the hoop problem Am. J. Phys. 80, 506 (2012) Embeddings and time evolution of the Schwarzschild wormhole Am.We show how the preexponential factor of the Feynman propagator for a large class of potentials can be calculated using contour integrals. This factor is relevant in the context of tunneling processes in quantum systems. The prerequisites for this analysis involve only introductory courses in ordinary differential equations and complex variables.
Abstract.In this article we analyze the resolvent, the heat kernel and the spectral zeta function of the operator −d 2 /dr 2 −1/(4r 2 ) over the finite interval. The structural properties of these spectral functions depend strongly on the chosen self-adjoint realization of the operator, a choice being made necessary because of the singular potential present. Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed. In particular, for k ∈ N we find terms like (log t) −k in the small-t asymptotic expansion of the heat kernel. Furthermore, the zeta function has s = 0 as a logarithmic branch point.
Abstract. In this article we consider the zeta regularized determinant of Laplace-type operators on the generalized cone. For arbitrary self-adjoint extensions of a matrix of singular ordinary differential operators modelled on the generalized cone, a closed expression for the determinant is given. The result involves a determinant of an endomorphism of a finite-dimensional vector space, the endomorphism encoding the self-adjoint extension chosen. For particular examples, like the Friedrich's extension, the answer is easily extracted from the general result. In combination with [13], a closed expression for the determinant of an arbitrary self-adjoint extension of the full Laplace-type operator on the generalized cone can be obtained.
We describe the structure of the resolvent kernel of an elliptic cone (or Fuchs type) differential operator and give a precise description of the asymptotics of the kernel as the spectral parameter tends to infinity. The structure of the resolvent is investigated through a class of parameter-dependent pseudodifferential operators that incorporate the particular degeneracies of cone operators and their resolvents.
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