The Very Unusual Properties of the Resolvent, Heat Kernel, and Zeta Function for the Operator
نویسنده
چکیده
In this article we analyze the resolvent, the heat kernel and the spectral zeta function of the operator −d/dr−1/(4r) over the finite interval. The structural properties of these spectral functions depend strongly on the chosen self-adjoint extension of the operator, a choice being made necessary because of the singular potential present. Only for the Friedrichs extension standard properties are reproduced, for all other extensions highly nonstandard properties are observed. In particular, for k ∈ N we find terms like (log t) in the small-t asymptotic expansion of the heat kernel. Furthermore, the zeta function has s = 0 as a logarithmic branch point.
منابع مشابه
A more accurate half-discrete Hardy-Hilbert-type inequality with the best possible constant factor related to the extended Riemann-Zeta function
By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the rever...
متن کاملA Hybrid Proximal Point Algorithm for Resolvent operator in Banach Spaces
Equilibrium problems have many uses in optimization theory and convex analysis and which is why different methods are presented for solving equilibrium problems in different spaces, such as Hilbert spaces and Banach spaces. The purpose of this paper is to provide a method for obtaining a solution to the equilibrium problem in Banach spaces. In fact, we consider a hybrid proximal point algorithm...
متن کاملParametrized Pseudodifferential Operators and Geometric Invariants
This is based on joint work with R. T. Seeley. The introduction presents the problem of parameter-dependent calculi for do's and the question of trace asymptotics for Atiyah-Patodi-Singer operators. Chapter 2 establishes relations between the three operator functions: resolvent, heat operator and power operator (zeta function). Chapter 3 explains our parameter-dependent do calculus with weak po...
متن کاملThe spectral properties of differential operators with matrix coefficients on elliptic systems with boundary conditions
Let $$(Lv)(t)=sum^{n} _{i,j=1} (-1)^{j} d_{j} left( s^{2alpha}(t) b_{ij}(t) mu(t) d_{i}v(t)right),$$ be a non-selfadjoint differential operator on the Hilbert space $L_{2}(Omega)$ with Dirichlet-type boundary conditions. In continuing of papers [10-12], let the conditions made on the operator $ L$ be sufficiently more general than [11] and [12] as defined in Section $1$. In this paper, we estim...
متن کاملTrace of Heat Kernel, Spectral Zeta Function and Isospectral Problem for Sub-laplacians Dedicated to Professor Tongde Zhong on His 80th Birthday
In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in Cn+1. Then we survey some results on the spectral zeta function which induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.
متن کامل