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Register a new userBiorthogonal systems on unit interval and zeta dilation operators
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Added by
Dorje C. Brody Professor
Publication date
2018
fields
Physics
and research's language is
English
Authors
Dorje C Brody
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An elementary quantum-mechanical derivation of the conditions for a system of functions to form a Reisz basis of a Hilbert space on a finite interval is presented.
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Let $Z_n(s; a_1,..., a_n)$ be the Epstein zeta function defined as the meromorphic continuation of the function sum_{kinZ^nsetminus{0}}(sum_{i=1}^n [a_i k_i]^2)^{-s}, text{Re} s>frac{n}{2} to the complex plane. We show that for fixed $s eq n/2$, the function $Z_n(s; a_1,..., a_n)$, as a function of $(a_1,..., a_n)in (R^+)^n$ with fixed $prod_{i=1}^n a_i$, has a unique minimum at the point $a_1=...=a_n$. When $sum_{i=1}^n c_i$ is fixed, the function $$(c_1,..., c_n)mapsto Z_n(s; e^{c_1},..., e^{c_n})$$ can be shown to be a convex function of any $(n-1)$ of the variables ${c_1,...,c_n}$. These results are then applied to the study of the sign of $Z_n(s; a_1,..., a_n)$ when $s$ is in the critical range $(0, n/2)$. It is shown that when $1leq nleq 9$, $Z_n(s; a_1,..., a_n)$ as a function of $(a_1,..., a_n)in (R^+)^n$, can be both positive and negative for every $sin (0,n/2)$. When $ngeq 10$, there are some open subsets $I_{n,+}$ of $sin(0,n/2)$, where $Z_{n}(s; a_1,..., a_n)$ is positive for all $(a_1,..., a_n)in(R^+)^n$. By regarding $Z_n(s; a_1,..., a_n)$ as a function of $s$, we find that when $ngeq 10$, the generalized Riemann hypothesis is false for all $(a_1,...,a_n)$.
We present a summary of recent and older results on Bessel integrals and their relation with zeta numbers.
The differential-equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using Fourier and WKB analysis. The Fourier analysis leads to a challenging open problem concerning the formulation of the eigenvalue problem in the momentum space. The WKB analysis gives the exact asymptotic behavior of the eigenfunction.
In this paper, whose aims are mainly pedagogical, we illustrate how to use the local zeta regularization to compute the stress-energy tensor of the Casimir effect. Our attention is devoted to the case of a neutral, massless scalar field in flat space-time, on a space domain with suitable (e.g., Dirichlet) boundary conditions. After a simple outline of the local zeta method, we exemplify it in the typical case of a field between two parallel plates, or outside them. The results are shown to agree with the ones obtained by more popular methods, such as point splitting regularization. In comparison with these alternative methods, local zeta regularization has the advantage to give directly finite results via analitic continuation, with no need to remove or subtract divergent quantities.
We study fractality of unbounded sets of finite Lebesgue measure at infinity by introducing the notions of Minkowski dimension and content at infinity. We also introduce the Lapidus zeta function at infinity, study its properties and demonstrate its use in analysis of fractal properties of unbounded sets at infinity.
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