Read Online The Heat Kernel and Theta Inversion on SL2(C) (Springer Monographs in Mathematics) - Jay Jorgenson file in ePub
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From a different point of view, one constructs the heat kernel on the group space using an eigenfunction, or spectral, expansion, which then leads to a theta function and a theta inversion formula by equating the two realizations of the heat kernel on the quotient space.
The heat kernel on the quotient space sl(2,z[i]) sl(2,c) is gotten through periodization, and further expanded in an eigenfunction expansion. A theta inversion formula is obtained by studying the trace of the heat kernel. Following the author's previous work, the inversion formula then leads to zeta functions through the gauss transform.
3 the second formula is now a consequence of the fourier inversion theorem applied in the the theta function (s) also extends to complex values of s when.
Which is often called the “functional equation” of the theta function.
Heat kernel, analytic continuation, symmetric spaces, segal it is tempting to think of (14) as the inversion formula for the heat operator.
The first expansion you wrote is the fourier series of the theta function. Using the heat kernel you sum over a lattice of periods to match the periodicity of the theta function. Notice the fourier transform connection between the two exponentials in your two equations.
Jay jorgenson department of mathematics city college of new york new york, ny 10031 usa jjorgenson@mindspring.
Lang the purpose of the text is to provide a complete, self-contained development of the trace formula and theta inversion formula for sl(2,z[i])\sl(2,c). Unlike other treatments of the theory, the approach taken here is to begin with the heat.
[sel 56], gangolli’s construction of the heat kernel on general g/k’s with cocompact discrete γ ) that theta inversion formulas can be viewed as part of a much larger context, stemming from the theory of semisimple lie groups, symmetric spaces, and the heat kernel. The spherical inversion then provides an essential background for this.
Mar 18, 2015 in this video, we derive the full nonlinear equations of motion for the classic inverted pendulum problem.
The heat kernel on the quotient space sl(2,z[i])sl(2,c) is arrived at through periodization, and further expanded in an eigenfunction expansion. A theta inversion formula is obtained by studying the trace of the heat kernel. Following the author's previous work, the inversion formula then leads to zeta functions through the gauss transform.
I recommend the book a brief introduction to theta functions by richard bellman reprinted by dover publications.
The heat kernel k(x, x\ t) of the iterated dirac operator on an n- dimensional sum is not a jacobi theta function and can not be inverted in terms of elementary.
The heat kernel on the quotient space sl(2,z[i])\sl(2,c) is arrived at through periodization, and further expanded in an eigenfunction expansion. A theta inversion formula is obtained by studying the trace of the heat kernel. Following the author's previous work, the inversion formula then leads to zeta functions through the gauss transform.
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