Outline: Uniformization by Spectral Determinant

2021 ◽  
pp. 155-157
Author(s):  
Norbert A’Campo
Keyword(s):  
Spectral Determinant

scholarly journals Spectral Determinants and an Ambarzumian Type Theorem on Graphs

2020 ◽  
Vol 92 (3) ◽  
Author(s):  
Márton Kiss
Keyword(s):  
Inverse Problem ◽  
Asymptotic Distribution ◽  
Type Theorem ◽  
Mean Value ◽  
Type Result ◽  
The Mean ◽  
Zero Potential ◽  
Roots Of A Polynomial ◽  
Specific Part ◽  
Spectral Determinant

Abstract We consider an inverse problem for Schrödinger operators on connected equilateral graphs with standard matching conditions. We calculate the spectral determinant and prove that the asymptotic distribution of a subset of its zeros can be described by the roots of a polynomial. We verify that one of the roots is equal to the mean value of the potential and apply it to prove an Ambarzumian type result, i.e., if a specific part of the spectrum is the same as in the case of zero potential, then the potential has to be zero.

Periodic orbit resummation and the quantization of chaos

1992 ◽  
Vol 436 (1896) ◽  
pp. 99-108 ◽  
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Energy Levels ◽  
Convergence Properties ◽  
Classical Motion ◽  
Quantum Energy ◽  
Formal Derivation ◽  
Spectral Determinant

We study the spectral determinant ∆ ( E ), which has, by construction, zeros at the quantum energy levels of a given system. If the classical motion of the system in question is chaotic then ∆ ( E ) has a semiclassical representation as a sum over combinations of periodic orbits. There are, however, a number of fundamental problems associated with its convergence properties. Imposing upon the sum the condition that, like ∆ ( E ) itself, it is real for real E , we obtain formal resummation equations relating the contributions from asymptotically long orbits to those of the short orbits. These then lead to a formal derivation of the previously conjectured ‘Riemann-Siegel lookalike’ formula, which involves only a finite orbit sum and thus represents, in principle, a semiclassical rule for quantizing chaos.

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