On functional determinants of matrix differential operators with multiple zero modes

Falco, G. M.; Fedorenko, Andrei A.; Gruzberg, Ilya A.

doi:10.1088/1751-8121/aa9205

Cited by 14 publications

(34 citation statements)

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“…Zero modes of operators which appear in quantum field theory have crucial meanings related with nonperturbative aspects of the theory (see for example, the first section of Ref. [14]). In our present paper, we considered mass terms in almost all examples and the cases with zero modes can be considered as the limit that the value of mass goes to zero.…”

Section: Resultsmentioning

confidence: 99%

Deconstructing the Gel’fand–Yaglom Method and Vacuum Energy from a Theory Space

Kan

Shiraishi

2019

Advances in Mathematical Physics

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The discrete Gel'fand-Yaglom theorem was studied several years ago. In the present paper, we generalize the discrete Gel'fand-Yaglom method to obtain the determinants of mass matrices which appear current works in particle physics, such as dimensional deconstruction and clockwork theory. Using the results, we show the expressions for vacuum energies in such various models. PACS: 02.10.Ox, 04.60.Nc, 11.10.Kk, 11.25.Mj.After completion of the first version of the manuscript of the present paper (arXiv:1711.06806), a paper which treats the determinants of discrete Laplace operators appeared [13]. Their method is substantially the same as ours, because the author also relies on the recurrence relation among three variables on a lattice (see Sec. 3 in the present paper and below). We recently become aware of another similar paper on the determinants of matrix differential operators [14]. They studied generalization of Gel'fand-Yaglom method to obtain the functional determinants. Their work differs essentially from ours because they considered differential operators while we treat matrices as operators. We also point out that they did not consider the matrices of large size which have certain continuum limits.The organization of this paper is as follows. In order to make the present paper selfcontained, we show a short review of the Gel'fand-Yaglom method for a differential operator, along with the Dunne's review [2], in Sec. 2. In Sec. 3, we give the method to obtain determinants of tridiagonal matrices with repeated structure. This is a straightforward generalization of description in Ref. [12]. In Sec. 4, we give the method to obtain determinants of periodic tridiagonal matrices. Determinants of extended periodic tridiagonal matrices are obtained in Sec. 5. The rest of the present paper is devoted to applications to deconstructed theories and discrete systems. In Sec. 6, free energy on a graph is discussed by using the results of previous sections. In Sec. 7, we show the method of calculation for evaluating one-loop vacuum energy in deconstructed models from the determinants of mass matrices. In Sec. 8, we show a few more examples of one-loop vacuum energies for slightly complicated theory spaces. We give conclusions in the last section, Sec. 9.

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Section: Resultsmentioning

confidence: 99%

Deconstructing the Gel’fand–Yaglom Method and Vacuum Energy from a Theory Space

Kan

Shiraishi

2019

Advances in Mathematical Physics

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“…For the case of determinants of general higher-order differential operators (with matrix-valued coefficients) and general boundary conditions on compact intervals we refer to Burghelea, Friedlander, and Kappeler [4], Dreyfus and Dym [11], Falco, Fedorenko, and Gruzberg [12], and Forman [13,Sect. 3].…”

Section: Sturm-liouville Operators On Bounded Intervalsmentioning

confidence: 99%

Effective computation of traces, determinants, and ζ-functions for Sturm–Liouville operators

Gesztesy

Kirsten

2019

Journal of Functional Analysis

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Dedicated with great pleasure to Konstantin Makarov on the occasion of his 60th birthday.Abstract. The principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents (and resolvent differences), Fredholm determinants, ζ-functions, and ζ-function regularized determinants associated with linear operators in a Hilbert space. In particular, we detail the connection between Fredholm and ζ-function regularized determinants.Concrete applications of our formalism to general (i.e., three-coefficient) regular Sturm-Liouville operators on compact intervals with various (separated and coupled) boundary conditions, and Schrödinger operators on a halfline, are provided and further illustrated with an array of examples.2) for 0 < ε < ε 0 and D(z 0 ; ε 0 )\{z 0 } ⊂ ρ(T ); here D(z 0 ; r 0 ) ⊂ C is the open disk with center z 0 and radius r 0 > 0, and C(z 0 ; r 0 ) = ∂D(z 0 ; r 0 ) the corresponding circle. The geometric multiplicity m g (z 0 ; T ) of an eigenvalue z 0 ∈ σ p (T ) is defined by m g (z 0 ; T ) = dim(ker((T − z 0

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“…determinant ( of a matrix differential operator [28,29] in the case of the SSP). We do this by using a technique developed in [28], which is based on the spectral -ζ functions of Sturm-Liouville type operators.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of work distributions in a stochastically driven system

Manikandan

Krishnamurthy

2017

Eur. Phys. J. B

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Abstract. We determine the asymptotic forms of work distributions at arbitrary times T , in a class of driven stochastic systems using a theory developed by Nickelsen and Engel (EN theory) [D. Nickelsen and A. Engel, Eur. Phys. J. B 82, 207 (2011)], which is based on the contraction principle of large deviation theory. In this paper, we extend the theory, previously applied in the context of deterministically driven systems, to a model in which the driving is stochastic. The models we study are described by overdamped Langevin equations and the work distributions in path integral form, are characterised by having quadratic augmented actions. We first illustrate EN theory, for a deterministically driven system -the breathing parabola model, and show that within its framework, the Crooks fluctuation theorem manifests itself as a reflection symmetry property of a certain characteristic polynomial, which also determines the exact moment-generating-function at arbitrary times. We then extend our analysis to a stochastically driven system, studied in references [S. Sabhapandit, EPL 89, 60003 (2010) (2014)], for both equilibrium and non-equilibrium steady state initial distributions. In both cases we obtain new analytic solutions for the asymptotic forms of (dissipated) work distributions at arbitrary T . For dissipated work in the steady state, we compare the large T asymptotic behaviour of our solution to the functional form obtained in reference [New J. Phys. 16, 095001 (2014)]. In all cases, special emphasis is placed on the computation of the pre-exponential factor and the results show excellent agreement with numerical simulations. Our solutions are exact in the low noise (β → ∞) limit.

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On functional determinants of matrix differential operators with multiple zero modes

Cited by 14 publications

References 50 publications

Deconstructing the Gel’fand–Yaglom Method and Vacuum Energy from a Theory Space

Deconstructing the Gel’fand–Yaglom Method and Vacuum Energy from a Theory Space

Effective computation of traces, determinants, and ζ-functions for Sturm–Liouville operators

Asymptotics of work distributions in a stochastically driven system

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