In present paper, using some methods of approximation theory, the trace formulas for eigenvalues of a eigenvalue problem are calculated under the periodic condition and the decaying condition at x∞.
This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas fo...This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas for the eigenvalues of a simple boundary problem generated in a finite interval by the considered Sturm-Liouville equation.展开更多
This paper revisits the classical problem“Can we hear the density of a string?”,which can be formulated as an inverse spectral problem for a Sturm-Liouville operator.Instead of inverting the map from density to spec...This paper revisits the classical problem“Can we hear the density of a string?”,which can be formulated as an inverse spectral problem for a Sturm-Liouville operator.Instead of inverting the map from density to spectral data directly,we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations.Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm.The impact of different parameters involved in the algorithm is also discussed.展开更多
The purposes of this article are to discuss the symplectic transformation laws on theta series and to give some explicit formulas for the trace of the symplectic operator.
In this paper we study the eigenvalue problem for integro-differential operators on a lasso graph.The trace formula of the operator is established by applying the residual technique in complex analysis.
This paper deals with a Dirac operator with periodic and finite-bands potentials.Taking advantage of the commutativity of the monodromy operator and the Dirac operator, we define the Bloch functions and multiplicator ...This paper deals with a Dirac operator with periodic and finite-bands potentials.Taking advantage of the commutativity of the monodromy operator and the Dirac operator, we define the Bloch functions and multiplicator curve, which leads to the formula of DubrovinNovikov＇s type. Further, by calculation of residues on the complex sphere and via gauge transformation, we get the trace formulae of eigenfunctions corresponding to the left endpoints and right end-points of the spectral bands, respectively. As an application, we obtain a completely integrable Hamiltonian system in Liouville sense through nonlinearization of the Dirac spectral problem.展开更多
Let G be the semidirect product A1 ■ A of the adeles and the norm 1 ideles of a global field k. Let г be the discrete subgroup kx ■ k. In this paper the trace formula for this setting is established and used to giv...Let G be the semidirect product A1 ■ A of the adeles and the norm 1 ideles of a global field k. Let г be the discrete subgroup kx ■ k. In this paper the trace formula for this setting is established and used to give the complete decomposition of the G-representation on L2(г\G). It turns out that every character of the norm-1 idele class group gives a one dimensional isotype and the complement of those consists of one irreducible representation.展开更多
This paper introduces a unified operator theory approach to the abstract Fourier analysis over homogeneous spaces of compact groups. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset ...This paper introduces a unified operator theory approach to the abstract Fourier analysis over homogeneous spaces of compact groups. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset space of H in G and μ be the normalized G-invariant measure on G/H associated to the Weil＇s formula. Then, we present a generalized abstract framework of Fourier analysis for the Hilbert function space L^2 （G / H, μ）.展开更多
In this paper, we define the localization operator associated with the spherical mean operator, and show that the localization operator is not only bounded, but also in Schatten-Von Neumann class. We also give a trace...In this paper, we define the localization operator associated with the spherical mean operator, and show that the localization operator is not only bounded, but also in Schatten-Von Neumann class. We also give a trace formula when the symbol function is a nonnegative function.展开更多
We use B. Randol’s method to improve the error term in the prime geodesic theorem for a noncompact Riemann surface having at least one cusp. The case considered is a general one, corresponding to a Fuchsian group of ...We use B. Randol’s method to improve the error term in the prime geodesic theorem for a noncompact Riemann surface having at least one cusp. The case considered is a general one, corresponding to a Fuchsian group of the first kind and a multiplier system with a weight on it.展开更多
We study the linear periods on GL2n twisted by a character using a new relative trace formula.We establish the relative fundamental lemma and the transfer of orbital integrals.Together with the spectral isolation tech...We study the linear periods on GL2n twisted by a character using a new relative trace formula.We establish the relative fundamental lemma and the transfer of orbital integrals.Together with the spectral isolation technique of Beuzart-Plessis-Liu-Zhang-Zhu,we are able to compare the elliptic part of the relative trace formulae and to obtain new results generalizing Waldspurger’s theorem in the n=1 case.展开更多
In this paper we study the eigenvalue problems of Schrödinger equations with energy-dependent potential on a lasso graph,and obtain a new regularized trace for this class of differential operators.
In this paper, we consider the eigenvalue problem for integro-differential operators with separated boundary conditions on the finite interval and find a trace formula for the integro-differential operator.
We show that the cuspidal component of the stable trace formula of a split special odd orthogonal group over a number field, satisfies a weak form of beyond endoscopic decomposition. We also study the r-stable trace f...We show that the cuspidal component of the stable trace formula of a split special odd orthogonal group over a number field, satisfies a weak form of beyond endoscopic decomposition. We also study the r-stable trace formula, when r is the standard or the second fundamental representation of the dual group, and show that they satisfy a similar kind of beyond endoscopic decomposition. The results are consequences of Arthur's works(2013) on endoscopic classification of automorphic representations, together with known results concerning a class of Langlands L-functions for special odd orthogonal groups.展开更多
Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg's trace formula is successful in solving the...Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg's trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.展开更多
In this paper,we give a survey on the Hill-type formula and its applications.Moreover,we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary condi...In this paper,we give a survey on the Hill-type formula and its applications.Moreover,we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions,which include the standard Neumann,Dirichlet and periodic boundary conditions.The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions.Further,based on the Hill-type formula,we derive the Krein-type trace formula.As applications,we give nontrivial estimations for the eigenvalue problem and the relative Morse index.展开更多
A local Hankel transformation of order 1/2 is defined for every finite place of the field of rational numbers.Its inversion formula and the Plancherel type theorem are obtained.A Connes type trace formula is given for...A local Hankel transformation of order 1/2 is defined for every finite place of the field of rational numbers.Its inversion formula and the Plancherel type theorem are obtained.A Connes type trace formula is given for each local Hankel transformation of order 1/2.An S-local Connes type trace formula is derived for the S-local Hankel transformation of order 1/2.These formulas are generalizations of Connes' corresponding trace formulas in 1999.展开更多
The two gate formula for a diffusion X_t in R^n is considered.The formula gives an expressionof the density function of X_t in terms of the path integral of the Brownian bridge starting from xand ending on y at time t.
We study an explicit exponential scheme for the time discretisation of stochastic SchrS- dinger Equations Driven by additive or Multiplicative It6 Noise. The numerical scheme is shown to converge with strong order 1 i...We study an explicit exponential scheme for the time discretisation of stochastic SchrS- dinger Equations Driven by additive or Multiplicative It6 Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Sehr6dinger equations satisfy trace formulas for the expected mass, energy, and momentum （i. e., linear drifts in these quantities）. Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.展开更多
Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even ...Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even weight k.Denote by S_(X)(f×g,α,β)a smoothly weighted sum of A_(f)(1,n)λ_(g)(n)e(αn~β)for X 0 are fixed real numbers.The subject matter of the present paper is to prove non-trivial bounds for a sum of S_(X)(f×g,α,β)over g as k tends to∞with X.These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec,Luo,and Sarnak.展开更多
文摘In present paper, using some methods of approximation theory, the trace formulas for eigenvalues of a eigenvalue problem are calculated under the periodic condition and the decaying condition at x∞.
文摘This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas for the eigenvalues of a simple boundary problem generated in a finite interval by the considered Sturm-Liouville equation.
基金partly supported by NSFC grant No.11621101,12071430the Fundamental Research Funds for the Central Universitiespartially supported by Research Grant Council of Hong Kong,China(GRF grailt 16305018).
文摘This paper revisits the classical problem“Can we hear the density of a string?”,which can be formulated as an inverse spectral problem for a Sturm-Liouville operator.Instead of inverting the map from density to spectral data directly,we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations.Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm.The impact of different parameters involved in the algorithm is also discussed.
文摘The purposes of this article are to discuss the symplectic transformation laws on theta series and to give some explicit formulas for the trace of the symplectic operator.
基金Supported by the National Natural Science Foundation of China(11871031)the National Natural Science Foundation of Jiang Su(BK20201303).
文摘In this paper we study the eigenvalue problem for integro-differential operators on a lasso graph.The trace formula of the operator is established by applying the residual technique in complex analysis.
基金Supported by the National Natural Science Foundation of China(Grant No.61473332)the Natural Science Foundation of Zhejiang Province(Grant No.LQ14A010009)the Natural Science Foundation of Huzhou City(Grant No.2013YZ06)
文摘This paper deals with a Dirac operator with periodic and finite-bands potentials.Taking advantage of the commutativity of the monodromy operator and the Dirac operator, we define the Bloch functions and multiplicator curve, which leads to the formula of DubrovinNovikov＇s type. Further, by calculation of residues on the complex sphere and via gauge transformation, we get the trace formulae of eigenfunctions corresponding to the left endpoints and right end-points of the spectral bands, respectively. As an application, we obtain a completely integrable Hamiltonian system in Liouville sense through nonlinearization of the Dirac spectral problem.
文摘Let G be the semidirect product A1 ■ A of the adeles and the norm 1 ideles of a global field k. Let г be the discrete subgroup kx ■ k. In this paper the trace formula for this setting is established and used to give the complete decomposition of the G-representation on L2(г\G). It turns out that every character of the norm-1 idele class group gives a one dimensional isotype and the complement of those consists of one irreducible representation.
文摘This paper introduces a unified operator theory approach to the abstract Fourier analysis over homogeneous spaces of compact groups. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset space of H in G and μ be the normalized G-invariant measure on G/H associated to the Weil＇s formula. Then, we present a generalized abstract framework of Fourier analysis for the Hilbert function space L^2 （G / H, μ）.
文摘In this paper, we define the localization operator associated with the spherical mean operator, and show that the localization operator is not only bounded, but also in Schatten-Von Neumann class. We also give a trace formula when the symbol function is a nonnegative function.
文摘We use B. Randol’s method to improve the error term in the prime geodesic theorem for a noncompact Riemann surface having at least one cusp. The case considered is a general one, corresponding to a Fuchsian group of the first kind and a multiplier system with a weight on it.
文摘We study the linear periods on GL2n twisted by a character using a new relative trace formula.We establish the relative fundamental lemma and the transfer of orbital integrals.Together with the spectral isolation technique of Beuzart-Plessis-Liu-Zhang-Zhu,we are able to compare the elliptic part of the relative trace formulae and to obtain new results generalizing Waldspurger’s theorem in the n=1 case.
基金supported by the National Natural Science Foundation of China(Grant No.11871031)the Natural Science Foundation of the Jiangsu Province of China(Grant No.BK 20201303)Graduate Education and Teaching Reform Project of Nanjing University of Science and Technology(Grant No.KT2024_B08)。
文摘In this paper we study the eigenvalue problems of Schrödinger equations with energy-dependent potential on a lasso graph,and obtain a new regularized trace for this class of differential operators.
基金Supported by the National Natural Science Foundation of China(No.11171152)the Natural Science Foundation of Jiangsu(No.BK 2010489)Scientific Research Project Unit of the Firat University(No.1881)
文摘In this paper, we consider the eigenvalue problem for integro-differential operators with separated boundary conditions on the finite interval and find a trace formula for the integro-differential operator.
文摘We show that the cuspidal component of the stable trace formula of a split special odd orthogonal group over a number field, satisfies a weak form of beyond endoscopic decomposition. We also study the r-stable trace formula, when r is the standard or the second fundamental representation of the dual group, and show that they satisfy a similar kind of beyond endoscopic decomposition. The results are consequences of Arthur's works(2013) on endoscopic classification of automorphic representations, together with known results concerning a class of Langlands L-functions for special odd orthogonal groups.
基金supported by National Natural Science Foundation of China(Grant No.11471154)
文摘Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg's trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.
基金The first author is partially supported by NSFC(Nos.12071255 and 11790271)National Key R&D Program of China(2020YFA0713300)+1 种基金The second authors is partially supported by NSFC(No.11801583)The third author is Partially supported by NSFC(Nos.11471189,and 11871308).
文摘In this paper,we give a survey on the Hill-type formula and its applications.Moreover,we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions,which include the standard Neumann,Dirichlet and periodic boundary conditions.The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions.Further,based on the Hill-type formula,we derive the Krein-type trace formula.As applications,we give nontrivial estimations for the eigenvalue problem and the relative Morse index.
文摘A local Hankel transformation of order 1/2 is defined for every finite place of the field of rational numbers.Its inversion formula and the Plancherel type theorem are obtained.A Connes type trace formula is given for each local Hankel transformation of order 1/2.An S-local Connes type trace formula is derived for the S-local Hankel transformation of order 1/2.These formulas are generalizations of Connes' corresponding trace formulas in 1999.
基金This project is supported by the National Natural Science Foundation of China
文摘The two gate formula for a diffusion X_t in R^n is considered.The formula gives an expressionof the density function of X_t in terms of the path integral of the Brownian bridge starting from xand ending on y at time t.
文摘We study an explicit exponential scheme for the time discretisation of stochastic SchrS- dinger Equations Driven by additive or Multiplicative It6 Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Sehr6dinger equations satisfy trace formulas for the expected mass, energy, and momentum （i. e., linear drifts in these quantities）. Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.
文摘Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even weight k.Denote by S_(X)(f×g,α,β)a smoothly weighted sum of A_(f)(1,n)λ_(g)(n)e(αn~β)for X 0 are fixed real numbers.The subject matter of the present paper is to prove non-trivial bounds for a sum of S_(X)(f×g,α,β)over g as k tends to∞with X.These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec,Luo,and Sarnak.
