Exceptional Eigenvalues Research Articles

First order structure-preserving perturbation theory for eigenvalues of symplectic matrices: Part II

The first order eigenvalue perturbation theory for structure-preserving perturbations of real or complex symplectic matrices started in [21] is continued and completed. While in [21] all eigenvalues were considered that qualitatively behave similarly under structure-preserving and structure-ignoring perturbations, the focus of this paper is on one particular case when the behavior under structure-preserving perturbations differs significantly from the one under structure-ignoring perturbations. This case occurs for the exceptional eigenvalues ±1 and is caused by restrictions in the possible Jordan structure of symplectic matrices. The main result gives asymptotic expansions of specific perturbed eigenvalues that are not yet covered by the results from [21]. While the main theorem requires that the rank of the perturbation is equal to the rank of its first order term, it is also discussed what can be said if this hypothesis is not satisfied.

Uniform bounds for Kloosterman sums of half-integral weight with applications

Abstract Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to x with implied constants depending on m and n. Recently, in 2009, Sarnak and Tsimerman obtained a bound uniformly in x, m and n. The generalized Kloosterman sums are defined with multiplier systems and on congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their error term is a power of x with implied constants depending on all the other factors. In this paper, for a wide class of half-integral weight multiplier systems, we get the same bound with the error term uniformly in x, m and n. Such uniform bounds have great applications. For the eta-multiplier, Ahlgren and Andersen obtained a uniform and power-saving bound with respect to m and n, which resulted in a convergent error estimate on the Rademacher exact formula of the partition function p ⁢ ( n ) {p(n)} . We also establish a Rademacher-type exact formula for the difference of partitions of rank modulo 3, which allows us to apply our power-saving estimate to the tail of the formula for a convergent error bound.

Multiple ground-state instabilities in the anisotropic quantum Rabi model

In this work, the anisotropic variant of the quantum Rabi model with different coupling strengths of the rotating and counter-rotating wave terms is studied by the Bogoliubov operator approach. The anisotropy preserves the parity symmetry of the original model. We derive the corresponding $G$-function, which yields both the regular and exceptional eigenvalues. The exceptional eigenvalues correspond to the crossing points of two energy levels with different parities and are doubly degenerate. We find analytically that the ground-state and the first excited state can cross several times, indicating multiple first-order phase transitions as function of the coupling strength. These crossing points are related to manifest parity symmetry of the Hamiltonian, in contrast to the level crossings in the asymmetric quantum Rabi model which are caused by a hidden symmetry.

Determinant Expressions of Constraint Polynomials and the Spectrum of the Asymmetric Quantum Rabi Model

AbstractThe purpose of this paper is to study the exceptional eigenvalues of the asymmetric quantum Rabi models (AQRMs), specifically, to determine the degeneracy of their eigenstates. Here, the Hamiltonian $H_{\textrm{Rabi}}^{\varepsilon }$ of the AQRM is defined by adding the fluctuation term $\varepsilon \sigma _x$, with $\sigma _x$ being the Pauli matrix, to the Hamiltonian of the quantum Rabi model, breaking its $\mathbb{Z}_{2}$-symmetry. The spectrum of $H_{\textrm{Rabi}}^{\varepsilon }$ contains a set of exceptional eigenvalues, considered to be remains of the eigenvalues of the uncoupled bosonic mode, which are further classified in two types: Juddian, associated with polynomial eigensolutions, and non-Juddian exceptional. We explicitly describe the constraint relations for allowing the model to have exceptional eigenvalues. By studying these relations we obtain the proof of the conjecture on constraint polynomials previously proposed by the third author. In fact we prove that the spectrum of the AQRM possesses degeneracies if and only if the parameter $\varepsilon $ is a halfinteger. Moreover, we show that non-Juddian exceptional eigenvalues do not contribute any degeneracy and we characterize exceptional eigenvalues by representations of $\mathfrak{s}\mathfrak{l}_2$. Upon these results, we draw the whole picture of the spectrum of the AQRM. Furthermore, generating functions of constraint polynomials from the viewpoint of confluent Heun equations are also discussed.

The mixed quantum Rabi model

The analytical exact solutions to the mixed quantum Rabi model (QRM) including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4 × 4 determinants responsible for the exact solutions are derived. These so-called G-functions with pole structures can be reduced to the previous ones in the unmixed QRMs. The zeros of G-functions reproduce completely the regular spectra. The exceptional eigenvalues can also be obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point. All energy levels only collapse to the lower one, which diverges negatively. The level crossings in the unmixed QRMs are relaxed to avoided crossings in the present mixed QRM due to absence of parity symmetry. In the weak two-photon coupling regime, the mixed QRM is equivalent to an one-photon QRM with an effective positive bias, suppressed photon frequency and enhanced one-photon coupling, which may pave a highly efficient and economic way to access the deep-strong one-photon coupling regime.

Quantum Rabi–Stark model: solutions and exotic energy spectra

The quantum Rabi–Stark model, where the linear dipole coupling and the nonlinear Stark-like coupling are present on an equal footing, are studied within the Bogoliubov operators approach. Transcendental functions responsible for the exact solutions are derived in a compact way, much simpler than previous ones obtained in the Bargmann representation. The zeros of transcendental functions reproduce completely the regular spectra. In terms of the explicit pole structure of these functions, two kinds of exceptional eigenvalues are obtained and distinguished in a transparent manner. Very interestingly, a first-order quantum phase transition indicated by level crossing of the ground state and the first excited state is induced by the positive nonlinear Stark-like coupling, which is however absent in any previous isotropic quantum Rabi models. When the absolute value of the nonlinear coupling strength is equal to twice the cavity frequency, this model can be reduced to an effective quantum harmonic oscillator, and solutions are then obtained analytically. The spectra collapse phenomenon is observed at a critical coupling, while below this critical coupling, infinite discrete spectra accumulate into a finite energy from below.

Cutoff on hyperbolic surfaces

In this paper, we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least $$\nicefrac {1}{4}$$ , then the distances on the surface are highly concentrated around the minimal possible value of the diameter, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres (Geom Funct Anal 26(4):1190–1216, 2016. https://doi.org/10.1007/s00039-016-0382-7 ) from the setting of graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from Sarnak and Xue (Duke Math J 64(1):207–227, 1991), we are able to show that the results apply to congruence subgroups of $$SL_{2}\left( \mathbb {Z}\right) $$ and other arithmetic lattices, without relying on the well-known conjecture of Selberg (Proc Symp Pure Math 8:1–15, 1965), thus relaxing the condition on the Laplace spectrum of a surface. Conceptually, we show the close relation between the cutoff phenomenon and temperedness of representations of algebraic groups over local fields, partly answering a question of Diaconis (Proc Natl Acad Sci 93(4):1659–1664, 1996), who asked under what general phenomena cutoff exists.

Noncongruence subgroups and Maass waveforms

The main topic of the paper is spectral theory for noncongruence subgroups of the modular group. We have studied a selection of the main conjectures in the area: the Roelcke–Selberg and Phillips–Sarnak conjectures and Selberg's conjecture on exceptional eigenvalues. The first two concern the existence and nonexistence of an infinite discrete spectrum for certain types of Fuchsian groups and last states that there are no exceptional eigenvalues for congruence subgroups, or in other words, there is a specific spectral gap in the cuspidal spectrum.Our main theoretical result states that if the corresponding surface has a reflectional symmetry which preserves the cusps then the Laplacian on this surface has an infinite number of “new” discrete eigenvalues. We define old and new spaces of Maass cusp forms for noncongruence subgroups in a way that provides a natural generalization of the usual definition from congruence subgroups and give a method for determining the structure of the old space algorithmically.In addition to the theoretical result we also present computational data, including a table of subgroups of the modular group and eigenvalues of Maass forms for noncongruence subgroups. We also present, for the first time, numerical examples of both exceptional and (non-trivial) residual eigenvalues. To be able to (even heuristically) certify lists of computed eigenvalues we also proved an explicit average Weyl's law in this setting.

Quantum Chaos on Random Cayley Graphs of

ABSTRACTWe investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of as the prime number p goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs, i.e., the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of and the explicit LPS Ramanujan projective graphs of have optimal spectral gap and diameter as the prime number p goes to infinity.

Applications of the Kuznetsov formula on GL(3) II: the level aspect

We develop an explicit Kuznetsov formula on GL(3) for congruence subgroups. Applications include a Lindelöf on average type bound for the sixth moment of GL(3) L-functions in the level aspect, an automorphic large sieve inequality, density results for exceptional eigenvalues and density results for Maaß forms violating the Ramanujan conjecture at finite places.

Solution of the two-mode quantum Rabi model using extended squeezed states

The two-mode quantum Rabi model with bilinear coupling is studied using extended squeezed states. We derive G-functions for each Bargmann index q. They share a common structure with the G-function of the one-photon and two-photon quantum Rabi models. The regular spectrum is given by zeros of the G-function while the conditions for the presence of doubly degenerate (exceptional) eigenvalues are obtained in closed form through the lifting property. The simple singularity structure of the G-function allows to draw conclusions about the distribution of eigenvalues along the real axis and to understand the spectral collapse phenomenon when the coupling reaches a critical value.

Exact solvability, non-integrability, and genuine multipartite entanglement dynamics of the Dicke model

In this paper, the finite-size Dicke model of arbitrary number of qubits is solved analytically in a unified way within extended coherent states. For the or Dicke models (k is an integer), the G-function, which is only an energy dependent determinant, is derived in a transparent manner. The regular spectrum is completely and uniquely given by stable zeros of the G-function. The closed-form exceptional eigenvalues are also derived. The level distribution controlled by the pole structure of the G-functions suggests non-integrability for model at any finite coupling in the sense of recent criteria found in the literature. A preliminary application to the exact dynamics of genuine multipartite entanglement in the finite-N Dicke model is presented using the obtained exact solutions.

Concise analytic solutions to the quantum Rabi model with two arbitrary qubits

Using extended coherent states, an analytical exact study has been carried out for the quantum Rabi model (QRM) with two arbitrary qubits in a very concise way. The G-functions with 2×2 determinants are generally derived. For the same coupling constants, the simplest G-function, resembling that in the one-qubit QRM, can be obtained. Zeros of the G-function yield the whole regular spectrum. The exceptional eigenvalues, which do not belong to the zeros of the G function, are obtained in the closed form. The Dark states in the case of the same coupling can be detected clearly in a continued-fraction technique. The present concise solution is conceptually clear and practically feasible to the general two-qubit QRM and therefore has many applications.

A numerical study on exceptional eigenvalues of certain congruence subgroups of $$\mathrm {SO}(n,1)$$ SO ( n , 1 ) and $$\mathrm {SU}(n,1)$$ SU ( n , 1 )

In a previous work we apply lattice point theorems on hyperbolic spaces obtaining asymptotic formulas for the number of integral representations of negative integers by quadratic and hermitian forms of signature (n,1) lying in Euclidean balls of increasing radius. This formula involves an error term that depends on the first nonzero eigenvalue of the Laplace-Beltrami operator on the corresponding congruence hyperbolic manifolds. The aim of this paper is to compare the error term obtained by experimental computations, with the error term mentioned above, for several choices of quadratic and hermitian forms. Our numerical results give evidences of the existence of exceptional eigenvalues for some arithmetic subgroups of SU(3,1), SU(4,1) and SU(5,1), thus they contradicts the generalized Selberg (and Ramanujan) conjecture in these cases. Furthermore, for several arithmetic subgroups of SO(4,1), SO(6,1), SO(8,1) and SU(2,1), there are evidences of a lower bound on the first nonzero eigenvalue better than the already known lower bound for congruences subgroups.

A large sieve inequality of Elliott–Montgomery–Vaughan type for Maass forms with applications to Linnikʼs problem

In this paper, we establish a large sieve inequality of Elliott–Montgomery–Vaughan type over short intervals and explore its applications to Linnikʼs problem for Hecke eigenvalues of primitive Maass forms. One major obstacle is that the generalized Ramanujan conjecture is still unknown for primitive Maass forms and a generalization of Sarnakʼs result on the number of exceptional Hecke eigenvalues will get around this difficulty.

Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$

We prove new bounds for weighted mean values of sums involving Fourier coefficients of cusp forms that are automorphic with respect to a~Hecke congruence subgroup $\Gamma\leq SL(2,{\mathbb Z}[i])$, and correspond to exceptional eigenvalues of the Laplace operator on the space $L^2(\Gamma\backslash SL(2,{\mathbb C})/SU(2))$. These results are, for certain applications, an effective substitute for the generalised Selberg eigenvalue conjecture. We give a~proof of one such application, which is an upper bound for a~sum of generalised Kloosterman sums (of significance in the study of certain mean values of Hecke zeta-functions with groessencharakters). Our proofs make extensive use of Lokvenec-Guleska's generalisation of the Bruggeman-Motohashi summation formulae for $PSL(2,{\mathbb Z}[i])\backslash PSL(2,{\mathbb C})$. We also employ a~bound of Kim and Shahidi for the first eigenvalues of the relevant Laplace operators, and an `unweighted' spectral large sieve inequality (our proof of which is to appear separately).

Applications of the Kuznetsov formula on GL(3).

We develop a fairly explicit Kuznetsov formula on GL(3) and discuss the analytic behavior of the test functions on both sides. Applications to Weyl’s law, exceptional eigenvalues, a large sieve and L-functions are given.

On exceptional eigenvalues of the Laplacian for Γ₀(𝑁)

An explicit Dirichlet series is obtained, which represents an analytic function of s s in the half-plane ℜ s > 1 / 2 \Re s>1/2 except for having simple poles at points s j s_{j} that correspond to exceptional eigenvalues λ j \lambda _{j} of the non-Euclidean Laplacian for Hecke congruence subgroups Γ 0 ( N ) \Gamma _{0}(N) by the relation λ j = s j ( 1 − s j ) \lambda _{j}=s_{j}(1-s_{j}) for j = 1 , 2 , ⋯ , S j=1,2,\cdots , S . Coefficients of the Dirichlet series involve all class numbers h d h_{d} of real quadratic number fields. But, only the terms with h d ≫ d 1 / 2 − ϵ h_{d}\gg d^{1/2-\epsilon } for sufficiently large discriminants d d contribute to the residues m j / 2 m_{j}/2 of the Dirichlet series at the poles s j s_{j} , where m j m_{j} is the multiplicity of the eigenvalue λ j \lambda _{j} for j = 1 , 2 , ⋯ , S j=1,2,\cdots , S . This may indicate (I’m not able to prove yet) that the multiplicity of exceptional eigenvalues can be arbitrarily large. On the other hand, by density theorem the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on N N .

Sums of Kloosterman sums for real quadratic number fields

We estimate sums of Kloosterman sums for a real quadratic number field F of the type S = ∑ c |N(c)| −1/2 S F(r,r 1;c) where c runs through the integers of F that satisfy C⩽| N( c)|<2 C, A⩽| c/ c′|< B, with A< B fixed and C→∞. By x↦ x′ we indicate the non-trivial automorphism of F. The Kloosterman sums are given by S F(r,r 1;c)= ∑ d|c ∗e 2πi Tr F/ Q (ra+r 1d)/c , with ad≡1| (c) . In the absence of exceptional eigenvalues for the corresponding Hilbert modular forms, our estimate implies that S =O F,ε,r,r 1,A,B C 5/6+ε for each ε>0. An estimate not taking cancellation between Kloosterman sums into account would yield O C . The exponent 5 6 +ε is less sharp than occurs in the bound O F,r,r 1,ε C 3/4+ε , obtained in our paper in J. reine angew. Math. 535 (2001) 103–164 for sums of Kloosterman sums where c runs over integers satisfying C ⩽|c|⩽2 C , C ⩽|c′|⩽2 C . The proof is based on the Kloosterman-spectral sum formula for the corresponding Hilbert modular group. The Bessel transform in this formula has a product structure corresponding to the infinite places of F. This does not fit well to the bounds depending on N( c) and c/ c′. Nevertheless, we do obtain non-trivial bounds for S.

On the spectral gap for infinite index “congruence” subgroups of SL2(Z)

A celebrated theorem of Selberg states that for congruence subgroups of SL2(Z) there are no exceptional eigenvalues below 3/16. Extending the work of Sarnak and Xue for cocompact arithmetic lattices, we prove a generalization of Selberg’s theorem for infinite index “congruence” subgroups of SL2(Z). For such subgroups with a high enough Hausdorff dimension of the limit set we establish a spectral gap property and consequently solve a problem of Lubotzky pertaining to expander graphs.