Leonid Friedlander

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Journals/Publications

• Friedlander, L. (2024). On the spectral asymptotics in domains with long boundary.
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  I discuss a simple toy problem for the Dirichlet Laplacian in a sequence ofdomains where the contribution of the boundary to the spectral asymptotics isof the same order as the contribution from the interior[Journal_ref: ]
• Capoferri, M., Friedlander, L., Levitin, M., & Vassiliev, D. (2023). Two-term spectral asymptotics in linear elasticity. Journal of Geometric Analysis, 33(8), 242--286.
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  Motivated in part by the erroneous results in "Geometric invariants of spectrum of the Navier-Lam\'{e} operator" by Genqian Liu published in the Journal of Geometric Analysis 31 (2021), 10164--10193, we establish the two-term spectral asymptotics for boundary value problems of linear elasticity on a smooth compact Riemannian manifold of arbitrary dimension. We also present some illustrative examples and give a historical overview of the subject. [Journal_ref: ]
• , M. B., , A. D., , L. F., , M. G., , V. I., , Y. K., , P. K., , V. M., , R. M., , T. S., & , A. Z. (2021). Mikhail Aleksandrovich Shubin. December 19, 1944 -- May 13, 2020.
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  The article is dedicated to thye memory of a distinguished mathematicianProfessor Misha Shubin[Journal_ref: ]
• , M. B., , B. M., , M. G., , V. I., , Y. A., , P. K., , V. M., , S. P., , T. S., , L. F., & , A. G. (2021). Misha Shubin. 1944 -- 2020.
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  The article describes the biography and manifold contributions to research inmathematics of Mikhail Aleksandrovich Shubin.[Journal_ref: ]
• Friedlander, L. (2019). On multiplicative properties of determinants. Contemporary Mathematics, 734, 123--127.
• Friedlander, L. (2018). The Dirichlet-to-Neumann operator for quantum graphs.
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  For a compact, connected metric graphs with a boundary that consists of $k$vertices, we prove that an arbitrary symmetric $k\times k$ matrix with realentries can be realized as the Dirichlet-to-Neumann operator for the Laplacianplus a constant.[Journal_ref: ]
• Friedlander, L. (2013). Generic eigenvalue multiplicities for two-parameter families of operators. Russian Journal of Mathematical Physics, 20(4), 424-437.
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  Abstract: A theorem about two-parameter families of Schrödinger operators in proved; the potential is parameter dependent. © 2013 Pleiades Publishing, Ltd.
• Friedlander, L., & Solomyak, M. (2009). On the spectrum of the Dirichlet Laplacian in a narrow strip. Israel Journal of Mathematics, 170(1), 337-354.
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  Abstract: We consider the Dirichlet Laplacian Δ in a family of bounded domains {-a < x < b, 0 < y < εh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in ε → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ. © 2009 Hebrew University Magnes Press.
• Friedlander, L., & Guillemin, V. (2008). Determinants of zeroth order operators. Journal of Differential Geometry, 78(1), 1-12.
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  Abstract: For compact Riemannian manifolds all of whose geodesics are closed (aka Zoll manifolds) one can define the determinant of a zeroth order pseudodifferential operator by mimicking Szego's definition of this determinant for the operator: multiplication by a bounded function, on the Hilbert space of square-integrable functions on the circle. In this paper we prove that the non-local contribution to this determinant can be computed in terms of a much simpler "zeta-regularized" determinant.
• Friedlander, L., & Solomyak, M. (2008). On the spectrum of narrow periodic waveguides. Russian Journal of Mathematical Physics, 15(2), 238-242.
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  Abstract: This is a continuation of [1] and [2]. We consider the spectrum of the Dirichlet Laplacian on the domain {(x, y) : 0 < y < εh(x)}, where h(x) is a positive periodic function. The main assumption is that h(x) has one point of global maximum on the period interval. We study the location of bands and prove that the band lengths decay exponentially as ε → 0. © 2008 Pleiades Publishing, Ltd.
• Friedlander, L. (2007). Problems in teaching through applications [3]. Notices of the American Mathematical Society, 54(7), 822-823.
• Friedlander, L. (2005). Extremal properties of eigenvalues for a metric graph. Annales de l'Institut Fourier, 55(1), 199-211+VII+XI.
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  Abstract: We establish the sharp lower bound for eigenvalues of a metric graph.
• Friedlander, L. (2005). Genericity of simple eigenvalues for a metric graph. Israel Journal of Mathematics, 146, 149-156.
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  Abstract: We prove that, for a metric graph different from a polygon, the spectrum of the Laplacian is generically simple.
• Friedlander, L. (2002). On the density of states of periodic media in the large coupling limit. Communications in Partial Differential Equations, 27(1-2), 355-380.
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  Abstract: Let Ω0 be a domain in the cube (0, 2π)n, and let χτ(x) be a function that equals 1 inside Ω0, equals τ in (0, 2π)n/Ω0, and that is extended periodically to Rn. It is known that, in the limit τ → ∞, the spectrum of the operator - ∇χτ(x)∇ exhibits the band-gap structure. We establish the asymptotic behavior of the density of states function in the bands.
• Friedlander, L. (2002). On the spectrum of a class of second order periodic elliptic differential operators. Communications in Mathematical Physics, 229(1), 49-55.
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  Abstract: Under an additional symmetry condition, we prove that the spectrum of a second order self-adjoint elliptic differential operator with periodic coefficients is purely absolutely continuous.
• Burghelea, D., Friedlander, L., & Kappeler, T. (2001). Relative torsion. Communications in Contemporary Mathematics, 3(1), 15-85.
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  Abstract: This paper achieves, among other things, the following: • It frees the main result of [9] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations. • It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [3] from finite dimensional representations of Γ to representations on an script A sign-Hilbert module of finite type (script A sign a finite von Neumann algebra). The result of [3] corresponds to script A sign = ℂ. • It provides interesting real valued functions on the space of representations of the fundamental group Γ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M. These objectives are achieved with the help of the relative torsion script R sign, first introduced by Carey, Mathai and Mishchenko [12] in special cases. The main result of this paper calculates explicitly this relative torsion (cf. Theorem 1.1). © World Scientific Publishing Company.
• Friedlander, L. (2001). An inequality between dirichlet and neumann eigenvalues in a centrally symmetric domain. Proceedings of the American Mathematical Society, 129(7), 2057-2060.
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  Abstract: We prove an inequality between Dirichlet and Neumann eigenvalues of the Laplacian in a centrally symmetric Euclidean domain. © 2000 American Mathematical Society.
• Burghelea, D., Friedlander, L., & Kappeler, T. (1999). Torsions for manifolds with boundary and glueing formulas. Mathematische Nachrichten, 208, 31-91.
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  Abstract: We extend the definition of analytic and Reidemeister torsion for closed compact Riemannian manifolds, to compact Riemannian manifolds with boundary (M, ∂M), given a parallel flat bundle F of A - Hilbert modules of finite type and a decomposition of the boundary ∂M = ∂_M ∪∂+M into disjoint components. When F is induced from the universal covering of M, and the fundamental group of M is infinite (cf. [BFKM]), these torsions are known as the L2-analytic resp. L2-Reidemeister torsions. If the system (M, ∂_M, ∂+M,F) is of determinant class we compute the quotient of the analytic and the Reidemeister torsion and prove gluing formulas for both of them. In particular we answer positively Conjecture 7.6 in [LL]. If F is induced from a Γ principal covering where Γ is a residually finite group, we derive from work of LÜCK (cf. [L3]), that, the system (M, ∂_M, ∂+M, F) is of determinant class (cf. Theorem 5.1 in Appendix A).
• Friedlander, L., & Nadirashvili, N. (1999). A Differential Invariant Related to the First Eigenvalue of the Laplacian. International Mathematics Research Notices, 1999(17), X3-952.
• Burghelea, D., Friedlander, L., & Kappeler, T. (1996). Asymptotic expansion of the Witten deformation of the analytic torsion. Journal of Functional Analysis, 137(2), 320-363.
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  Abstract: Given a compact Riemannian manifold (Md, g), a finite dimensional representation ρ:π1(M) → GL(V) of the fundamental group π1(M) on a vector space V of dimension l and a Hermitian structure μ on the flat vector bundle ℰ → p M associated to ρ, Ray-Singer [RS] have introduced the analytic torsion T = T(M,ρ,g,μ) > 0. Witten's deformation dq(t) of the exterior derivative dq, dq(t) = e-htdqeht, with h: M → R a smooth Morse function, can be used to define a deformation T(h, t) > 0 of the analytic torsion T with T(h, 0) = T. The main results of this paper are to provide, assuming that grad gh is Morse Smale, an asymptotic expansion for log T(h, t) for t → ∞ of the form Σd+1j=0 ajtj + b log t + O(1/√t) and to present two different formulae for a0. As an application we obtain a shorter derivation of results due to Ray-Singer [RS], Cheeger [Ch], Müller [Mu1, 2] which, in increasing generality, concern the equality for odd dimensional manifolds of the analytic torsion with the average of the Reidemeister torsion corresponding to the triangulation script capital T sign = (h, g) and the dual triangulation script capital T sign script D = (d-h, g). © 1996 Academic Press, Inc.
• Burghelea, D., Friedlander, L., Kappeler, T., & Mcdonald, P. (1996). Analytic and Reidemeister torsion for representations in finite type Hilbert modules. Geometric and Functional Analysis, 6(5), X-859.
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  Abstract: For a closed Riemannian manifold (M, g) we extend the definition of analytic and Reidemeister torsion associated to a unitary representation of π1(M) on a finite dimensional vector space to a representation on a A-Hilbert module W of finite type where A is a finite von Neumann algebra. If (M, W) is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L2-analytic and L2-Reidemeister torsions are equal.
• Burghelea, D., Kappeler, T., Mcdonald, P., & Friedlander, L. (1994). On the Functional logdet and Related Flows on the Space of Closed Embedded Curves on S². Journal of Functional Analysis, 120(2), 440-466.
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  Abstract: For any two-dimensional Riemannian manifold (M, g) we introduce a new functional, hg, on the space of closed simple nonparametrized curves on M. This functional associates to any simple curve Γ the regularize determinant of the Laplace operator on the manifold obtained by cutting M along Γ and imposing Dirichlet boundary conditions. When M is of genus zero we derive a formula for the variation of hg, we prove that the critical points are conformal circles (i.e., the curves which, with respect to the unique metric of constant curvature 1 in the conformal class {e2αg:α ∈ C∞(S2, R)} of g, have constant geodesic curvature), and that the hessian of the functional at a critical point is nondegenerate in directions normal the critical submanifold (Theorem 1.1). We also construct smooth flows on the space of nonparametrized curves retracting the space onto the critical sub-manifold and show that they are gradient-like for our function. These flows deform a given closed embedded curve on S2 to a conformal circle keeping the area of the domain bounded by each curve of the deformation constant (Theorem 1.3). © 1994 Academic Press. All rights reserved.
• Burghelea, D., Friedlander, L., & Kappeler, T. (1993). Regularized determinants for pseudodifferential operators in vector bundles over S¹. Integral Equations and Operator Theory, 16(4), 496-513.
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  Abstract: We express the ζ-regularized determinant of an elliptic pseudodifferential operator A over S1 with strongly invertible principal symbol in terms of the Fredholm determinant of an operator of determinant class, canonically associated to A, and local invariants. These invariants are given by explicit formulae involving the principal and subprincipal symbol of the operator. We remark that, generically, elliptic pseudodifferential operators have a strongly invertible principal symbol. © 1993 Birkhäuser Verlag.
• Burghelea, D., Friedlander, L., & Kappeler, T. (1992). Meyer-vietoris type formula for determinants of elliptic differential operators. Journal of Functional Analysis, 107(1), 34-65.
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  Abstract: For a closed codimension one submanifold Γ of a compact manifold M, let MΓ be the manifold with boundary obtained by cutting M along Γ. Let A be an elliptic differential operator on M and B and C be two complementary boundary conditions on Γ. If (A, B) is an elliptic boundary valued problem on MΓ, then one defines an elliptic pseudodifferential operator R of Neumann type on Γ and prove the following factorization formula for the ζ-regularized determinants: DetA Det(A, B) = KDetR, with K a local quantity depending only on the jets of the symbols of A, B and C along Γ. The particular case when M has dimension 2, A is the Laplace-Beltrami operator, and B resp. C is the Dirichlet resp. Neumann boundary condition is considered. © 1992.
• Burghelea, D., Friedlander, L., & Kappeler, T. (1992). On the determinant of elliptic differential and finite difference operators in vector bundles over S¹. Communications in Mathematical Physics, 150(2), 431-.
• Friedlander, L. (1992). On the second eigenvalue of the dirichlet laplacian. Israel Journal of Mathematics, 79(1), 23-32.
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  Abstract: The multiplicity of the second eigenvalue of the Dirichlet Laplacian on smooth Riemannian surfaces with boundary that satisfy certain convexity condition is at most two. The proof is based on variational formulas for eigenvalues under the change of the domain. © 1992 The Magnes Press.
• Burghelea, D., Friedlander, L., & Kappeler, T. (1991). On the determination of elliptic differential and finite difference operators in vector bundles over S¹. Communications in Mathematical Physics, 138(1), 1-18.
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  Abstract: For an elliptic differential operator A over S1, {Mathematical expression}, with Ak(x) in END(ℂr) and θ as a principal angle, the ζ-regularized determinant DetθA is computed in terms of the monodromy map PA, associated to A and some invariant expressed in terms of An and An-1. A similar formula holds for finite difference operators. A number of applications and implications are given. In particular we present a formula for the signature of A when A is self adjoint and show that the determinant of A is the limit of a sequence of computable expressions involving determinants of difference approximation of A. © 1991 Springer-Verlag.
• Friedlander, L. (1991). Some inequalities between Dirichlet and neumann eigenvalues. Archive for Rational Mechanics and Analysis, 116(2), 153-160.
• Friedlander, L. (1985). A summation method for the Rayleigh-Schrödinger series for the anharmonic oscillator. Journal of Mathematical Physics, 26(5), 961-964.
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  Abstract: We approximate the energy levels of the anharmonic oscillator with any coupling constant by eigenvalues λj(g,T) of the operator - d2/dx2 + x2 + gVT(x) with V T(x) = x4 when |x|≤T and VT(x) = T 4 when |x| > T. The functions λj(g,T) are holomorphic with respect to g in a neighborhood of the non-negative half-axis. The conformal transformation maps this neighborhood onto the unit circle of the complex plane. It gives the summation method for the Rayleigh-Schrödinger series for every g>0. © 1985 American Institute of Physics.
• Friedlander, L. (1985). An invariant measure for the equation u_tt-u_xx+u³=0. Communications in Mathematical Physics, 98(1), 1-16.
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  Abstract: Numerical studies of the initial boundary-value problem of the semilinear wave equation utt-uxx+u3=0 subject to periodic boundary conditions u(t, 0)=u(t, 2 π), ut(t, 0)=ut(t, 2 π) and initial conditions u(0, x)=u0(x), ut(0, x)=v0(x), where u0(x) and v0(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial state u0(x), v0(x) after undergoing a possibly chaotic evolution. In this paper an appropriate abstract space is considered. In this space a finite measure is constructed. This measure is invariant under the flow generated by the Hamiltonian system which corresponds to the original equation. This enables one to verify the above "returning" property. © 1985 Springer-Verlag.

Presentations

• Friedlander, L. (2023, May). Spectral asymptotics in domains with long bounday. Conference in spectral theory. Quebec City: CRM.
• Friedlander, L. (2023, October). Spectral asymptotics in domains with long bounday. Conference in spectral theory. College Station, TX: Texas A&M University.
• Friedlander, L. (2019, June). Multiplicative anomalies for determinants. International conference in differential equations. Baku, Azerbaijan: Khazar University.
• Friedlander, L. (2019, Spring). Eigenvalues of quantum graphs with different boundary conditions. Mathematical Physics Seminar. Texas A&M University: Texas A&M UUniversity.
• Friedlander, L. (2018, Fall). Determinants of Elliptic Operators. Conference in Functional Analysis. Kent State University.
• Friedlander, L. (2018, June). The Dirichlet-to-Neumann operator for quantum graphs. Analysis Seminar. King's College, london.
• Friedlander, L. (2018, October). Determinants og Elliptic Operators. Recent Advances in Functional Analysis. Kent State University.
• Friedlander, L. (2017, Nov). On multiplicative properties of determinants. Symplectic geometry and microlocal analysis. MIT: MIT.
• Friedlander, L. (2014, July). Spectrum of the Robin problem with a large parameter. International Conference in Spectral theory. St. Pererburg, Russia.