Leonid Friedlander
 Professor, Mathematics
 Professor, Applied Mathematics  GIDP
 Member of the Graduate Faculty
Contact
 (520) 6212742
 Environment and Natural Res. 2, Rm. S351
 Tucson, AZ 85719
 lfriedla@arizona.edu
Bio
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Interests
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Courses
202425 Courses

Real Analy One Variable
MATH 425A (Fall 2024) 
Real Analy One Variable
MATH 525A (Fall 2024) 
Real Analysis
MATH 523A (Fall 2024)
202324 Courses

Banach+Hilbert Spaces
MATH 528A (Fall 2023) 
Calculus I
MATH 125 (Fall 2023)
202223 Courses

TopologyGeometry
MATH 534B (Spring 2023) 
Linear Algebra
MATH 413 (Fall 2022) 
TopologyGeometry
MATH 534A (Fall 2022)
202122 Courses

Banach + Hilbert Spaces
MATH 528B (Spring 2022) 
Banach+Hilbert Spaces
MATH 528A (Fall 2021)
202021 Courses

Calculus II
MATH 129 (Spring 2021) 
Complex Analysis
MATH 520B (Spring 2021) 
Complex Analysis
MATH 520A (Fall 2020)
201920 Courses

Real Analysis
MATH 523B (Spring 2020) 
Calculus II
MATH 129 (Fall 2019) 
Real Analysis
MATH 523A (Fall 2019)
201819 Courses

Dissertation
MATH 920 (Spring 2019) 
Independent Study
MATH 499 (Spring 2019) 
Principles Of Analysis
MATH 527B (Spring 2019) 
Calculus II
MATH 129 (Fall 2018) 
Independent Study
MATH 499 (Fall 2018) 
Principles Of Analysis
MATH 527A (Fall 2018)
201718 Courses

Directed Research
MATH 492 (Summer I 2018) 
Dissertation
MATH 920 (Spring 2018) 
Linear Algebra
MATH 413 (Spring 2018) 
Calculus II
MATH 129 (Fall 2017) 
Complex Analysis
MATH 520A (Fall 2017) 
Dissertation
MATH 920 (Fall 2017)
201617 Courses

Dissertation
MATH 920 (Spring 2017) 
TopologyGeometry
MATH 534B (Spring 2017) 
Dissertation
MATH 920 (Fall 2016) 
Independent Study
MATH 599 (Fall 2016) 
Topics Modern Analysis
MATH 529 (Fall 2016) 
TopologyGeometry
MATH 534A (Fall 2016)
201516 Courses

Banach + Hilbert Spaces
MATH 528B (Spring 2016)
Scholarly Contributions
Journals/Publications
 Friedlander, L. (2024). On the spectral asymptotics in domains with long boundary.More infoI 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). Twoterm spectral asymptotics in linear elasticity. Journal of Geometric Analysis, 33(8), 242286.More infoMotivated in part by the erroneous results in "Geometric invariants of spectrum of the NavierLam\'{e} operator" by Genqian Liu published in the Journal of Geometric Analysis 31 (2021), 1016410193, we establish the twoterm 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.More infoThe 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.More infoThe 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, 123127.
 Friedlander, L. (2018). The DirichlettoNeumann operator for quantum graphs.More infoFor 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 DirichlettoNeumann operator for the Laplacianplus a constant.[Journal_ref: ]
 Friedlander, L. (2013). Generic eigenvalue multiplicities for twoparameter families of operators. Russian Journal of Mathematical Physics, 20(4), 424437.More infoAbstract: A theorem about twoparameter 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), 337354.More infoAbstract: 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 twoterm asymptotics in ε → 0 of the eigenvalues and the oneterm 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), 112.More infoAbstract: 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 squareintegrable functions on the circle. In this paper we prove that the nonlocal contribution to this determinant can be computed in terms of a much simpler "zetaregularized" determinant.
 Friedlander, L., & Solomyak, M. (2008). On the spectrum of narrow periodic waveguides. Russian Journal of Mathematical Physics, 15(2), 238242.More infoAbstract: 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), 822823.
 Friedlander, L. (2005). Extremal properties of eigenvalues for a metric graph. Annales de l'Institut Fourier, 55(1), 199211+VII+XI.More infoAbstract: 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, 149156.More infoAbstract: 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(12), 355380.More infoAbstract: 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 bandgap 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), 4955.More infoAbstract: Under an additional symmetry condition, we prove that the spectrum of a second order selfadjoint 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), 1585.More infoAbstract: 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 signHilbert 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), 20572060.More infoAbstract: 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, 3191.More infoAbstract: 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 L2analytic resp. L2Reidemeister 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), X3952.
 Burghelea, D., Friedlander, L., & Kappeler, T. (1996). Asymptotic expansion of the Witten deformation of the analytic torsion. Journal of Functional Analysis, 137(2), 320363.More infoAbstract: 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, RaySinger [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) = ehtdqeht, 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 RaySinger [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 = (dh, 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), X859.More infoAbstract: 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 AHilbert module W of finite type where A is a finite von Neumann algebra. If (M, W) is of determinant class we prove, generalizing the CheegerMüller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive NovikovShubin invariants, the L2analytic and L2Reidemeister 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 S^{2}. Journal of Functional Analysis, 120(2), 440466.More infoAbstract: For any twodimensional 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 submanifold and show that they are gradientlike 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 S^{1}. Integral Equations and Operator Theory, 16(4), 496513.More infoAbstract: We express the ζregularized 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). Meyervietoris type formula for determinants of elliptic differential operators. Journal of Functional Analysis, 107(1), 3465.More infoAbstract: 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 ζregularized 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 LaplaceBeltrami 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 S^{1}. Communications in Mathematical Physics, 150(2), 431.
 Friedlander, L. (1992). On the second eigenvalue of the dirichlet laplacian. Israel Journal of Mathematics, 79(1), 2332.More infoAbstract: 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 S^{1}. Communications in Mathematical Physics, 138(1), 118.More infoAbstract: For an elliptic differential operator A over S1, {Mathematical expression}, with Ak(x) in END(ℂr) and θ as a principal angle, the ζregularized determinant DetθA is computed in terms of the monodromy map PA, associated to A and some invariant expressed in terms of An and An1. 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 SpringerVerlag.
 Friedlander, L. (1991). Some inequalities between Dirichlet and neumann eigenvalues. Archive for Rational Mechanics and Analysis, 116(2), 153160.
 Friedlander, L. (1985). A summation method for the RayleighSchrödinger series for the anharmonic oscillator. Journal of Mathematical Physics, 26(5), 961964.More infoAbstract: 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 nonnegative halfaxis. The conformal transformation maps this neighborhood onto the unit circle of the complex plane. It gives the summation method for the RayleighSchrödinger series for every g>0. © 1985 American Institute of Physics.
 Friedlander, L. (1985). An invariant measure for the equation u_{tt}u_{xx}+u^{3}=0. Communications in Mathematical Physics, 98(1), 116.More infoAbstract: Numerical studies of the initial boundaryvalue problem of the semilinear wave equation uttuxx+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 SpringerVerlag.
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 DirichlettoNeumann 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.