# Spectral Theory of Laplace and Schroedinger Operators (13w5059)

Arriving in Banff, Alberta Sunday, July 28 and departing Friday August 2, 2013

## Organizers

Mark Ashbaugh (University of Missouri-Columbia)

Rafael Benguria (Pontificia Universidad Catolica de Chile)

Richard Laugesen (University of Illinois)

Iosif Polterovich (Université de Montréal)

Timo Weidl (Universität Stuttgart)

## Objectives

Our Workshop will center on three topics of opportunity.

1. Isoperimetric inequalities and shape optimization

Recent progress on geometrically sharp "isoperimetric'' estimates of eigenvalues includes work on the ground state energy (lowest eigenvalue), on excited energies (higher eigenvalues), and spectral gaps (which control the rate of decay to equilibrium).

The classical Rayleigh-Faber-Krahn inequality asserts that among all domains of fixed volume in a $d$-dimensional Euclidean space, the ball minimizes the first eigenvalue of the Dirichlet Laplacian (that is, the fundamental tone of the drum). Upper and lower inequalities of such isoperimetric type are now known for the first (positive) eigenvalue under all the major boundary conditions: Dirichlet, Robin, and Neumann, for Schroedinger operators, and for the Laplace-Beltrami operator on surfaces without boundary.

Isoperimetric results for higher eigenvalues tend to be either trivial or extremely difficult to prove. The second Dirichlet eigenvalue is minimized by the union of two disjoint identical balls (trivial, given Faber-Krahn), but it was only in the 1990's that Ashbaugh and Benguria resolved the Payne-P'{o}lya-Weinberger conjecture that the ball maximizes the ratio of the first two Dirichlet eigenvalues. The maximizing domain for the third Dirichlet eigenvalue is unknown. For Neumann boundary conditions, Girouard, Nadirashvili, and Polterovich showed in 2009 that among all simply connected plane domains, the second nonzero Neumann eigenvalue is maximized by the union of two disjoint disks (connected by an infinitesimal passage). It is a challenging open problem to remove the ``simply-connected'' hypothesis, and to extend this Neumann result to higher dimensions.

Sharp inequalities on eigenvalues of the Steklov problem (corresponding to a membrane with all its mass concentrated on the boundary) were obtained by Girouard and Polterovich, but analogous questions stand open for surfaces without boundary. Interesting results were obtained by Fraser and Schoen in recent work.

On closed surfaces, Hersch proved long ago that among all metrics on the sphere with fixed area, the first nonzero eigenvalue attains its maximum for the round sphere. Can one identify the extremal metrics on surfaces of higher genus? To date, maximizing metrics for the first eigenvalue are known or conjectured only for genus $leq 2$. In higher genera the question is wide open. In particular, a problem of fundamental importance is to understand the regularity of extremal metrics on higher genus surfaces. Under an additional restriction of a fixed conformal class, Kokarev, Nadirashvili, and Sire have made significant progress in this direction.

Returning to planar domains, new types of isoperimetric inequality have been discovered in the last decade for sums of eigenvalues (Laugesen and Siudeja) and spectral zeta functions, heat traces and probabilistic exit times (Laugesen and Morpurgo, Ba~nuelos and Mendez-Hernandez), and for Schroedinger operators with magnetic fields. Much progress has been stimulated by numerical calculation of specific cases (such as in work of Henrot), and by large-scale numerical investigations of generic cases (work of Freitas and Antunes). Numerical evidence matters particularly for shape optimization problems. Outstanding such problems are the minimization of the second Dirichlet eigenvalue under diameter normalization (is the minimizer a disk?), and P'{o}lya's ``polygonal Faber-Krahn'' conjecture that in the class of n-gons of fixed area, the fundamental tone is minimized on a regular polygon. P'{o}lya's conjecture is unsolved even for pentagons!

Another current problem in shape optimization is to understand connectivity of minimizers for higher Dirichlet eigenvalues subject to a perimeter constraint. Intriguing recent results are due to van den Berg and Iversen. It remains a challenge to obtain satisfactory results under an area constraint.

Lastly, shape optimization for the spectral gap has seen some spectacular recent progress. The Fundamental Gap Conjecture of van den Berg and Yau asserts that the difference between the first two eigenvalues of a Schroedinger operator with convex potential on a convex domain should be minimized by the degenerate rectangular box (line segment) with zero potential. Impressive parabolic comparison techniques involving the modulus of convexity enabled Andrews and Clutterbuck to prove this Gap Conjecture earlier this year. Interestingly, their proof has a certain affinity with the alternative proof of the Faber-Krahn inequality given by Lieb in the 1970s in connection with the Choquard problem, and their proof also yields a refinement of the Brascamp--Lieb inequality.

2. Universal and semi-classical inequalities

Payne conjectured in 1955 that the $(k+1)$-th Neumann eigenvalue is smaller than the $k$-th Dirichlet eigenvalue for all $k=1,2, dots$ on any Euclidean domain. Friedlander proved this conjecture for $C^1$-domains in 1991 by using Dirichlet-to-Neumann operators. Mazzeo generalized the result to certain domains in Riemannian manifolds, and then some years ago, Filonov found a striking, elementary proof for general Euclidean domains. Extensions to the Heisenberg Laplacian (by Frank and Laptev) and to mixed Steklov problems (by Ba~nuelos, Kulczycki, Polterovich, and Siudeja) show the continuing power of this method.

Friedlander conjectured a much stronger inequality, that the $(k+d)$-th Neumann eigenvalue is smaller than the $k$-th Dirichlet eigenvalue, for any $d$-dimensional Euclidean domain. This was shown for convex domains by Levine and Weinberger, but remains completely open for general domains.

Recently, Benguria, Levitin, and Parnovski discovered a surprising link between the Dirichlet and Neumann eigenvalues of a convex Euclidean domain and the zeros of the Fourier transform of its characteristic function. This connection has given rise to some intriguing conjectures and open problems.

Semi-classical estimates too have seen substantial developments. Lieb-Thirring estimates treat eigenvalue sums under localized potential wells. In recent years, Ekholm and Frank discovered how to include Hardy-type terms in Lieb-Thirring inequalities, Harrell and Stubbe found a new monotonicity approach yielding ``universal inequalities'' and sharp constants in certain cases, and their idea was applied to quantum graphs by Harrell and Demirel. For the Dirichlet Laplacian we have seen progress on Berezin-Li-Yau bounds capturing both the first sharp Weyl term and an additional second term of the expected order (by Weidl, Vougalter, and Kovarik, and by Geisinger, Laptev, and Weidl). In this context one should also mention some new work by Geisinger and Frank which computes the second Weyl term for eigenvalue sums of non-integer powers of the Laplacian in the case of non-smooth boundaries. Finally, we begin to see interesting initial results that relate Lieb-Thirring type bounds to Schroedinger operators with emph{complex} potentials (Frank, Laptev and Frank).

The P'{o}lya Conjecture concerning the Weyl asymptotic for the Dirichet Laplacian for a domain in Euclidean space (claiming that the asymptotic in fact provides a lower bound) remains famously open, although a magnetic version was resolved in the negative by Frank, Loss, and Weidl.

3. Nodal geometry of eigenfunctions

The study of nodal sets (i.e., zero sets) and nodal domains (components of the complement of the zero set) of eigenfunctions began more than two centuries ago with investigations of Chladni, who observed nodal patterns of vibrating plates. He reported that the patterns became more complicated as the frequency of vibration increased. A bound on nodal complexity was given by Courant, who proved that the $n$-th eigenfunction can have at most $n$ nodal domains. As $n$ tends to infinity the correct count is better, namely $cn$ for a certain constant $c<1,$ as Pleijel showed in 1956 under Dirichlet boundary conditions. Polterovich recently proved the same bound for Neumann boundary conditions, on plane domains, by employing estimates of Toth and Zelditch on the number of boundary zeros of eigenfunctions. A notable open challenge is to extend this nodal domain counting estimate to higher dimensions.

Estimates on the inradius of nodal domains were proved lately by Mangoubi. In dimension two his result is optimal, but in higher dimensions the optimal lower bound on the inradius remains unknown.

Turning now to the nodal sets, we recall that their asymptotic size was determined by Donnelly and Fefferman, on real-analytic Riemannian manifolds. For smooth manifolds, the optimal upper and lower bounds on the size of the nodal set were conjectured by Yau. An optimal lower bound in two dimensions was proved by Bruening back in 1978. In higher dimensions, a significant improvement in understanding of lower bounds was achieved this year, due to Sogge, Zelditch, Colding, Minicozzi, Mangoubi, Hezari, and Wang. However, the results obtained to date are still far from optimal.

A further emerging direction of research in geometric spectral theory is the study of spectral minimal partitions, which are closely linked to nodal domains. This subject was developed by Helffer, Hoffmann-Ostenhof, Terrachini, Bonnaillie-Noel, and their collaborators. Many challenging problems lie open in this area. For instance, it is conjectured that as $k to infty$, the minimal $k$-partition of an arbitrary planar domain converges (in an appropriate sense) to a hexagonal tiling.

-----

We want the Workshop to stimulate progress on major unsolved problems. Towards that objective, each morning of the workshop will begin with an expository talk on a major recent problem. For example, good candidates for these presentations include: (i) Andrews or Clutterbuck on the methods used in their proof of the Fundamental Gap Conjecture, including their refinement of the Brascamp-Lieb inequality and their applications, (ii) Henrot on isoperimetric inequalities, (iii) Laptev or Geisinger on their results on geometrical versions of improved Berezin-Li-Yau inequalities, (iv) Helffer on spectral minimal partitions or Zelditch on nodal geometry.

Given all these recent developments to exploit in related fields, and with the possibility of bringing many of the world's experts together in Banff to exchange and explore ideas for future progress, we anticipate hosting an intense and exciting workshop.

REFERENCES

B. Andrews and J. Clutterbuck. "Proof of the fundamental gap conjecture.'' J. Amer. Math. Soc. 24 (2011), no. 3, 899-916.

P. Antunes and P. Freitas. "New bounds for the principal Dirichlet eigenvalue of planar regions.'' Experiment. Math. 15 (2006), no. 3, 333-342.

P. Antunes and P. Freitas. "A numerical study of the spectral gap.'' J. Phys. A 41 (2008), no. 5, 055201, 19 pp.

M. S. Ashbaugh and R. D. Benguria. "A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions.'' Ann. of Math. (2) 135 (1992), no. 3, 601-628.

R. Ba~{n}uelos, T. Kulczycki, I. Polterovich and B. Siudeja. "Eigenvalue inequalities for mixed Steklov problems.'' Operator Theory and Its Applications, In Memory of V. B. Lidskii (1924-2008), Amer. Math. Soc. Transl., 231, 2010.

R. Ba~{n}uelos and P. J. M'{e}ndez-Hern'{a}ndez. "Symmetrization of L'{e}vy processes and applications.'' J. Funct. Anal. 258 (2010), no. 12, 4026-4051.

R. Benguria, M. Levitin and L. Parnovski. "Fourier transform, null variety, and Laplacian's eigenvalues.'' J. Funct. Anal. 257 (2009), no. 7, 2088-2123.

M. van den Berg and M. Iversen. "On the minimization of Dirichlet eigenvalues of the Laplace operator.'' arXiv:0905.4812

D. Bucur, G. Buttazzo and A. Henrot. "Minimization of $lambda_2$ with a perimeter constraint.'' Indiana Univ. Math. J. 58 (2009), no. 6, 2709-2728.

S. Demirel and E. M. Harrell II. "On semiclassical and universal inequalities for eigenvalues of quantum graphs.'' Rev. Math. Phys. 22 (2010), no. 3, 305-329.

T. Ekholm and R. L. Frank. "Lieb-Thirring inequalities on the half-line with critical exponent.'' J. Eur. Math. Soc. (JEMS) 10 (2008), no. 3, 739-755.

T. Ekholm and R. L. Frank. "On Lieb-Thirring inequalities for Schroedinger operators with virtual level.'' Comm. Math. Phys. 264 (2006), no. 3, 725-740.

R. Frank. "Eigenvalue bounds for Schroedinger operators with complex potentials.'' Bull. Lond. Math. Soc., to appear.

R. Frank. "A simple proof of Hardy-Lieb-Thirring inequalities.'' Comm. Math. Phys. 290 (2009), no. 2, 789-800.

R. L. Frank and A. Laptev. "Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group.'' Int. Math. Res. Not. IMRN 2010, no. 15, 2889-2902.

R. L. Frank, A. Laptev, E. H. Lieb, and R. Seiringer. "Lieb-Thirring inequalities for Schroedinger operators with complex-valued potentials.'' Lett. Math. Phys. 77 (2006), no. 3, 309-316.

A. Fraser and R. Schoen. "The first Steklov eigenvalue, conformal geometry, and minimal surfaces.'' Adv. Math. 226 (2011), no. 5, 4011-4030.

L. Friedlander. ``Some inequalities between Dirichlet and Neumann eigenvalues.'' Arch. Rational Mech. Anal. 116 (1991), no. 2, 153-160.

L. Geisinger and R. Frank. "Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain.'' Submitted.

A. Girouard, N. Nadirashvili and I. Polterovich. "Maximization of the second positive Neumann eigenvalue for planar domains.'' J. Differential Geom. 83 (2009), no. 3, 637-661.

E. M. Harrell II and J. Stubbe. "Universal bounds and semiclassical estimates for eigenvalues of abstract Schroedinger operators.'' SIAM J. Math. Anal. 42 (2010), no. 5, 2261--2274.

B. Helffer, T. Hoffmann-Ostenhof and S. Terracini."`Nodal domains and spectral minimal partitions.'' Ann. Inst. H. Poincar'{e} Anal. Non Li'{e}aire 26 (2009), no. 1, 101-138.

G. Kokarev and N. Nadirashvili. "On first Neumann eigenvalue bounds for conformal metrics.'' Around the research of Vladimir Maz'ya. II, 229-238, Int. Math. Ser. (N. Y.), 12, Springer, New York, 2010.

H. Kovarik, S. Vugalter, and T. Weidl. "Two-dimensional Berezin-Li-Yau inequalities with a correction term.'' Comm. Math. Phys. 287 (2009) no. 3, 959-981.

A. Laptev, L. Geisinger, and T. Weidl. "Geometrical versions of improved Berezin-Li-Yau inequalities.'' J. Spectral Th. 1 (2011), 87-109.

R. S. Laugesen, and B. A. Siudeja. "Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane.'' J. Funct. Anal. 260 (2011), no. 6, 1795-1823.

D. Mangoubi. "A remark on recent lower bounds for nodal sets.'' To appear in Comm. Partial Differential Equations, arXiv:math/1010.4579.

L. Ni. "Estimates on the Modulus of Expansion for Vector Fields Solving Nonlinear Equations.'' 2011 preprint.

I. Polterovich. "Pleijel's nodal domain theorem for free membranes.'' Proc. Amer. Math. Soc. 137 (2009), no. 3, 1021-1024.

C. D. Sogge and S. Zelditch. "Lower bounds on the Hausdorff measure of nodal sets.'' Math. Res. Lett. 18 (2011), no. 01, 25-37.

J. Stubbe. "Universal monotonicity of eigenvalue moments and sharp Lieb-Thirring inequalities.'' J. Eur. Math. Soc. (JEMS) 12 (2010), no. 6, 1347-1353.

T. Weidl. "Improved Berezin-Li-Yau inequalities with a remainder term.'' In: Spectral Theory of Differential Operators, Amer. Math. Soc. Transl. (2) 225 (2008), 253-263.

1. Isoperimetric inequalities and shape optimization

Recent progress on geometrically sharp "isoperimetric'' estimates of eigenvalues includes work on the ground state energy (lowest eigenvalue), on excited energies (higher eigenvalues), and spectral gaps (which control the rate of decay to equilibrium).

The classical Rayleigh-Faber-Krahn inequality asserts that among all domains of fixed volume in a $d$-dimensional Euclidean space, the ball minimizes the first eigenvalue of the Dirichlet Laplacian (that is, the fundamental tone of the drum). Upper and lower inequalities of such isoperimetric type are now known for the first (positive) eigenvalue under all the major boundary conditions: Dirichlet, Robin, and Neumann, for Schroedinger operators, and for the Laplace-Beltrami operator on surfaces without boundary.

Isoperimetric results for higher eigenvalues tend to be either trivial or extremely difficult to prove. The second Dirichlet eigenvalue is minimized by the union of two disjoint identical balls (trivial, given Faber-Krahn), but it was only in the 1990's that Ashbaugh and Benguria resolved the Payne-P'{o}lya-Weinberger conjecture that the ball maximizes the ratio of the first two Dirichlet eigenvalues. The maximizing domain for the third Dirichlet eigenvalue is unknown. For Neumann boundary conditions, Girouard, Nadirashvili, and Polterovich showed in 2009 that among all simply connected plane domains, the second nonzero Neumann eigenvalue is maximized by the union of two disjoint disks (connected by an infinitesimal passage). It is a challenging open problem to remove the ``simply-connected'' hypothesis, and to extend this Neumann result to higher dimensions.

Sharp inequalities on eigenvalues of the Steklov problem (corresponding to a membrane with all its mass concentrated on the boundary) were obtained by Girouard and Polterovich, but analogous questions stand open for surfaces without boundary. Interesting results were obtained by Fraser and Schoen in recent work.

On closed surfaces, Hersch proved long ago that among all metrics on the sphere with fixed area, the first nonzero eigenvalue attains its maximum for the round sphere. Can one identify the extremal metrics on surfaces of higher genus? To date, maximizing metrics for the first eigenvalue are known or conjectured only for genus $leq 2$. In higher genera the question is wide open. In particular, a problem of fundamental importance is to understand the regularity of extremal metrics on higher genus surfaces. Under an additional restriction of a fixed conformal class, Kokarev, Nadirashvili, and Sire have made significant progress in this direction.

Returning to planar domains, new types of isoperimetric inequality have been discovered in the last decade for sums of eigenvalues (Laugesen and Siudeja) and spectral zeta functions, heat traces and probabilistic exit times (Laugesen and Morpurgo, Ba~nuelos and Mendez-Hernandez), and for Schroedinger operators with magnetic fields. Much progress has been stimulated by numerical calculation of specific cases (such as in work of Henrot), and by large-scale numerical investigations of generic cases (work of Freitas and Antunes). Numerical evidence matters particularly for shape optimization problems. Outstanding such problems are the minimization of the second Dirichlet eigenvalue under diameter normalization (is the minimizer a disk?), and P'{o}lya's ``polygonal Faber-Krahn'' conjecture that in the class of n-gons of fixed area, the fundamental tone is minimized on a regular polygon. P'{o}lya's conjecture is unsolved even for pentagons!

Another current problem in shape optimization is to understand connectivity of minimizers for higher Dirichlet eigenvalues subject to a perimeter constraint. Intriguing recent results are due to van den Berg and Iversen. It remains a challenge to obtain satisfactory results under an area constraint.

Lastly, shape optimization for the spectral gap has seen some spectacular recent progress. The Fundamental Gap Conjecture of van den Berg and Yau asserts that the difference between the first two eigenvalues of a Schroedinger operator with convex potential on a convex domain should be minimized by the degenerate rectangular box (line segment) with zero potential. Impressive parabolic comparison techniques involving the modulus of convexity enabled Andrews and Clutterbuck to prove this Gap Conjecture earlier this year. Interestingly, their proof has a certain affinity with the alternative proof of the Faber-Krahn inequality given by Lieb in the 1970s in connection with the Choquard problem, and their proof also yields a refinement of the Brascamp--Lieb inequality.

2. Universal and semi-classical inequalities

Payne conjectured in 1955 that the $(k+1)$-th Neumann eigenvalue is smaller than the $k$-th Dirichlet eigenvalue for all $k=1,2, dots$ on any Euclidean domain. Friedlander proved this conjecture for $C^1$-domains in 1991 by using Dirichlet-to-Neumann operators. Mazzeo generalized the result to certain domains in Riemannian manifolds, and then some years ago, Filonov found a striking, elementary proof for general Euclidean domains. Extensions to the Heisenberg Laplacian (by Frank and Laptev) and to mixed Steklov problems (by Ba~nuelos, Kulczycki, Polterovich, and Siudeja) show the continuing power of this method.

Friedlander conjectured a much stronger inequality, that the $(k+d)$-th Neumann eigenvalue is smaller than the $k$-th Dirichlet eigenvalue, for any $d$-dimensional Euclidean domain. This was shown for convex domains by Levine and Weinberger, but remains completely open for general domains.

Recently, Benguria, Levitin, and Parnovski discovered a surprising link between the Dirichlet and Neumann eigenvalues of a convex Euclidean domain and the zeros of the Fourier transform of its characteristic function. This connection has given rise to some intriguing conjectures and open problems.

Semi-classical estimates too have seen substantial developments. Lieb-Thirring estimates treat eigenvalue sums under localized potential wells. In recent years, Ekholm and Frank discovered how to include Hardy-type terms in Lieb-Thirring inequalities, Harrell and Stubbe found a new monotonicity approach yielding ``universal inequalities'' and sharp constants in certain cases, and their idea was applied to quantum graphs by Harrell and Demirel. For the Dirichlet Laplacian we have seen progress on Berezin-Li-Yau bounds capturing both the first sharp Weyl term and an additional second term of the expected order (by Weidl, Vougalter, and Kovarik, and by Geisinger, Laptev, and Weidl). In this context one should also mention some new work by Geisinger and Frank which computes the second Weyl term for eigenvalue sums of non-integer powers of the Laplacian in the case of non-smooth boundaries. Finally, we begin to see interesting initial results that relate Lieb-Thirring type bounds to Schroedinger operators with emph{complex} potentials (Frank, Laptev and Frank).

The P'{o}lya Conjecture concerning the Weyl asymptotic for the Dirichet Laplacian for a domain in Euclidean space (claiming that the asymptotic in fact provides a lower bound) remains famously open, although a magnetic version was resolved in the negative by Frank, Loss, and Weidl.

3. Nodal geometry of eigenfunctions

The study of nodal sets (i.e., zero sets) and nodal domains (components of the complement of the zero set) of eigenfunctions began more than two centuries ago with investigations of Chladni, who observed nodal patterns of vibrating plates. He reported that the patterns became more complicated as the frequency of vibration increased. A bound on nodal complexity was given by Courant, who proved that the $n$-th eigenfunction can have at most $n$ nodal domains. As $n$ tends to infinity the correct count is better, namely $cn$ for a certain constant $c<1,$ as Pleijel showed in 1956 under Dirichlet boundary conditions. Polterovich recently proved the same bound for Neumann boundary conditions, on plane domains, by employing estimates of Toth and Zelditch on the number of boundary zeros of eigenfunctions. A notable open challenge is to extend this nodal domain counting estimate to higher dimensions.

Estimates on the inradius of nodal domains were proved lately by Mangoubi. In dimension two his result is optimal, but in higher dimensions the optimal lower bound on the inradius remains unknown.

Turning now to the nodal sets, we recall that their asymptotic size was determined by Donnelly and Fefferman, on real-analytic Riemannian manifolds. For smooth manifolds, the optimal upper and lower bounds on the size of the nodal set were conjectured by Yau. An optimal lower bound in two dimensions was proved by Bruening back in 1978. In higher dimensions, a significant improvement in understanding of lower bounds was achieved this year, due to Sogge, Zelditch, Colding, Minicozzi, Mangoubi, Hezari, and Wang. However, the results obtained to date are still far from optimal.

A further emerging direction of research in geometric spectral theory is the study of spectral minimal partitions, which are closely linked to nodal domains. This subject was developed by Helffer, Hoffmann-Ostenhof, Terrachini, Bonnaillie-Noel, and their collaborators. Many challenging problems lie open in this area. For instance, it is conjectured that as $k to infty$, the minimal $k$-partition of an arbitrary planar domain converges (in an appropriate sense) to a hexagonal tiling.

-----

We want the Workshop to stimulate progress on major unsolved problems. Towards that objective, each morning of the workshop will begin with an expository talk on a major recent problem. For example, good candidates for these presentations include: (i) Andrews or Clutterbuck on the methods used in their proof of the Fundamental Gap Conjecture, including their refinement of the Brascamp-Lieb inequality and their applications, (ii) Henrot on isoperimetric inequalities, (iii) Laptev or Geisinger on their results on geometrical versions of improved Berezin-Li-Yau inequalities, (iv) Helffer on spectral minimal partitions or Zelditch on nodal geometry.

Given all these recent developments to exploit in related fields, and with the possibility of bringing many of the world's experts together in Banff to exchange and explore ideas for future progress, we anticipate hosting an intense and exciting workshop.

REFERENCES

B. Andrews and J. Clutterbuck. "Proof of the fundamental gap conjecture.'' J. Amer. Math. Soc. 24 (2011), no. 3, 899-916.

P. Antunes and P. Freitas. "New bounds for the principal Dirichlet eigenvalue of planar regions.'' Experiment. Math. 15 (2006), no. 3, 333-342.

P. Antunes and P. Freitas. "A numerical study of the spectral gap.'' J. Phys. A 41 (2008), no. 5, 055201, 19 pp.

M. S. Ashbaugh and R. D. Benguria. "A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions.'' Ann. of Math. (2) 135 (1992), no. 3, 601-628.

R. Ba~{n}uelos, T. Kulczycki, I. Polterovich and B. Siudeja. "Eigenvalue inequalities for mixed Steklov problems.'' Operator Theory and Its Applications, In Memory of V. B. Lidskii (1924-2008), Amer. Math. Soc. Transl., 231, 2010.

R. Ba~{n}uelos and P. J. M'{e}ndez-Hern'{a}ndez. "Symmetrization of L'{e}vy processes and applications.'' J. Funct. Anal. 258 (2010), no. 12, 4026-4051.

R. Benguria, M. Levitin and L. Parnovski. "Fourier transform, null variety, and Laplacian's eigenvalues.'' J. Funct. Anal. 257 (2009), no. 7, 2088-2123.

M. van den Berg and M. Iversen. "On the minimization of Dirichlet eigenvalues of the Laplace operator.'' arXiv:0905.4812

D. Bucur, G. Buttazzo and A. Henrot. "Minimization of $lambda_2$ with a perimeter constraint.'' Indiana Univ. Math. J. 58 (2009), no. 6, 2709-2728.

S. Demirel and E. M. Harrell II. "On semiclassical and universal inequalities for eigenvalues of quantum graphs.'' Rev. Math. Phys. 22 (2010), no. 3, 305-329.

T. Ekholm and R. L. Frank. "Lieb-Thirring inequalities on the half-line with critical exponent.'' J. Eur. Math. Soc. (JEMS) 10 (2008), no. 3, 739-755.

T. Ekholm and R. L. Frank. "On Lieb-Thirring inequalities for Schroedinger operators with virtual level.'' Comm. Math. Phys. 264 (2006), no. 3, 725-740.

R. Frank. "Eigenvalue bounds for Schroedinger operators with complex potentials.'' Bull. Lond. Math. Soc., to appear.

R. Frank. "A simple proof of Hardy-Lieb-Thirring inequalities.'' Comm. Math. Phys. 290 (2009), no. 2, 789-800.

R. L. Frank and A. Laptev. "Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group.'' Int. Math. Res. Not. IMRN 2010, no. 15, 2889-2902.

R. L. Frank, A. Laptev, E. H. Lieb, and R. Seiringer. "Lieb-Thirring inequalities for Schroedinger operators with complex-valued potentials.'' Lett. Math. Phys. 77 (2006), no. 3, 309-316.

A. Fraser and R. Schoen. "The first Steklov eigenvalue, conformal geometry, and minimal surfaces.'' Adv. Math. 226 (2011), no. 5, 4011-4030.

L. Friedlander. ``Some inequalities between Dirichlet and Neumann eigenvalues.'' Arch. Rational Mech. Anal. 116 (1991), no. 2, 153-160.

L. Geisinger and R. Frank. "Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain.'' Submitted.

A. Girouard, N. Nadirashvili and I. Polterovich. "Maximization of the second positive Neumann eigenvalue for planar domains.'' J. Differential Geom. 83 (2009), no. 3, 637-661.

E. M. Harrell II and J. Stubbe. "Universal bounds and semiclassical estimates for eigenvalues of abstract Schroedinger operators.'' SIAM J. Math. Anal. 42 (2010), no. 5, 2261--2274.

B. Helffer, T. Hoffmann-Ostenhof and S. Terracini."`Nodal domains and spectral minimal partitions.'' Ann. Inst. H. Poincar'{e} Anal. Non Li'{e}aire 26 (2009), no. 1, 101-138.

G. Kokarev and N. Nadirashvili. "On first Neumann eigenvalue bounds for conformal metrics.'' Around the research of Vladimir Maz'ya. II, 229-238, Int. Math. Ser. (N. Y.), 12, Springer, New York, 2010.

H. Kovarik, S. Vugalter, and T. Weidl. "Two-dimensional Berezin-Li-Yau inequalities with a correction term.'' Comm. Math. Phys. 287 (2009) no. 3, 959-981.

A. Laptev, L. Geisinger, and T. Weidl. "Geometrical versions of improved Berezin-Li-Yau inequalities.'' J. Spectral Th. 1 (2011), 87-109.

R. S. Laugesen, and B. A. Siudeja. "Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane.'' J. Funct. Anal. 260 (2011), no. 6, 1795-1823.

D. Mangoubi. "A remark on recent lower bounds for nodal sets.'' To appear in Comm. Partial Differential Equations, arXiv:math/1010.4579.

L. Ni. "Estimates on the Modulus of Expansion for Vector Fields Solving Nonlinear Equations.'' 2011 preprint.

I. Polterovich. "Pleijel's nodal domain theorem for free membranes.'' Proc. Amer. Math. Soc. 137 (2009), no. 3, 1021-1024.

C. D. Sogge and S. Zelditch. "Lower bounds on the Hausdorff measure of nodal sets.'' Math. Res. Lett. 18 (2011), no. 01, 25-37.

J. Stubbe. "Universal monotonicity of eigenvalue moments and sharp Lieb-Thirring inequalities.'' J. Eur. Math. Soc. (JEMS) 12 (2010), no. 6, 1347-1353.

T. Weidl. "Improved Berezin-Li-Yau inequalities with a remainder term.'' In: Spectral Theory of Differential Operators, Amer. Math. Soc. Transl. (2) 225 (2008), 253-263.