Insubria Summer School in Mathematical Physics

Spectral and scattering theory: from selfadjoint operators to boundary value problems

Jussi Behrndt (Technische Universität Graz) 

Spectral theory for Schrödinger operators with singular interactions supported on curves and surfaces

In this course we discuss an approach via extension theory of symmetric operators to define and study selfadjoint Schrödinger operators with delta-type potentials supported on curves and hypersurfaces. Our main ingredient is the abstract method of quasi boundary triples and their Weyl functions - with the help of suitable boundary mappings and analytic properties of Dirichlet-to-Neumann maps we derive a Birman-Schwinger principle and a variant of Krein's resolvent formula, which in turn allow a detailed description of the spectrum and lead to Schatten-von Neumann type estimates for the resolvent differences of Schrödinger operators with delta potentials and the unperturbed free Laplacian. The introductory material in this course is partly based on the textbook [W00], for more details on the abstract theory of quasi boundary triples see [BL07,BL12], and [BLL13]

for applications to Schrödinger operators with singular interactions supported on hypersurfaces. Further references and a brief discussion of other related methods will be given during the course.


[BL07] J.Behrndt and M.Langer,Boundary value problems for elliptic partial differential operators on bounded domains,J. Funct. Anal. 243 (2007), 536--565.

[BL12] J.Behrndt and M.Langer, Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples,

in Operator Methods for Boundary Value Problems,  London Math. Soc., Lecture Note Series, Vol. 404, 2012, pp.121-160.

[BLL13] J.Behrndt, M.Langer, and V.Lotoreichik, Schrödinger operators with  δ  and δ'-potentials supported on hypersurfaces, Ann. Henri Poincarè 14 (2013), 385-423.

[W00] J.Weidmann, Lineare Operatoren in Hilbertraumen. Teil I,Teubner, Stuttgart, 2000.

A. Komech (Universität Wien)

On global attractors of Hamilton nonlinear PDEs

Our main goal is a survey of our results [1]-[14] and open problems in the theory of global attractors of Hamilton nonlinear PDEs in infinite space. Main results mean the following global attraction:

Each finite energy solution converges to a finite-dimensional attractor A in a local convergence as t → ±∞

The structure of the global attractor crucially depends on the symmetry group of the equation:

A. For a generic equation, the attractor A is the set of all stationary states ψ(x).

B. For generic U(1)-invariant equations, the attractor A is the set of all solitary waves e−iωtψ(x).

C. For generic translation-invariant equations, the attractor A is the set of all solitons ψ(x − vt).

The global attraction in the cases A, B, and C give the first mathematical model of Bohr’s transitions between quantum stationary states and wave-particle duality.


[1] A.I. Komech, H. Spohn, M. Kunze, Long-time asymptotics for a classical particle interacting with a scalar wave field, Comm. Partial Diff. Equs. 22 (1997), no.1/2, 307-335.

[2] A.I. Komech, H. Spohn, Soliton-like asymptotics for a classical particle interacting with a scalar wave field, Nonlinear Analysis 33 (1998), no.1, 13-24.

[3] A.I. Komech, On transitions to stationary states in one-dimensional nonlinear wave equations, Arch. Rat. Mech. Anal. 149 (1999), no.3, 213-228.

[4] A.I. Komech, Attractors of non-linear Hamiltonian one-dimensional wave equations, Russ. Math. Surv. 55 (2000), no. 1, 43-92.

[5] A.I. Komech, H. Spohn, Long-time asymptotics for the coupled Maxwell-Lorentz equations, Comm. Partial Diff. Equs. 25 (2000), no.3/4, 558-585.

[6] V. Imaikin, A.I. Komech, N. Mauser, Soliton-type asymptotics for the coupled Maxwell-Lorentz equations, Ann. Inst. Poincar ́e, Phys. Theor. 5 (2004), 1117-1135.

[7] A.I. Komech, N. Mauser, A. Vinnichenko, On attraction to solitons in relativistic nonlinear wave equations, Russ. J. Math. Phys. 11 (2004), no. 3, 289-307.

[8] A.I. Komech, A.A. Komech, On global attraction to solitary waves for the Klein-Gordon equation coupled to nonlinear oscillator, C. R., Math., Acad. Sci. Paris 343 (2006), no. 2, 111-114.

[9] A.I. Komech, A.A. Komech, Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field, Arch. Rat. Mech. Anal. 185 (2007), 105-142.

[10] A.I. Komech, A.A. Komech, Global attraction to solitary waves for Klein-Gordon equation with mean field interaction, Annales de l’IHP-ANL 26 (2009), no. 3, 855-868. arXiv:math-ph/0711.1131

[11] A.I. Komech, A.A. Komech, Global attraction to solitary waves for nonlinear Dirac equation with mean field interaction, SIAM J. Math. Analysis, 42 (2010), no. 6, 2944-2964. arXiv:0910.0517

[12] A. Comech, Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator, Discrete Contin. Dyn. Syst. 33 (2013), no. 7, 2711-2755.

[13] A.A. Komech, A.I Komech, A variant of the Titchmarsh convolution theorem for distributions on the circle, Funktsional. Anal. i Prilozhen. 47 (2013), no. 1, 26-32 [Russian]; translation in Funct. Anal. Appl. 47 (2013), no. 1, 21-26.

[14] A.I Komech, Attractors of nonlinear Hamilton PDEs, Discrete and Continuous Dynamical Systems A 36 (2016), no. 11, 6201-6256. arXiv:1409.2009

Gianfausto Dell'Antonio (SISSA)

Contact interactions in N-particle systems.

We study a system of N interacting particles in the limit in which  the radius of the interaction potentials goes to zero.

For a range of masses and given symmetries, the limit exist in a strong sense and coincides with a self-adjoint operator, that can defined independently within the theory of self-adjoint extensions (contact interaction).

For another range of masses and symmetries the limit exists only in a  very weak sense for  quadratic forms and corresponds to a family of self-adjoint operators, all bounded below, with a negative point spectrum that may accumulate at zero.

This will be recognised to be a Weyl limit circle effect.

The proof uses the theory of self-adjoint extensions and a map that has the same role as the Weyl map for boundary value problems.

We also comment on the case in which there are zero-energy resonances and point out the difference between “contact interactions” and “point interaction”.  

Elena Kopylova (Universität Wien)

Dispersion decay for the Schrödinger and Klein-Gordon equations

We give an introduction to spectral methods for the Schrödinger and Klein-Gordon equations with a potential, and their applications to dispersion time-decay and scattering theory. The methods have been introduced by Agmon, Jensen and Kato in the context of the Schrödinger equation. The methods were successfully developed for the Klein-Gordon and wave equations with a potential, and the obtained time-decay plays a crucial role in proving of an asymptotic stability for relativistic nonlinear equations [14]-[15].


[1] Dispersion decay and scattering theory. John Willey & Sons, Hoboken, New Jersey, 2012. Co-authored by A. Komech

[2] Dispersion estimates for one-dimensional Schrödinger and Klein-Gordon equation revisited, Russian Math. Surveys, 71 (2016), no.3, 391-415, Co-authored by I. Egorova, V.A. Marchenko, G. Teschl.

[3] Dispersion estimates for one-dimensional discrete Dirac equations, J. Math. Anal. Appl. 434 (2016), no. 1, 191-208. Co-authored by G. Teschl

[4] Dispersion estimates for one-dimensional discrete Schrödinger and wave equations, Journal of Spectral Theory 5 (2015), no. 4, 663-696. Co-authored by I.Egorova, G.Teschl.

[5] Weighted energy decay for magnetic Klein-Gordon equation, Applicable Analysis 94 (2015), no. 2, 219233. Co-authored by A. Komech.

[6] Dispersion decay for the magnetic Schrödinger equation, J. Funct. Analysis. 264 (2013), 735-751. Co-authored by A. Komech.

[7] Weighted energy decay for 1D Dirac equation, Dynamics of PDE. 8 (2011) no.2, 113-125.

[8] Weighted energy decay for 3D Klein-Gordon equation, J. of Diff. Eqns. 248 (2010), no. 3, 501-520. Co-authored by A. Komech.

[9] Weighted energy decay for 1D Klein-Gordon equation, Comm. PDE 35 (2010), no. 2, 353-374. Co-authored by A. Komech.

[10] Dispersion estimates for Schrödinger and Klein-Gordon equation,Russian Math. Survey 65 (2010), no. 1, 95-142.

[11] Long time decay for 2D Klein-Gordon equation, J. Funct. Anal. 259,(2010), no. 2, 477-502. Co-authored by A. Komech.

[12] Weighted energy decay for 1D wave equation, J. Math. Anal. Appl. 366 (2010), no. 2, 494-505.

[13] On dispersion estimates for discrete 3D Schrödinger and Klein-Gordon

equations, St. Petersburg Math. J. 21 (2010), 743-760.

[14] On asymptotic stability of moving kink for relativistic Ginsburg- Landau equation, Comm. Math. Phys. 302 (2011), no. 1, 225-252. Co-authored by A. Komech.

[15] On asymptotic stability of kink for relativistic Ginsburg-Landau equation, Arch. Rat. Mech. and Analysis 202 (2011), no. 1, 213-245. Co- authored by A. Komech.

Mark Malamud (National Academy of Science of Ukraine)

Extension Theory and its Applications to Boundary-Value Problems

During four last decades a new approach to the extension theory of symmetric operators has been intensively developed. This approach is based on the concepts of a boundary triplet and the corresponding Weyl function. In the framework of this approach different applications to the spectral theory of various classes of differential operators of mathematical physics have been found. We will discuss the main results of this theory as well as certain of the above mentioned applications.

The following topics will be in focus of the planned lectures.

1. Classical aspects of extension theory of symmetric operators.

2. Boundary triplets approach to the extension theory and abstract Weyl function.

3. The role of the Weyl function in Krein’s theory of semibounded extensions of non-negative operators.

4. Spectral properties (e.g., selfadjointness, semiboundedness from below, discreteness property, discreteness and finiteness of negative spectrum, spectral types, etc.) of 1D-Schrödinger operators with point interactions.

5. Spectral properties of 1D-Dirac operators with point interactions.

6. Spectral theory of 3D-Schrödinger operators with point interactions.

The lectures will be based on recent results published in [1]–[8].


[1] R. Carlone, M. Malamud, A. Posilicano, On the spectral theory of Gesztesy–Šeba realizations of 1–D Dirac operators with point interactions on a discrete set, J. Diff. Equat., v. 254 (2013), p. 3835–3902.

[2] S. Albeverio, A. Kostenko, M. Malamud, and H.Neidhardt, Spherical Schrödinger operators with δ-type interactions, J. of Math. Physics v. 54, 052103, (2013), 24 p.

[3] M.M. Malamud, H. Neidhardt, Sturm-Liouville boundary value problems with operator potentials and unitary equivalence, J. Diff. Equations, v.252 (2012), p. 5875-5922.

[4] M.M. Malamud, K. Schmuedgen, Spectral Theory of Schrödinger operators with infinitely many point interactions and radial positive definite functions, J. Func. Anal., v. 263, No 10 (2012), p. 3144–3194.

[5] M.M. Malamud, H. Neidhardt, On the unitary equivalence of absolutely continuous parts of self-adjoint extensions, J. Funct. Anal., v. 260, No 3 (2011), p. 613–638.

[6] A.S. Kostenko, M.M. Malamud, 1–D Schrödinger operators with local point interactions on a discrete set, J. Differential Equations, v. 249, (2010), 253–304.

[7] M. M. Malamud, Spectral theory of elliptic operators in exterior domains, Russian J. Math. Phys. v. 17 , No 1 (2010), 97–126.

[8] V.A. Derkach and M.M. Malamud, Generalized Resolvents and Boundary Value Problems for Hermitian Operators with Gaps, J. Funct. Anal., v. 95, No 1 (1991), 1–95.