## Research

## Research interests

I am interested in spectral theory of differential operators. I have mainly worked on the magnetic Schrödinger operator. Lately I have also studied the problem of counting nodal domains.

## List of publications

#### 19 * *The distribution of superconductivity near a magnetic barrier

with Wafaa Assaad and Ayman Kachmar

We consider the Ginzburg–Landau functional, defined on a twodimensional simply connected domain with smooth boundary, in the situation when the applied magnetic field is piecewise constant with a jump discontinuity along a smooth curve. In the regime of large Ginzburg–Landau parameter and strong magnetic field, we study the concentration of minimizing configurations along this discontinuity.

#### 18 * *On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator III: Magnetic fields that change sign

with Bernard Helffer and Hynek Kovařík

We consider the Dirichlet Pauli operator in bounded connected domains in the plane, with a semiclassical parameter. We show that the ground state energy of the Pauli operator is exponentially small as the semiclassical parameter tends to zero and estimate the decay rate. This extends our recent results discussing a recent paper by Ekholm–Kovařík–Portmann, including non-simply connected domains.

#### 17 * *On the Semiclassical Analysis of the Ground State Energy of the Dirichlet Pauli Operator in Non-Simply Connected Domains

with Bernard Helffer

We consider the Dirichlet Pauli operator in bounded connected domains in the plane, with a semiclassical parameter. We show that the ground state energy of the Pauli operator is exponentially small as the semiclassical parameter tends to zero and estimate the decay rate. This extends our recent results discussing a recent paper by Ekholm–Kovařík–Portmann, including non-simply connected domains.

#### 16 * *On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator

with Bernard Helffer

We discuss the results of a recent paper by Ekholm, Kovařík and Portmann in connection with a question of C. Guillarmou about the semiclassical expansion of the lowest eigenvalue of the Pauli operator with Dirichlet conditions. We exhibit connections with the properties of the torsion function in mechanics, the exit time of a Brownian motion and the analysis of the low eigenvalues of some Witten Laplacian.

#### 15 * *On nodal domains in Euclidean balls

with Bernard Helffer

Å. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues when the dimension is $\geq 2$. Recently Polterovich extended the result to the Neumann problem in two dimensions in the case when the boundary is piecewise analytic. A question coming from the theory of spectral minimal partitions has motivated the analysis of the cases when one has equality in Courant's theorem. We identify the Courant sharp eigenvalues for the Dirichlet and the Neumann Laplacians in balls in $\mathbb R^d$, $d\geq 2$. It is the first result of this type holding in any dimension. The corresponding result for the Dirichlet Laplacian in the disc in $\mathbb R^2$ was obtained by B. Helffer, T.Hoffmann-Ostenhof and S. Terracini.

#### 14 * *Band functions in the presence of magnetic steps

with Peter Hislop, Nicolas Popoff and Nicolas Raymond

We complete the analysis of the band functions for two-dimensional magnetic Schrödinger operators with piecewise constant magnetic fields. The discontinuity of the magnetic field can create edge currents that flow along the discontinuity, which have been described by physicists. Properties of these edge currents are directly related to the behavior of the band functions. The effective potential of the fiber operator is an asymmetric double well (eventually degenerated) and the analysis of the splitting of the bands incorporates the asymmetry. If the magnetic field vanishes, the reduced operator has essential spectrum and we provide an explicit description of the band functions located below the essential spectrum. For non-degenerate magnetic steps, we provide an asymptotic expansion of the band functions at infinity. We prove that when the ratio of the two magnetic fields is rational, a splitting of the band functions occurs and has a natural order, predicted by numerical computations.

#### 13 * *Nodal domains in the square—the Neumann case

with Bernard Helffer

Å. Pleijel has proved that in the case of the Laplacian on the square with Neumann condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues. We identify five Courant sharp eigenvalues for the Neumann Laplacian in the square, and prove that there are no other cases. (to appear in Moscow Mathematical Journal)

#### 12 * *A uniqueness theorem for higher order anharmonic oscillators

with Søren Fournais

We study for $\alpha\in\mathbb{R}$, $k\in\mathbb{N}\setminus\{0\}$ the family of self-adjoint operators $$-\frac{d^2}{dt^2}+\Bigl(\frac{t^{k+1}}{k+1}-\alpha\Bigr)^2$$ in $L^2(\mathbb{R})$ and show that if $k$ is even then $\alpha=0$ gives the unique minimum of the lowest eigenvalue of this family of operators. Combined with earlier results this gives that for any $k\geq 1$, the lowest eigenvalue has a unique minimum as a function of $\alpha$.

#### 11 * *Lack of Diamagnetism and the Little–Parks Effect

with Søren Fournais

When a superconducting sample is submitted to a sufficiently strong external magnetic field, the superconductivity of the material is lost. In this paper we prove that this effect does not, in general, take place at a unique value of the external magnetic field strength. Indeed, for a sample in the shape of a narrow annulus the set of magnetic field strengths for which the sample is superconducting is not an interval. This is a rigorous justification of the Little–Parks effect. We also show that the same oscillation effect can happen for disc-shaped samples if the external magnetic field is non-uniform. In this case the oscillations can even occur repeatedly along arbitrarily large values of the Ginzburg–Landau parameter κ. The analysis is based on an understanding of the underlying spectral theory for a magnetic Schrödinger operator. It is shown that the ground state energy of such an operator is not in general a monotone function of the intensity of the field, even in the limit of strong fields.

#### 10 * *On the essential spectrum of magnetic Schrödinger operators in exterior domains

with Ayman Kachmar

We establish equality between the essential spectrum of the Schrödinger operator with magnetic field in the exterior of a compact arbitrary dimensional domain and that of the operator defined in all the space, and discuss applications of this equality.

#### 9 * *The ground state energy of the three dimensional Ginzburg–Landau functional. Part II: Surface regime

with Søren Fournais and Ayman Kachmar

We study the Ginzburg–Landau model of superconductivity in three dimensions and for strong external magnetic fields. For magnetic field strengths above the phenomenologically defined second critical field it is known from Physics that superconductivity should be essentially restricted to a region near the boundary. We prove that the expected region does indeed carry superconductivity. Furthermore, we give precise energy estimates valid also in the regime around the second critical field which display the transition from bulk superconductivity to surface superconductivity.

#### 8 * *Strong diamagnetism for the ball in three dimensions

with Søren Fournais

In this paper we give a detailed asymptotic formula for the lowest eigenvalue of the magnetic Neumann Schrödinger operator in the ball in three dimensions with constant magnetic field, as the strength of the magnetic field tends to infinity. This asymptotic formula is used to prove that the eigenvalue is monotonically increasing for large values of the magnetic field.

#### 7 * *Superconductivity between $H_{C_2}$ and $H_{C_3}$

with Søren Fournais and Bernard Helffer

Superconductivity for Type II superconductors in external magnetic fields of magnitude between the second and third critical fields is known to be restricted to a narrow boundary region. The profile of the superconducting order parameter in the Ginzburg–Landau model is expected to be governed by an effective one-dimensional model. This is known to be the case for external magnetic fields sufficiently close to the third critical field. In this text we prove such a result on a larger interval of validity.

#### 6 * *Spectral properties of higher order anharmonic oscillators

with Bernard Helffer

We discuss spectral properties of the selfadjoint operator $$-\frac{d^2}{dt^2}+\Bigl(\frac{t^{k+1}}{k+1}-\alpha\Bigr)^2$$ in $L^2(\mathbb{R})$ for odd integers $k$. We prove that the minimum over $\alpha$ of the ground state energy of this operator is attained at a unique point which tends to zero as $k$ tends to infinity. We also show that the minimum is nondegenerate. These questions arise naturally in the spectral analysis of Schrödinger operators.

#### 5 * *A non-existence result for the Ginzburg–Landau equations

with Ayman Kachmar

We consider the stationary Ginzburg–Landau equations in $\mathbb{R}^d$, $d=2,3$. We exhibit a class of applied magnetic fields (including constant fields) such that the Ginzburg–Landau equations do not admit finite energy solutions.

#### 4 * *Eigenvalue asymptotics of the even-dimensional exterior Landau–Neumann Hamiltonian

We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in $\mathbb{R}^{2d}$, $d\geq 1$. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We give asymptotic formulas for the rate of accumulation of eigenvalues in these clusters. When the compact is a Reinhardt domain we are able to show a more precise asymptotic formula.

#### 3 * *Generalizations of the Aharonov–Casher formula to higher-dimensional spaces

We study the ground state of the Pauli Hamiltonian with a magnetic field in $\mathbb{R}^{2d}$, $d>1$. We consider the case where a scalar potential $W$ is present and the magnetic field $B$ is given by $B=2i\bar\partial\partial W$. The main result is that there are no zero modes if the magnetic field decays faster than quadratically at infinity. If the magnetic field decays quadratically then zero modes may appear, and we give a lower bound for the number of them. The results in this paper partly correct a mistake in a paper from 1993.

#### 2 * * On the Dirac and Pauli operators with several Aharonov–Bohm solenoids

We study the self-adjoint Pauli operators that can be realized as the square of a self-adjoint Dirac operator and correspond to a magnetic field consisting of a finite number of Aharonov–Bohm solenoids and a regular part, and prove an Aharonov–Casher type formula for the number of zero-modes for these operators. We also see that essentially only one of the Pauli operators are spin-flip invariant, and this operator does not have any zero-modes.

#### 1 * * On the Aharonov–Casher formula for different self-adjoint extensions of the Pauli operator with singular magnetic field

Two different self-adjoint Pauli extensions describing a spin-$1/2$ two-dimensional quantum system with singular magnetic field are studied. An Aharonov–Casher type formula is proved for the maximal Pauli extension and the possibility of approximation of the two different self-adjoint extensions by operators with regular magnetic fields is investigated.