Counterexample to strong diamagnetism for the magnetic Robin Laplacian

We determine a counterexample to strong diamagnetism for the Laplace operator in the unit disc with a uniform magnetic field and Robin boundary condition. The example follows from the accurate asymptotics of the lowest eigenvalue when the Robin parameter tends to $-\infty$.

1. Introduction 1.1. Magnetic Robin Laplacian. We denote by a fx P R P X jxj < Ig the open unit disk and by a @ a fx P R P X jxj a Ig its boundary. We study the lowest eigenvalue of the magnetic Robin Laplacian in L P @A, b a @r ibA H A P ; (1.1) with domain D@ b A a fu P H P @A X ¡ @ru ibA H Au C u a H on @g : (1.2) Here is the unit outward normal vector of , < H the Robin parameter and b > H is the intensity of the applied magnetic eld. The vector eld A H generates the unit magnetic eld and is dened as follows A H @x I ; x P A a I P @ x P ; x I A : (1.3) To be more precise, the operator b is dened as the Friedrichs extension, starting from the quadratic form [8,Ch. 4 has a compact resolvent, and thus its spectrum consists of an increasing sequence of eigenvalues. We are interested in examining the asymptotics of the principal eigenvalue I @b; A a inf uPH 1 @A b @uA kuk P L 2 @A (1.5) when b > H is xed and the Robin parameter tends to I. Theorem 1.1. Let b > H. Then, as 3 I, I @b; A a P C C inf mPZ m b P P I P C o@IA: The rst two terms in the asymptotic expansion given in Theorem 1.1 are well known after many contributions (see [15,16,17] for the case b a H and [12] for the case b > H); however, the third correction term is new for the disc geometry for b > H. The recent contribution [11,Thm. 1.5] shows that Theorem 1.1 continues to hold in the case b a H. 1 Besides its mathematical interest, the question of strong diamagnetism has applications to Physics, particularly in the context of superconductivity [2]. In the case of a simply connected domain subject to a uniform applied magnetic eld and Neumann boundary condition ( a H), strong diamagnetism holds [3,4]. Counter examples of strong diamagnetism exist for uniform magnetic elds in non-simply connected domains, or for non-uniform magnetic elds in simply connected domains [6,9]. Interestingly, the Robin boundary condition has the unique feature where strong diamagnetism fails for the disc (which is a simply connected domain) even when it is subject to a uniform applied magnetic eld. Corollary 1.2 results from the following statement. Given a positive real number A, there exist H < H and A < b I < b P < b Q such that, for all P @ I; H , I @b I ; A < I @b P ; A 8 I @b P ; A > I @b Q ; A : We can simply select the constants b i as follows b I a Pn H ; b P a Pn H C I ; b Q a Pn H C Q P ; where n H is the smallest natural number satisfying n H > A; the conclusion then follows from Theorem 1.1.
Using the periodicity of the function b U 3 e@bA, given a natural number N, we can select I < H such that b I;i Xa b I C i < b P;i Xa b P C i < b Q;i Xa b Q C i with the following two inequalities I @b I;i ; A < I @b P;i ; A ; I @b P;i ; A > I @b Q;i ; A ; holding for all I and i P fI; P; ¡ ¡ ¡ ; Ng.  T c 0 C o@IA ; (1.13) where e@bA is introduced in (1.6). It is worth noticing that T c @bA > T c 0 ; Up to approximation errors, T c @bA is a periodic function of b, which is consistent with the LittleParks eect ; For T < T c @bA, the global minimizer of i is non-trivial (in the sense £ T H) ; while for T > T c @bA, the normal solution is a local minimizer of i. Since we are not looking at the large magnetic eld limit, the terms in the potential that appears in polar coordinates is easier to handle since the angular momentum and magnetic eld strength do not compete against each other. The m in the right-hand side stands for the quantized angular momentum.
Our next step is to make a Fourier expansion that will reduce our study to the study of an innite family (parametrized by m P Z) of ordinary dierential operators.
2.4. Reduction to ber operators. We recall that b > H and P @H; I P A are considered to be xed constants. In polar coordinates (x I a r os , x P a r sin ) the quadratic form q b; D@r ;h A a fu P H P @@H; AA X u H @HA a u@HA nd u@A a Hg: (2.9) r b; m;h becomes self-adjoint in the weighted space L P @@H; A; @I h I=P A dA. We We will get the information needed by comparing with simpler operators. In fact, we will rst compare with the weighted Laplace obtained by ignoring the third term in the right-hand side of (2.8). To do this, we rst look at the simpler operator obtained by ignoring also the second term.
2.5. A 1D Laplacian. The spectrum of the operator d 2 d 2 in L P @R C A with domain fu P H P @R C A X u H @HA a u@HAg is explicitly known (see [10] D@r h A a fu P H P @@H; AA X u H @HA a u@HA nd u@A a Hg : (2.13) The operator r h is dened starting from the closed quadratic form q h @uA a H ju H @Aj P @I h I=P A d ju@HAj P : The increasing sequence of the eigenvalues of r h (counting multiplicities) is denoted by @ n @r h AA nPN . In [10,Lem. 4.4 & Prop. 4.5] it is proved that I @r h A a I h I=P C o@h I=P A nd P @r h A ! y@h A @h 3 H C A : (2.14) We are going to rene the expansion of I @r h A by determining the term of order h.