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Update: I intended this to be a complete answer originally, but my "counterexample" was based on a miscalculation. So this is now more a collection of remarks on what I think the question is about. W …

Write $P$ for the projection onto the $v_j$, and let $N$ denote the number of eigenvalues $\ge \delta/2$ of the matrix $PSP$. …

We cannot have strict inequalities in all cases since you could have $B=0$. After this adjustment, we can obtain the claim as follows. Let me slightly change notations and consider $A(s)=A-sB$ (so $s …

this way by looking at the zero set of eigenfunctions of the square (in particular, the reflection symmetry of the region can be removed also) and perhaps also starting from other regions with multiple eigenvalues …

There is a more general reason why any such statement will fail: If you consider a $P$ consisting of $N$ separate copies of the same basic region $P_0$, then $|P|=N|P_0|$, while everything else in you …

I'll assume that $n$ is large and $i\ll n$. However, I believe it should be possible to treat the general situation in the same way; as we'll see, we only need a certain simple property of the recursi …

It is an immediate consequence of min-max that restriction does not increase eigenvalues, $\lambda_j(A_0)\le \lambda_j(A)$, and this gives (1). …

The question of whether we have finitely or infinitely many eigenvalues below zero is probably answerable, but seems a bit tricky. … EDIT 2: I'm reasonably confident now that there are only finitely many eigenvalues, though to show it properly would probably require some work. …

Here, the $\mu_j$ are the eigenvalues of $CA+AC$, compressed to $L(v_1,\ldots, v_k)$. … In other words, they are the eigenvalues of the $k\times k$ matrix with entries $$ \langle v_j, (CA+AC) v_m\rangle = 2\lambda \langle v_j, Cv_m\rangle . $$ Now we only need to make sure that not all the …

No, this isn't working at all. Since $N,P$ commute (you would have to be more specific what exactly you mean by this since the operators are unbounded, but let's just assume we have the right version) …

For a problem on a half line or a bounded (= finite) interval (let's say with $n=0$ as its left endpoint) $$ (Ju)_n = \begin{cases} a_0 u_1 + b_0 u_0 & n=0\\ a_n u_{n+1} + a_{n-1} u_{n-1} + b_n u_n & …

Next attempt: By min-max, it's clear that the largest eigenvalue satisfies $\lambda_1=M_{11}n^{2(h_3-h_1)}(1+o(1))$ (test on $e_1$; nothing else in the matrix is as large as the $11$ entry, so this pr …

. $$ This matrix is already in (complex) Schur form, and the obvious procedure to make the eigenvalues real and similar in spirit to what you propose would be to make the diagonal entries real (that is … (So the general message is that even eigenvalues that aren't close to the real axis might need only a small perturbation to get them there.) …

Yes, this works. Or, to be more honest, I'm fairly confident it does, but I'm only going to give a sketch. The basic step is: a given symmetric $2\times 2$ matrix $A$ is unitarily equivalent to one w …

As a general rule of thumb, it's usually most convenient in one-dimensional problems to work with solutions of the ODE $-y''+Vy=Ey$ rather than operator theoretic methods. Here, everything follows fro …